Appendix B Programming with Scilab Module contents B.1 Introduction .................... 226 B.2 Getting started with Scilab ........... 227 B.3 Variables ....................... 229 B.4 Arithmetic expressions .............. 232 B.4.1 Precedence .................... 232 B.4.2 Matrix arithmetic ................ 233 B.4.3 Array arithmetic ................. 235 B.5 Built–in functions ................. 237 B.5.1 Arithmetic functions ............... 237 B.5.2 Utility functions ................. 240 B.6 Visualising data in Scilab ............. 241 B.6.1 Plotting mathematical functions from within Scilab ...................... 241 B.6.2 Adding titles,labels and legends to the plot . . 244 B.6.3 Plotting numeric data stored in an ASCII file . 247 B.7 Script files ...................... 248 B.7.1 Script input and output ............. 250 B.7.2 Formatting script files .............. 257 B.8 Function files .................... 260 B.8.1 Local variables .................. 265 B.9 Control structures ................. 265 B.9.1 Relational operations ............... 266 B.9.2 Logical operations ................ 267 B.9.3 Branch controls (conditional statements) .... 269 B.9.4 Iteration controls (explicit loops) ........ 273 B.9.5 Implicit loops ................... 278 B.9.6 Using functions and scripts together ...... 279 B.10 Numerical programming ............. 280 B.10.1 Modular Programs ................ 280 B.10.2 Debugging ..................... 281 This module will lead you through numerical programming using Scilab. When studying this module 1 , peruse all of the appendix first so that you have some idea of what can be found in it should you need it in the course of writing programs for the other modules. 1 Everything in this module will be required when implementing the numerical methods discussed in the rest of the course. Use this module as a reference throughout the rest of the course. 225
B61 Plotting mathematical functions from withinScilab 241
B62 Adding titleslabels and legends to the plot 244
B63 Plotting numeric data stored in an ASCII file 247
B7 Script files 248
B71 Script input and output 250
B72 Formatting script files 257
B8 Function files 260
B81 Local variables 265
B9 Control structures 265
B91 Relational operations 266
B92 Logical operations 267
B93 Branch controls (conditional statements) 269
B94 Iteration controls (explicit loops) 273
B95 Implicit loops 278
B96 Using functions and scripts together 279
B10 Numerical programming 280
B101 Modular Programs 280
B102 Debugging 281
This module will lead you through numerical programming usingScilab When studying this module1 peruse all of the appendixfirst so that you have some idea of what can be found in it shouldyou need it in the course of writing programs for the other modules
1 Everything in this module will be required when implementing the numericalmethods discussed in the rest of the course Use this module as a referencethroughout the rest of the course
225
226 Appendix B Programming with Scilab
Then in conjunction with the numerical modules in the StudyBook you should refer to this appendix repeatedly as you attemptthe programming exercises throughout the course This way yourprogramming skills will build up through writing programs Youcannot learn how to write programs through reading alone anymore than you can learn how to drive a car from a manual
B1 Introduction
Scilab is an interactive programming environment It allows theuser to interact directly with the Scilab interpreter through acommand window Commands typed in at the command windoware executed immediately Coupled with the rich graphical capa-bilities of Scilab the immediacy of the command window makesit a good tool for data visualisation and investigating ideas andconcepts
Using Scilab for numerical methods requires you to develop skillin writing functions to perform various algorithms Not only mustyou have it clear in your mind what has to be done (the algorithm)but you also need to analyse carefully which of the tasks will beuseful stand-alone tools
Functions are used repeatedly in their area of application either tosupply answers to well defined problems or to build other functionsto answer work specific problems The important thing about afunction is that having identified a well known task what dataare required to complete the task and a way (or algorithm) forcompleting the task all this information is stored away and can berecalled at will If written carefully functions are reusable acrossprojects and problems
The skill of writing an effective function cannot be acquired byjust reading text like this module To develop skills in writing nu-merical programs requires more than reading it requires you towrite programs and to learn from the experience of writing pro-grams We recommend that this module be used as a referencethroughout the course Everything in this module will be requiredwhen implementing the numerical methods discussed in the courseYour skill as a Scilab programmer should develop as the courseprogresses and you are called on to write more complex programsAs you write and use functions you will begin to appreciate thefeatures that make a particular function useful and what featureslimit a functionrsquos usefulness
When data is to be collected from the console or file there shouldbe special functions designed to do just that jobmdashthey may presenton-screen menus boxes to be filled in icons to be clicked on and soforth Designing a user interface is a whole area of study in itselfScilab does provide some interface tools which lead to quite a
B2 Getting started with Scilab 227
professional looking product but it would take us the whole courseto master these and for the most part we will leave this activity toother courses Ours will be a study of efficient numerical techniquesand display of results will be plain and simple that is why thiscourse is called High Performance Numerical Computing
B2 Getting started with Scilab
Starting Scilab will give a command window similar to the oneshown in Figure B1 Commands are entered at the cursor (--gt)Each command is executed when either the return or enter key ispressed Any command execution can to be interrupted by pressinginterrupt a running command
Ctrl amp C keys together To exit Scilab simply type exit orquit at the command prompt2 Earlier commands executed inquitexit
a session can be recalled using the up (uarr)down (darr) keys on thekeyboard Once recalled the command can either be edited usingthe keyboard or re-executed by pressing the returnenter key Itis also possible to combine multiple commands on the one commandline by separating the commands using either a comma () or asemi-colon () The semi-colon () suppress any output from thesuppressing output from a
command command being executed
Figure B1 The default Scilab Command Window
Help about commands and functions can be obtained by typinghelp at the command plot This will bring up a window where youhelp
2 Note as with all commands in Scilab this will not be executed until eitherthe return or entry is pressed
228 Appendix B Programming with Scilab
can search for help on version topics Note you can short-cut thiswindow if you know the name of the command or function for whichyou require help by typing help command or function name atthe command prompt For example to get help on the log func-tion you would simply type help log at the command promptIn Scilab it is also possible to search for commands andor func-tions which contain certain keywords using the apropos commandapropos
As example the command apropos(rsquologarithmrsquo) will list all thecommands in Scilab relating to logarithms
Example B1 Sample commands in Scilab
--gt3^2-5-632
ans =
0
--gt3^2-5-6(32)
ans =
3
--gt42^2+1
ans =
17
--gtsin(pi2)
ans =
1
--gtsqrt(25)
ans =
5
--gtsqrt(-4)
ans =
2i
--gta1=5b1=3c1=sqrt(a1+b1)
a1 =
5
b1 =
3
c1 =
28284271
--gtans
ans =
2i
--gtt=linspace(0pi26)
t =
0 03141593 06283185 09424778 12566371 15707963
--gty=cos(t)
y =
1 09510565 08090170 05877853 03090170 6123D-17
Example B1 illustrates an important feature of the Scilab com-putational engine If an expression is just typed in and evaluatedthe answer is stored in a special variable called ans This vari-ans
B3 Variables 229
able contains the value of last unassigned (see sectB3 for details onhow variables are assigned) evaluation performed Hence the valuestored in ans changes each time an unassigned calculation is per-formed in Scilab as illustrated in Example B1 Consequentlyif you intend to reuse values of calculations you should store theresults in a variable (see sectB3) to avoid the possible of using theincorrect result
Activity BA To familiarise yourself with some of the basic features of rarrScilab and how to use it for elementary arithmetic andas a sophisticated calculator try some of the examples inExample B1 Make sure you understand what Scilabdid to calculate the answer Also as you work throughthe appendix attempt all the examples and exercises asyou come to them
B3 Variables
Variables are used in a programming language as tags to memorylocations in the computer The data stored in the memory locationis accessed using the variable name in the program The variableis not attached to the data but the memory location This meansthat the value contained within a variable can change
In languages such as C every variable must be declared before it isused specifying both its name and its type The name and type ofvariables do not need to be pre-declared in Scilab In Scilab anyvariable can take integer real complex scalar string or matrixvalues Languages that pre-declare variables have clearly statedat the start of a program or function the type of every variableused The Scilab interpreterrsquos liberal approach to variable typeshas both advantages and disadvantages
The equality sign (=) is used to assign the values to a variableassigning values to
variables This is an unfortunate choice of symbol and means that when wewant to test whether two quantities are equal or not we cannot usethe equal symbol = but must use the more cumbersome == Thus5+2 == 3+4 is true but 5+2 = 3+4 is a syntax error In Scilabprograms the equals sign does not have its mathematical meaningIt is an assignment operator The statement to the right of theequals sign is evaluated and the result is placed in the memorylocation tagged by the variable name on the left For example thestatement x = 3 + 5 means Scilab evaluates the statement onthe right and stores the result in the memory location tagged bythe variable x This means that the statement 3+ 5 = x whichis equivalent to x= 3 + 5 in mathematics will lead to an errormessage in Scilab
The following are examples of variable types in Scilab (it is notcomplete but contains all the variable types that you will encounter
230 Appendix B Programming with Scilab
in this course)
x = 3 Assign the scalar integer value 3 to the variable x
x = 7297e17 Assign the scalar real value 7297 times 1017 to the variable xThe number has been defined using the scientific exponentnotation The number following the e is the power of 10 tomultiply the number in front of the e Either e or E can beused
x = rsquoNorthrsquo Assign the string North to the variable x Strings of char-acters are defined by surrounding the characters with apos-trophes The construct rsquoSouthrsquo defines a constant stringcontaining the characters South
x = 05100 Assign the row vector containing 21 elements from 0 to 100in steps of 5 to the variable x
x = [12 13 25 67] Assign the 2times 2 matrix to the variable x so that
x =
[12 13
25 67
]
x = 51+i63 Assign the complex number 51+ i63 to the variable x
Variable names can be of any length but must be unique withinthe first 31 characters Variable names in Scilab are case sensi-tive that is the variables time and Time are distinct Howeverfunctions and scripts (which we discuss later) are stored in sepa-rate files (with extension sce) outside the interpreter and may ormay not be case sensitive depending on the outside environment(in a dos environment for example they will not be)
Scilab allows overloading of names This means that if you choosea name that has been predefined by Scilab you loose the builtin meaning For example cos = 634 defines a scalar variablecos but it is also the same name for the Scilab cosine functionThe variable assignment means that the cosine function will nolonger be accessible in this session All references to the name coswill assume the variable not the function Take care not to usevariable names that might disable built in functions or constantsthat you may need later When writing scripts for others to usethe names of all variables that will be over-written by the runningof the script must be listed for easy reference (and one shouldread these references before running any script) This is one ofthe reasons why it is nearly always preferable to write functionsrather than scripts Variables defined in the course of runninga functionrsquos algorithm remain local to the function and do notover-write workspace variables
The results of all calculations should be assigned to a variable sothat this value can be used later If this is not done the result
B3 Variables 231
Table B1 Special variables and constants in ScilabName Description
pi The value of π to the precession of the machineis returned
i Recognised as theradicminus1
inf This is Scilabrsquos representation of infin This is arecognised value in a variable and typically ap-pears as a result of division by zero or logarithmof zero
nan Stands for Not a Number This is a recognisedvalue in a variable and appears as a result ofa mathematically undefined operation such asdivision of zero by zero or infinminusinfin
eps Floating point relative accuracy That is epsreturns the smallest difference resolvable be-tween two floating point numbers For examplethe next largest representable number beyond10 is 10+eps Scilab cannot represent anyfloating point number that falls between 10 and10+eps
ans This variable contains the last computed expres-sion that was not stored in a variable
date This variable contains a string representation oftodayrsquos date in the format dd-mmm-yyyy Forexample 01-Apr-2000
232 Appendix B Programming with Scilab
Table B2 Scilab Arithmetic OperatorsOperator Meaning Scilab
^ Exponentiation xy x^y
Multiply xtimes y xy
Right Division xy = xy xy
Left Division yx = xy yx
+ Addition x+ y x+y
- Subtraction xminus y x-y
of a calculation will be stored in a special variable called ans (seeans
Table B1) The value stored in this variable is overridden the nexttime a calculation is done that is not assigned to a variable Thismeans that the result from the previous calculation is then lost
B4 Arithmetic expressions
Table B2 lists the basic arithmetic operators to be found in ScilabWith the exception of the left division operator the arithmetic op-erators in Scilab follow standard notation
B41 Precedence
When more than one operation appears in a mathematical state-ment the programming language needs to define in which order theoperations must be performed or the precedence That is whichparts are evaluated first
If precedence was not defined then the same statement can be eval-uated to a different result for example is 186+3 equal to 2 oris 186+3 equal to 6 The first statement is evaluated from theright using the precedence rule that usually applies in mathemat-ical statements that is work from right to left as for example inlsquosin log tan xrsquo where we start with a value of x compute the tan-gent of that value (tan x) then take the logarithm of the resultingnumber (log(tan x)) and finally take the sine of what this deliv-ers However it is usual (because it is much easier to write poly-nomials with this convention) to give multiplication and divisionprecedence over addition and subtraction The second statementrecognises the precedence of the division over addition and evalu-ates it first Right to left precedence has a much simpler parsingtree than schemes which give precedence to particular operatorsbut the common arithmetic precedences are so widely acceptedand expected that most computer language designers will not riskthe possible lsquoconsumer backlashrsquo that breaking with this traditionwould involve So with every computer language one must searchout and learn the precedence rules that it adopts The usual arith-metic precedences will normally be observed but note that even
B4 Arithmetic expressions 233
Table B3 Precedence of arithmetic expressionsOperator Precedence Order of evaluation
() 1 Left to right innermost first 2 Left to right 3 Left to right 4 Right to left
+ minus 5 Left to right
they are not unambiguous For example which divide has prece-dence in 41632 Working right to left one would get the answerof 8 but Scilab returns (the decimal for) 1128 mdashit works fromleft to right Table B3 lists the operators and their precedencevalue
The rules for evaluation of operators are as follows
bull Operators having high precedence (low numerical value) willbe evaluated before those having a low precedence (high nu-merical value)
bull If an expression contains more than one operator at the samelevel of precedence then the left-to-right rule of evaluationapply
Parentheses () always have the highest precedence Whateveris within the parentheses is evaluated first This means when indoubt about the order of any statement place parentheses aroundthe statement to ensure that the operations are performed in theorder you expect
B42 Matrix arithmetic
One of the strengths of Scilab is its ability to execute matrixarithmetic3 Matrix arithmetic must be implemented in other lan-guages with many lines of code or special functions Scilab isespecially designed to handle matrix arithmetic The representsmatrix multiplication or scalar multiplication of a matrix if one ofthe arguments is a scalar Since scalars are 1times1 matrices ordinaryscalar multiplication can also be performed by the operator Thedot product of two vectors however requires that the first vectorbe a row vector and the second a column vector To multiply thecorresponding elements in two vectors together requires a differentoperator so that
gt a = [2 3 4] b = [6 7 8]
a b
ans =
12 21 32
3 Remember vectors are examples of matrices A column vector of length nis an example of a ntimes 1 matrix
234 Appendix B Programming with Scilab
However because it is a matrix operation
gt a + b
ans =
8 10 40
The standard arithmetic operators described in Table B2 are alsoused with matrices The normal rules of matrix addition subtrac-tion and multiplication apply
The left and right division of matrices are also generalisations ofscalar division Right division BA is roughly the same as Binv(A)Left division AB is roughly the same as inv(A)B The differenceis in how the division is performed Different algorithms are usedwhen A or B are square or not
Scilab also defines a transpose operator for matrices The trans-pose of the matrix A is Arsquo For example if
A =
[25 63
12 09
] then A prime =
[25 12
63 09
]
Accessing matrix elements
Individual elements of a matrix can accessed and used in state-ments by specifying the matrix index of an element The index ofan element is its position in the matrix For example consider thefollowing matrix
A =
6 minus1 4
3 5 7
minus3 6 1
Define this in Scilab with the command
A=[ 6 -1 4 3 5 7 -3 6 7]
to display the contents of matrix A in Scilab just type the variablename A To display or use one element from the matrix specify theelement For example A(13) specifies the element in the first rowand third column that is 4 for the above matrix
To change a specific entry just assign a value to it For example
A(31)=-9
changes the value of the third row first column element from -3
to -9
To display or use a whole row or column use the operator Forexample
gt A(2)
ans =
3 5 7
gt A(2)
B4 Arithmetic expressions 235
ans =
-1
5
6
It is also possible to pick out a sub-matrix by the specifying the rowand column entries required For example To pick out the sub-matrix of A containing the elementsA11 A13 A21 A23 the commandA(12123) could be used as shown below
gt A(12123)
ans =
6 4
3 7
Note 123 means from 1 to 3 in steps of 2 This technique alongwith the assignment operator can be used for values of whole rowscolumns or sub-matrices in one command For example
gt A(12123)=[10111213]
A =
10 -1 11
12 5 13
-3 6 7
Note that the elements A11 A13 A21 A23 now contain the values10 11 12 and 13 respectively instead of their previous values
Index notation is logically an unfortunate phenomenon since itclashes with standard function notation On the face of it A(3)
should be the value of a function called A with input 3 but whenit is interpreted as index notation A is a variable However indexnotation is so familiar and useful to those familiar with classicalmathematical notation that many developers of mathematical soft-ware are not prepared to risk a ldquoconsumer backlashrdquo to providinga functional way of extracting elements from an array such as forexample 3 pick A to pick out the third element of A Similarlywe as teachers feel we would be doing students a disservice if wefailed to introduce them to the index notation that abounds inmathematical texts usually in the form of subscripting as in A3Logic does not always win an argument In the commercial worldof computing an established user base carries more weight thanmathematical logic just as in mathematics tradition may well alsofly in the face of logical consistency
B43 Array arithmetic
Scilab has two forms of arithmetic operations that can be usedwith matrices matrix arithmetic or array arithmetic Matrix arith-metic follows the normal rules of matrix algebra Array arithmeticwhich is also called element-by-element arithmetic is where oper-ations are performed on matrices element by element
236 Appendix B Programming with Scilab
Table B4 Scilab array operatorsOperator Meaning Scilab
^ Exponentiation x^y
Multiply xy
Right Division xy
Left Division yx
+ Addition x+y
- Subtraction x-y
Table B4 lists the Scilab array operators
The array addition and subtraction are identical to matrix additionor subtraction Both array and matrix addition and subtractionoperate element by element Array addition and subtraction re-quire that both arrays have the same size The only exception tothis rule in Scilab occurs when we add or subtract a scalar to orfrom an array In this case the scalar is added or subtracted fromeach element in the array
For example
gt [6 5 6] + 3
ans =
[9 8 9]
gt [6 4 5 8] + [3 1 2 4]
ans =
[9 5 7 12]
gt [4 3] - 7
ans =
[-3 -4]
Array multiplication and division is very different to matrix mul-tiplication and division Array multiplication or division is an ele-ment by element multiplication or division that produces an arraywith the same dimensions as the starting arrays As with addi-tion and subtraction array multiplication and division require thatboth arrays have the same dimensions
Example B2 Consider the two row vectors x=[2 4 7 1] andy=[6 1 3 2] Scilab does not recognise the possibility ofmatrix multiplication of these two vectors They are not com-patible the number of columns of the pre-multiplier must bethe same as the number of rows of the post-multiplier Anattempt to execute x y will bring an error message
Obtain the usual dot product of the two row vectors by trans-posing the post-multiplier as in xyrsquo or yxrsquo The matrixproduct xrsquoy produces a matrix of 4 rows and 4 columns
B5 Builtndashin functions 237
Array multiplication produces the result
gt xy
ans =
[12 4 21 2]
by multiplying corresponding elements of each vector Theresult is a row vector with the same dimensions as x and ythat is a 1times 4 vector
Example B3 Consider the row vectors x=[2 4 7 1] and col-umn vector y=[6 1 3 2]rsquo Array multiplication of thesetwo matrices is not possible because their dimensions do notagree
Matrix multiplication xy produces the result 39 that is theinner product or dot product of the vectors Note vectorsas such do not exist in Scilab there are only single rowedor single columned arrays
Scilab not only allows arrays to be raised to powers but alsoallows scalars and arrays to be raised by array powers To performexponentiation on an element-by-element basis use the ^ symbolFor example if x=[3 5 8] then x^3 produces the matrix [33 53
83] = [27 125 512]
A scalar can be raised to an array power For example if p=[2 4 5]then typing 3^p produces the array [32 34 35] = [9 81 243]
With element-by-element array operations it is important to re-member that the dot () is a part of a two-character symbol for aparticular array operator such as () () () (^) Sometimesthis can cause some confusion the dot in 3^p is not a decimalpoint associated with the number 3 The following operations withthe value of p=[2 4 5] are equivalent
3^p
30^p
3^p
(3)^p
3^[2 4 5]
B5 Builtndashin functions
B51 Arithmetic functions
Arithmetic expressions often require computations other than ad-dition subtraction multiplication division and exponentiationFor example many expressions require the use of logarithms ex-ponentials and trigonometric functions Scilab has predefined alarge number of functions that perform these types of computations(see Tables B5 and B6)
238 Appendix B Programming with Scilab
An example of a predefined function is sin which returns the sineof a given angle
m = sin(angle)
where the variable angle contains the angle in radians and theresult is stored in the variable m The variable angle is called theargument or parameter of the function If the angle is stored indegrees it must be converted to radians for example
m = sin(anglepi180)
or
radians =anglepi180
m = sin(radians)
In Scilab the arguments or parameters of the function are con-tained in parentheses following the name of the function This isreferred to as pre-fix notation Many common functions in math-ematics have a syntax using in-fix notation where two argumentsappear to the right and left of the function name as for examplein x+y atimes b nmod r etc Scilab does not support in-fix nota-tion for any but the arithmetic functions Classical mathematicalnotation also has examples of post-fix notationmdash n x squared mdashand other hybrids like |x| while Scilab has ahb to confound thepurist
A function may contain no arguments one argument or many ar-guments depending on its definition If a function contains morethan one argument it is vital to give the arguments in the cor-rect order For example consider the problem of finding θ in thefollowing triangle using the atan command
5
8θ
Now by hand calculation the answer is 05586 radians If we useatan as below we get the answers shown
--gt atan(58)
ans =
05586
--gt atan(85)
ans =
10122
The first answer is the correct one The second answer is incorrect
B5 Builtndashin functions 239
Table B5 Some common mathematical functions defined inScilabFunction Description
abs(x) Computes the absolute value of xsqrt(x) Square root of x The parameter can be nega-
tiveround(x) Round x to the nearest integerfix(x) Round x to the nearest integer toward zerofloor(x) Round x to the nearest integer toward minusinfinceil(x) Round x to the nearest integer toward +infinsign(x) Returns -1 if x is less than zero zero if x equals
zero and 1 otherwisemodulo(xy) Returns the remainder of xy For example
modulo(254) is 1exp(x) Returns ex
log(x) Returns the natural logarithm of x That isthe logarithm to the base e Parameter can benegative but cannot be zero
log10(x) Returns the common logarithm of x That isthe logarithm to the base 10 Parameter can benegative but cannot be zero
since we passed the values to the function in the incorrect order
Some functions also require that the arguments be in specific unitsFor example the trigonometric functions assume that the argu-ments are in radians
In Scilab some functions use the number of arguments to deter-mine the output of the function For example the zeros functioncan have one or two arguments which then determine the output(check this with the command help zeros)
A function reference cannot be made on the left hand side of anassignment since a function call always returns a value(s) whereasan assignment is always to a variable Functions can of courseappear on the right side of an assignment since they return valuesto be assigned to some variable Functions may be applied tothe output of other functions and may be called in the algorithmdefining a function for example
logx = log(abs(x))
When one function is used to compute the argument of anotherfunction be sure to enclose the argument of each function in itsown set of parentheses
Scilab allows recursive functions that is functions that call them-selves in their defining algorithm
Exercise B4 Use examples to show the differences between the
240 Appendix B Programming with Scilab
Table B6 Some common trigonometric functions defined inScilab
Function Description
sin(x) Sine of x where x is in radianscos(x) Cosine of x where x is in radianstan(x) Tangent of x where x is in radiansasin(x) Arcsine or inverse sine of x where x must be
between -1 and 1 The return angle is in radiansbetween minusπ2 and π2
acos(x) Arccosine or inverse cosine of x where x mustbe between minus1 and 1 The return angle is inradians between 0 and π
atan(x) Arctangent or inverse tangent of x The returnangle is in radians between minusπ2 and π2
atan(xy) Arctangent or inverse tangent of xy The re-turn angle is in radians between minusπ and π de-pending on the sign of x and y
sec(x) Secant (1(cos x)) of x where x is in radianscsc(x) Cosecant (1(sin x)) of x where x is in radianscotg(x) Cotangent (1(tan x)) of x where x is in radians
Table B7 Some of Scilabrsquos utility functionsFunction Description
clear Clear variables from memoryhelp Display the first set of comment lines from
scriptswho List the variables in memorywhos List the variables in memory with their sizeerror Terminates the execution of the Sci-file and out-
puts the supplied error message
functions round(x) fix(x) floor(x) and ceil(x)
Exercise B5 The functions listed in Tables B5 B6 are only asmall sample of all of Scilabrsquos mathematical functions
Using Scilabrsquos help command list all of the hyperbolic func-tions available Also list any restrictions on the values of thefunction arguments
B52 Utility functions
In addition to the arithmetic functions Scilab has several utilityfunctions (see Table B7) These functions allow users to trackvariables (whoswho) feedback an error message if an error occursin the code (error) clear the variables form the memory (clear)etc Essential these functions allow users to debug and performbookkeeping operations inside their code
B6 Visualising data in Scilab 241
6
5
4- 04- 8
3
- 02
- 6
00
2- 4 Y- 2
02
Z
10
04
X 02
06
4 - 1
08
6- 28
10
Figure B2 Plot of z = sin(radicx2 + y2)
radicx2 + y2 illustrating the
3d capabilities of Scilab
B6 Visualising data in Scilab
One of the big advantages in using Scilab is the ability to visualisedata and results of complex calculations (see Figure B2 as exampleof the 3d capabilities) While Scilab is capability of complex 3dgraphics (see help pages for mesh and surf) in this section we willmesh amp surf
focus on the 2d plotting command (plot) Hence in this sectionplot
we will focus on
1 plotting mathematical functions from within Scilab
2 changing stylistic elements of the plot
3 adding titles labels and legends to the plot
4 including mathematical notation in titles labels and legendsand
5 plotting data stored in an external ASCII file
B61 Plotting mathematical functions from within Scilab
To plot a mathematical function in Scilab three steps are requiredThese are
1 Define the domain of the function (eg x=linspace(02pi))
2 Calculate the values of function at the selected points in thedomain (eg y=sin(x))
242 Appendix B Programming with Scilab
3 Final step is to plot the function using the plot commandplot
(eg plot(xy))
Using these steps it is possible to produce a basic plot of any math-ematical function (see Figure B3) Note that the plot shown inFigure B3 does not have a title axes labels or a legend all ofwhich are crucial for a good plot The Scilab commands to addthese elements to the plot are given in the following sections
- 10
- 08
- 06
- 04
- 02
00
02
04
06
08
10
0 1 2 3 4 5 6 7
Figure B3 The default plot of y = sin(x) in Scilab Notethat by default axes are not labelled and there is no title orlegend These need to added using the additional commandsxlabelylabeltitle and legend
Changing stylistic elements of the plot
An optional argument can be given to the plot command to cus-tomise each new line aspect This argument is a concatenatedstring containing information about colour line style or markersbased on the specifiers shown in Table B8 To specify the plotbe drawn with a red longdash-dot line with diamond marker thestring that would be passed to plot command would be rsquor-drsquo Thestring which is a concatenation (in any order) of the three typesof properties shown in Table B8 must be unambiguous Further-more the string specification is not case sensitive An exampleof specifying various line styles is shown in Example B6 and Fig-ure B4
Example B6 The following commands illustrate how line stylescan be used to distinguish between the plots of sin(2πt) sin(2πt+π6) and sin(2πt+ π2)
t=linspace(0130)
B6 Visualising data in Scilab 243
Table B8 Plot specifiers required to change line styles coloursand markers in plotsSpecifier Line style Specifier Colour Specifier Marker stylendash Solid line (default) r Red + Plus signndash ndash Dashed line g Green o Circle Dotted line b Blue Asteriskndash Dash-dotted line c Cyan x Cross
m Magenta s Squarey Yellow rsquononersquo No marker(default)k Blackw White
Figure B4 Resulting plotting after issuing the commands givenin Example B6
244 Appendix B Programming with Scilab
Example B6 illustrate a number of key commands relating to thecreation of multiple plots in Scilab The mtlb hold4 commandmtlb hold
holds the current plot and axis properties so that all subsequentplots are added to the existing plot
To distinguish each plot a different line specification is passed eachplot command In the example y = sin(2πt) is plotted using a redsolid line with data points indicated using a red circle as indicatedby the string rsquor-orsquo y = sin(2πt + π6) is plotted using greendashed-dotted line with data points marked by a square (rsquog-srsquo)while y = sin(2πt+ π2) is plotted using a cyan dashed line usingdiamonds to mark the data points (rsquoc--drsquo) Finally to completethe plot axes labels and a legend has been added to plot the com-mands for doing this are discussed in sectB62
B62 Adding titleslabels and legends to the plot
All good plots should have the following properties
1 The style of figure is appropriate to the information beingconveyed
2 The contents are well organised so it is easy for readers tointerpret the information
3 Legends labels etc are clear and concise (eg contain unitsif required do not obstruct parts of the plot etc) and
4 A caption or title that summarises the information presented
Like most mathematical applications Scilab has a wide variety ofdifferent plotting styles that can be used (eg look up the help forbar polarplot etc) This allows you to select the most appropri-barpolarplot
ate style to display and organise your data
The Scilab commands xlabel ylabel title and legend al-xlabel ylabel title and
legend lows you to add axes labels titles and legends to plot As shownin Example B6 the commands xlabel ylabel and title takea string input containing the title or label text For the legend
command a string is required for each separate plot Note theorder of the strings in the legend command reflects the order inwhich the each of plots were made
Including mathematical notation in titles labels and legends
It is possible to include mathematical characters such as π inScilab plots This is done by inserting special commands intoeither the title legend or label string These commands are basedon the abbreviations for mathematical notation used in programssuch as LATEX MathType etc A brief summary of the commands
4 mtlb hold on is equivalent to the Scilab commandset(gca()auto clearoff) while mtlb hold off is equivalentto the Scilab command set(gca()auto clearon)
B6 Visualising data in Scilab 245
Table B9 Basic mathematical symbols available in Scilab forincluding mathematical notation in titles labels and legendscommand symbol command symbol command symbolalpha α upsilon υ sim sim
Im = cup cup wp weierpotimes otimes subseteq sube oslash cap cap in isin supseteq supesupset sup lceil d subset subint
intcdot middot o oslash
rfloor c neg 6= nabla nablalfloor b times times ldots perp perp surd
radicprime prime
wedge and varpi $ O Oslashrceil e rangle 〉 mid |
vee or copyright ccopy langle 〈
available in Scilab for typesetting mathematical notation is givenin Table B9 To insert these commands into a text string simplystart and end the string with $ signs as shown in Example B7
Example B7 Scilab commands to produce the labelled plot ofy = sin
There are a few key concepts to note about the Example B7 Thefirst can be seen in the string being passed to the title commandtitle
In this string sin is prefixed with a backslash () This is doneso that the letters for sin are not typeset in italic which shouldonly be used for variables and not common mathematical func-tions Table B10 shows the command sequences to use for themore common mathematical functions5 To get the fraction in thetitle the command fracab has been used to produce a
b Ifsuperscripts and subscripts are required in a title or label these aresuperscripts amp subscripts
obtained by adding the subscript character ( ) and the superscriptcharacter (^) to the string These characters modify the characteror substring defined in braces immediately following For examplex 2 gives x2 while e^2x gives e2x
Note in Example B7 the commands xlabel and ylabel has beenxlabelylabel
used to set the axes labels in Figure B5 Instead of using thesethree commands the title and the axes labels could have been setusing the single xtitle command (see help xtitle for details)xtitle
In the example we have also customised the font size to make thetext more readable by a) passing an option to the title xlabelylabel commands and b) by using the command xset to set thexset
default font size before executing the xstring command Finallythe xstring command adds the text string to plot at the locationxstring
specified by variables b and c In the example the coordinates wewant to add the text are obtained by executing the xclick functionxclick
and clicking on the point required in the plot window
5 Additional details can be found in books covering Mathematics in LATEX
B6 Visualising data in Scilab 247
Table B10 Command sequences for the common mathematicalfunctions in Scilabarccos arcsin arctan cos cot
csc exp lim ln log
max min sec sin tan
B63 Plotting numeric data stored in an ASCII file
As data can be stored in a wide variety of ways in external filesScilab has a wide variety of functions which can be used to importthe data stored in these files (see help for the commands read
fscanf and xlsread) Since data can be stored in such a wideread fscanf and xlsread
range of formats we will confine our discussions here to the situa-tion where numeric data is stored in a ASCII (text) file where eachrow of the file contains the same number of entries
To illustrate the procedure for plotting data stored in ASCII file wewill consider the data obtained measuring the temperature of a cupof coffee every minute for a total of 100 mins The data recorded isstored in the a file called coffeedatatxt6 which simply containsthe two columns of numbers shown in Table B11 As an examplethe first 10 lines of the text file contains the following numbers7
00000 1000000
10000 950000
20000 903125
30000 859180
40000 817981
50000 779357
60000 743147
70000 709201
80000 677376
90000 647540
Note the current version of Scilab has a builtin text editor whichcan be used to view such text files To access this editor simplytype the command edit at the command prompt Once open youedit
can use the graphical interface to open the required file
The first issue is to get the data into Scilab so it is available forplot command In Scilab the command read can used to importread
the data and assign it to a variable In the current example wewould use the following command to import the data and assignit to a variable called cdata assuming the data is in the currentdirectory8
6 The file coffeedatatxt has been attached and should be available for savingprovided you are using Adobe Reader To save the file right click on thepurple Windows version or the red MacOSLinux version in the margin andselect save file from the options Ensure you save the file as coffeedatatxt
7 normally you will have received the data file in this format8 You can check what directory Scilab is in by typing the command pwd
Figure B6 Temperature of coffee based on the data recorded inthe file coffeedatatxt
cdata=read(rsquocoffeedatatxtrsquo-12)
The -12 following the name of the file tells Scilab that you donrsquotknow how many rows you have but you have two columns of dataSince we know that we have 101 rows of data we could have alsoused the following command to load the data
cdata=read(rsquocoffeedatatxtrsquo1012)
We now have a variable called cdata which has two columns ofdata The first column contains the time while the second containsthe Temperature records Hence we can plot it using the followingcommand
plot(cdata(1)cdata(2))
This gives the plot shown in Figure B6 after appropriate labelshave been added as described above
B7 Script files
Executing commands entered at the keyboard or keypad is themost primitive use of a calculating device It is only when se-quences of operations (to perform complex tasks) are stored awayand re-called for repeated application with different inputs that wehave a programmed operation Complex programs must necessarilyhave an interface with the user The basic interface tool in Scilabcapable of executing sequences of commands that are stored in a fileis the script The file names must have an extension of sce forexample program1sce assignment1sce These files are calledScript Sci-files or simply scripts
A script file is a file of standard keyboard charactersthat define commands to drive a program A script filecan be generated using any text editor or word proces-sor
To invoke a script in Scilab type the commandexec
exec filenamesce
where filename denotes the actual name of the file containing thescript When the script runs it is exactly the same as if you typedthe commands into the command window from the keyboard Thismeans that the Sci-file can use any variables you have defined priorto invoking the script Any variables created in the script are alsoavailable when the script is finished and any variable whose namewas assigned a value in the script will be over-written
If you give a script the same name as any variable that it computesin the running of the script then Scilab will give precedence tothe workspace variable over a script As stated in sectB3 the variablename will hide the script name after the script has run Scilab willtake the name as referring to the variable and not to the definedscript
Scilab also needs to know where to find the script to executedIf Scilab cannot find the script it will report that you have anundefined function or variable
The exact procedure for creating and saving Sci-files depends onthe Scilab version the operating system (MS Windows Macin-tosh or UnixLinux ) and the text editor you use The currentversion of Scilab includes a built-in editor which can be used tocreate and edit Sci-files and text files
To create a new Sci-file under the current version of Scilab simplytype edit newfilenamesce This will open up the Scilab editoredit
with a blank document called newfilename
B71 Script input and output
A script is a named file (with the extension sce) containing asequence of Scilab instructions which when its name is executedin a session performs those instructions in order as though theywere entered at the console
There are two common uses for scripts Pragmatically we maywant to save a particular computational sequence for later use orreference If the tasks involved are to be repeated often thenthe nature of the tasks should be analysed formulated into usefulfunctions (which we will discuss later) and the whole procedurepresented in a single function But if the tasks are only likely toget occasional use then the script may be adequately commentedand stored as a script file It is a quick and dirty way of saving
B7 Script files 251
the instructions for doing a particular job when we do not wantto go to the trouble of writing the functions for the job It is nota highly recommended modus operandi but useful all the same
The second use is for the script to act like a storyboard presentinginformation to a user in a structured and controlled way permit-ting the user to see the steps of a computational procedure andtheir outputs up on the screen At a more sophisticated level thescript takes on the proportions of an application controlling a largeand complex interaction with advanced interface features We willnot take our study of Scilab programming to this level but workmostly at the storyboard level using the simplest of Scilabrsquos in-terface tools
All the variables created in a script become global variables definedin the workspace and available for perusal As an application itis assumed that a script controls the entire workspace and thevariables it creates are lsquothe datarsquo of the application When scriptsare used as storyboards on a small scale then it cannot be assumedthat the variables they create deserve global recognition and it iscourteous to erase these variables when the script is finished unlessthey have some possible further use An unfortunate side-effect isthat if variables with the same name already exist in the workspacebefore a script is run then these variables will be over-written andlost
Used as storyboards scripts allow the user to cut paste and re-trythe various steps of a procedure and to modify the script to suithis own needs Storyboards may be used for demonstrations oftoolbox utilities or extended documentationexplanation of func-tions However if there is any danger that variables defined bythe script may over-write workspace variables then the demo ordocumentation should be re-fashioned into a function that leavesthe workspace variables untouched (more on this later)
Scripts will often read data from pre-prepared files and we will dis-cuss the functions provided by Scilab to perform this task below
Gathering data entered by the user really falls into the realm ofinterface design but we only briefly consider the simplest interfacefunction provided by Scilab Similarly we briefly consider thefunctions for providing the simplest forms of output presentation
Keyboard input
Scilab has a built-in function input that can be used to promptinput
the user to enter the value of a variable from the keyboard Forexample
Tc = input(rsquoPlease enter the temperature in degrees Celsiusrsquo)
will print the message
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
226 Appendix B Programming with Scilab
Then in conjunction with the numerical modules in the StudyBook you should refer to this appendix repeatedly as you attemptthe programming exercises throughout the course This way yourprogramming skills will build up through writing programs Youcannot learn how to write programs through reading alone anymore than you can learn how to drive a car from a manual
B1 Introduction
Scilab is an interactive programming environment It allows theuser to interact directly with the Scilab interpreter through acommand window Commands typed in at the command windoware executed immediately Coupled with the rich graphical capa-bilities of Scilab the immediacy of the command window makesit a good tool for data visualisation and investigating ideas andconcepts
Using Scilab for numerical methods requires you to develop skillin writing functions to perform various algorithms Not only mustyou have it clear in your mind what has to be done (the algorithm)but you also need to analyse carefully which of the tasks will beuseful stand-alone tools
Functions are used repeatedly in their area of application either tosupply answers to well defined problems or to build other functionsto answer work specific problems The important thing about afunction is that having identified a well known task what dataare required to complete the task and a way (or algorithm) forcompleting the task all this information is stored away and can berecalled at will If written carefully functions are reusable acrossprojects and problems
The skill of writing an effective function cannot be acquired byjust reading text like this module To develop skills in writing nu-merical programs requires more than reading it requires you towrite programs and to learn from the experience of writing pro-grams We recommend that this module be used as a referencethroughout the course Everything in this module will be requiredwhen implementing the numerical methods discussed in the courseYour skill as a Scilab programmer should develop as the courseprogresses and you are called on to write more complex programsAs you write and use functions you will begin to appreciate thefeatures that make a particular function useful and what featureslimit a functionrsquos usefulness
When data is to be collected from the console or file there shouldbe special functions designed to do just that jobmdashthey may presenton-screen menus boxes to be filled in icons to be clicked on and soforth Designing a user interface is a whole area of study in itselfScilab does provide some interface tools which lead to quite a
B2 Getting started with Scilab 227
professional looking product but it would take us the whole courseto master these and for the most part we will leave this activity toother courses Ours will be a study of efficient numerical techniquesand display of results will be plain and simple that is why thiscourse is called High Performance Numerical Computing
B2 Getting started with Scilab
Starting Scilab will give a command window similar to the oneshown in Figure B1 Commands are entered at the cursor (--gt)Each command is executed when either the return or enter key ispressed Any command execution can to be interrupted by pressinginterrupt a running command
Ctrl amp C keys together To exit Scilab simply type exit orquit at the command prompt2 Earlier commands executed inquitexit
a session can be recalled using the up (uarr)down (darr) keys on thekeyboard Once recalled the command can either be edited usingthe keyboard or re-executed by pressing the returnenter key Itis also possible to combine multiple commands on the one commandline by separating the commands using either a comma () or asemi-colon () The semi-colon () suppress any output from thesuppressing output from a
command command being executed
Figure B1 The default Scilab Command Window
Help about commands and functions can be obtained by typinghelp at the command plot This will bring up a window where youhelp
2 Note as with all commands in Scilab this will not be executed until eitherthe return or entry is pressed
228 Appendix B Programming with Scilab
can search for help on version topics Note you can short-cut thiswindow if you know the name of the command or function for whichyou require help by typing help command or function name atthe command prompt For example to get help on the log func-tion you would simply type help log at the command promptIn Scilab it is also possible to search for commands andor func-tions which contain certain keywords using the apropos commandapropos
As example the command apropos(rsquologarithmrsquo) will list all thecommands in Scilab relating to logarithms
Example B1 Sample commands in Scilab
--gt3^2-5-632
ans =
0
--gt3^2-5-6(32)
ans =
3
--gt42^2+1
ans =
17
--gtsin(pi2)
ans =
1
--gtsqrt(25)
ans =
5
--gtsqrt(-4)
ans =
2i
--gta1=5b1=3c1=sqrt(a1+b1)
a1 =
5
b1 =
3
c1 =
28284271
--gtans
ans =
2i
--gtt=linspace(0pi26)
t =
0 03141593 06283185 09424778 12566371 15707963
--gty=cos(t)
y =
1 09510565 08090170 05877853 03090170 6123D-17
Example B1 illustrates an important feature of the Scilab com-putational engine If an expression is just typed in and evaluatedthe answer is stored in a special variable called ans This vari-ans
B3 Variables 229
able contains the value of last unassigned (see sectB3 for details onhow variables are assigned) evaluation performed Hence the valuestored in ans changes each time an unassigned calculation is per-formed in Scilab as illustrated in Example B1 Consequentlyif you intend to reuse values of calculations you should store theresults in a variable (see sectB3) to avoid the possible of using theincorrect result
Activity BA To familiarise yourself with some of the basic features of rarrScilab and how to use it for elementary arithmetic andas a sophisticated calculator try some of the examples inExample B1 Make sure you understand what Scilabdid to calculate the answer Also as you work throughthe appendix attempt all the examples and exercises asyou come to them
B3 Variables
Variables are used in a programming language as tags to memorylocations in the computer The data stored in the memory locationis accessed using the variable name in the program The variableis not attached to the data but the memory location This meansthat the value contained within a variable can change
In languages such as C every variable must be declared before it isused specifying both its name and its type The name and type ofvariables do not need to be pre-declared in Scilab In Scilab anyvariable can take integer real complex scalar string or matrixvalues Languages that pre-declare variables have clearly statedat the start of a program or function the type of every variableused The Scilab interpreterrsquos liberal approach to variable typeshas both advantages and disadvantages
The equality sign (=) is used to assign the values to a variableassigning values to
variables This is an unfortunate choice of symbol and means that when wewant to test whether two quantities are equal or not we cannot usethe equal symbol = but must use the more cumbersome == Thus5+2 == 3+4 is true but 5+2 = 3+4 is a syntax error In Scilabprograms the equals sign does not have its mathematical meaningIt is an assignment operator The statement to the right of theequals sign is evaluated and the result is placed in the memorylocation tagged by the variable name on the left For example thestatement x = 3 + 5 means Scilab evaluates the statement onthe right and stores the result in the memory location tagged bythe variable x This means that the statement 3+ 5 = x whichis equivalent to x= 3 + 5 in mathematics will lead to an errormessage in Scilab
The following are examples of variable types in Scilab (it is notcomplete but contains all the variable types that you will encounter
230 Appendix B Programming with Scilab
in this course)
x = 3 Assign the scalar integer value 3 to the variable x
x = 7297e17 Assign the scalar real value 7297 times 1017 to the variable xThe number has been defined using the scientific exponentnotation The number following the e is the power of 10 tomultiply the number in front of the e Either e or E can beused
x = rsquoNorthrsquo Assign the string North to the variable x Strings of char-acters are defined by surrounding the characters with apos-trophes The construct rsquoSouthrsquo defines a constant stringcontaining the characters South
x = 05100 Assign the row vector containing 21 elements from 0 to 100in steps of 5 to the variable x
x = [12 13 25 67] Assign the 2times 2 matrix to the variable x so that
x =
[12 13
25 67
]
x = 51+i63 Assign the complex number 51+ i63 to the variable x
Variable names can be of any length but must be unique withinthe first 31 characters Variable names in Scilab are case sensi-tive that is the variables time and Time are distinct Howeverfunctions and scripts (which we discuss later) are stored in sepa-rate files (with extension sce) outside the interpreter and may ormay not be case sensitive depending on the outside environment(in a dos environment for example they will not be)
Scilab allows overloading of names This means that if you choosea name that has been predefined by Scilab you loose the builtin meaning For example cos = 634 defines a scalar variablecos but it is also the same name for the Scilab cosine functionThe variable assignment means that the cosine function will nolonger be accessible in this session All references to the name coswill assume the variable not the function Take care not to usevariable names that might disable built in functions or constantsthat you may need later When writing scripts for others to usethe names of all variables that will be over-written by the runningof the script must be listed for easy reference (and one shouldread these references before running any script) This is one ofthe reasons why it is nearly always preferable to write functionsrather than scripts Variables defined in the course of runninga functionrsquos algorithm remain local to the function and do notover-write workspace variables
The results of all calculations should be assigned to a variable sothat this value can be used later If this is not done the result
B3 Variables 231
Table B1 Special variables and constants in ScilabName Description
pi The value of π to the precession of the machineis returned
i Recognised as theradicminus1
inf This is Scilabrsquos representation of infin This is arecognised value in a variable and typically ap-pears as a result of division by zero or logarithmof zero
nan Stands for Not a Number This is a recognisedvalue in a variable and appears as a result ofa mathematically undefined operation such asdivision of zero by zero or infinminusinfin
eps Floating point relative accuracy That is epsreturns the smallest difference resolvable be-tween two floating point numbers For examplethe next largest representable number beyond10 is 10+eps Scilab cannot represent anyfloating point number that falls between 10 and10+eps
ans This variable contains the last computed expres-sion that was not stored in a variable
date This variable contains a string representation oftodayrsquos date in the format dd-mmm-yyyy Forexample 01-Apr-2000
232 Appendix B Programming with Scilab
Table B2 Scilab Arithmetic OperatorsOperator Meaning Scilab
^ Exponentiation xy x^y
Multiply xtimes y xy
Right Division xy = xy xy
Left Division yx = xy yx
+ Addition x+ y x+y
- Subtraction xminus y x-y
of a calculation will be stored in a special variable called ans (seeans
Table B1) The value stored in this variable is overridden the nexttime a calculation is done that is not assigned to a variable Thismeans that the result from the previous calculation is then lost
B4 Arithmetic expressions
Table B2 lists the basic arithmetic operators to be found in ScilabWith the exception of the left division operator the arithmetic op-erators in Scilab follow standard notation
B41 Precedence
When more than one operation appears in a mathematical state-ment the programming language needs to define in which order theoperations must be performed or the precedence That is whichparts are evaluated first
If precedence was not defined then the same statement can be eval-uated to a different result for example is 186+3 equal to 2 oris 186+3 equal to 6 The first statement is evaluated from theright using the precedence rule that usually applies in mathemat-ical statements that is work from right to left as for example inlsquosin log tan xrsquo where we start with a value of x compute the tan-gent of that value (tan x) then take the logarithm of the resultingnumber (log(tan x)) and finally take the sine of what this deliv-ers However it is usual (because it is much easier to write poly-nomials with this convention) to give multiplication and divisionprecedence over addition and subtraction The second statementrecognises the precedence of the division over addition and evalu-ates it first Right to left precedence has a much simpler parsingtree than schemes which give precedence to particular operatorsbut the common arithmetic precedences are so widely acceptedand expected that most computer language designers will not riskthe possible lsquoconsumer backlashrsquo that breaking with this traditionwould involve So with every computer language one must searchout and learn the precedence rules that it adopts The usual arith-metic precedences will normally be observed but note that even
B4 Arithmetic expressions 233
Table B3 Precedence of arithmetic expressionsOperator Precedence Order of evaluation
() 1 Left to right innermost first 2 Left to right 3 Left to right 4 Right to left
+ minus 5 Left to right
they are not unambiguous For example which divide has prece-dence in 41632 Working right to left one would get the answerof 8 but Scilab returns (the decimal for) 1128 mdashit works fromleft to right Table B3 lists the operators and their precedencevalue
The rules for evaluation of operators are as follows
bull Operators having high precedence (low numerical value) willbe evaluated before those having a low precedence (high nu-merical value)
bull If an expression contains more than one operator at the samelevel of precedence then the left-to-right rule of evaluationapply
Parentheses () always have the highest precedence Whateveris within the parentheses is evaluated first This means when indoubt about the order of any statement place parentheses aroundthe statement to ensure that the operations are performed in theorder you expect
B42 Matrix arithmetic
One of the strengths of Scilab is its ability to execute matrixarithmetic3 Matrix arithmetic must be implemented in other lan-guages with many lines of code or special functions Scilab isespecially designed to handle matrix arithmetic The representsmatrix multiplication or scalar multiplication of a matrix if one ofthe arguments is a scalar Since scalars are 1times1 matrices ordinaryscalar multiplication can also be performed by the operator Thedot product of two vectors however requires that the first vectorbe a row vector and the second a column vector To multiply thecorresponding elements in two vectors together requires a differentoperator so that
gt a = [2 3 4] b = [6 7 8]
a b
ans =
12 21 32
3 Remember vectors are examples of matrices A column vector of length nis an example of a ntimes 1 matrix
234 Appendix B Programming with Scilab
However because it is a matrix operation
gt a + b
ans =
8 10 40
The standard arithmetic operators described in Table B2 are alsoused with matrices The normal rules of matrix addition subtrac-tion and multiplication apply
The left and right division of matrices are also generalisations ofscalar division Right division BA is roughly the same as Binv(A)Left division AB is roughly the same as inv(A)B The differenceis in how the division is performed Different algorithms are usedwhen A or B are square or not
Scilab also defines a transpose operator for matrices The trans-pose of the matrix A is Arsquo For example if
A =
[25 63
12 09
] then A prime =
[25 12
63 09
]
Accessing matrix elements
Individual elements of a matrix can accessed and used in state-ments by specifying the matrix index of an element The index ofan element is its position in the matrix For example consider thefollowing matrix
A =
6 minus1 4
3 5 7
minus3 6 1
Define this in Scilab with the command
A=[ 6 -1 4 3 5 7 -3 6 7]
to display the contents of matrix A in Scilab just type the variablename A To display or use one element from the matrix specify theelement For example A(13) specifies the element in the first rowand third column that is 4 for the above matrix
To change a specific entry just assign a value to it For example
A(31)=-9
changes the value of the third row first column element from -3
to -9
To display or use a whole row or column use the operator Forexample
gt A(2)
ans =
3 5 7
gt A(2)
B4 Arithmetic expressions 235
ans =
-1
5
6
It is also possible to pick out a sub-matrix by the specifying the rowand column entries required For example To pick out the sub-matrix of A containing the elementsA11 A13 A21 A23 the commandA(12123) could be used as shown below
gt A(12123)
ans =
6 4
3 7
Note 123 means from 1 to 3 in steps of 2 This technique alongwith the assignment operator can be used for values of whole rowscolumns or sub-matrices in one command For example
gt A(12123)=[10111213]
A =
10 -1 11
12 5 13
-3 6 7
Note that the elements A11 A13 A21 A23 now contain the values10 11 12 and 13 respectively instead of their previous values
Index notation is logically an unfortunate phenomenon since itclashes with standard function notation On the face of it A(3)
should be the value of a function called A with input 3 but whenit is interpreted as index notation A is a variable However indexnotation is so familiar and useful to those familiar with classicalmathematical notation that many developers of mathematical soft-ware are not prepared to risk a ldquoconsumer backlashrdquo to providinga functional way of extracting elements from an array such as forexample 3 pick A to pick out the third element of A Similarlywe as teachers feel we would be doing students a disservice if wefailed to introduce them to the index notation that abounds inmathematical texts usually in the form of subscripting as in A3Logic does not always win an argument In the commercial worldof computing an established user base carries more weight thanmathematical logic just as in mathematics tradition may well alsofly in the face of logical consistency
B43 Array arithmetic
Scilab has two forms of arithmetic operations that can be usedwith matrices matrix arithmetic or array arithmetic Matrix arith-metic follows the normal rules of matrix algebra Array arithmeticwhich is also called element-by-element arithmetic is where oper-ations are performed on matrices element by element
236 Appendix B Programming with Scilab
Table B4 Scilab array operatorsOperator Meaning Scilab
^ Exponentiation x^y
Multiply xy
Right Division xy
Left Division yx
+ Addition x+y
- Subtraction x-y
Table B4 lists the Scilab array operators
The array addition and subtraction are identical to matrix additionor subtraction Both array and matrix addition and subtractionoperate element by element Array addition and subtraction re-quire that both arrays have the same size The only exception tothis rule in Scilab occurs when we add or subtract a scalar to orfrom an array In this case the scalar is added or subtracted fromeach element in the array
For example
gt [6 5 6] + 3
ans =
[9 8 9]
gt [6 4 5 8] + [3 1 2 4]
ans =
[9 5 7 12]
gt [4 3] - 7
ans =
[-3 -4]
Array multiplication and division is very different to matrix mul-tiplication and division Array multiplication or division is an ele-ment by element multiplication or division that produces an arraywith the same dimensions as the starting arrays As with addi-tion and subtraction array multiplication and division require thatboth arrays have the same dimensions
Example B2 Consider the two row vectors x=[2 4 7 1] andy=[6 1 3 2] Scilab does not recognise the possibility ofmatrix multiplication of these two vectors They are not com-patible the number of columns of the pre-multiplier must bethe same as the number of rows of the post-multiplier Anattempt to execute x y will bring an error message
Obtain the usual dot product of the two row vectors by trans-posing the post-multiplier as in xyrsquo or yxrsquo The matrixproduct xrsquoy produces a matrix of 4 rows and 4 columns
B5 Builtndashin functions 237
Array multiplication produces the result
gt xy
ans =
[12 4 21 2]
by multiplying corresponding elements of each vector Theresult is a row vector with the same dimensions as x and ythat is a 1times 4 vector
Example B3 Consider the row vectors x=[2 4 7 1] and col-umn vector y=[6 1 3 2]rsquo Array multiplication of thesetwo matrices is not possible because their dimensions do notagree
Matrix multiplication xy produces the result 39 that is theinner product or dot product of the vectors Note vectorsas such do not exist in Scilab there are only single rowedor single columned arrays
Scilab not only allows arrays to be raised to powers but alsoallows scalars and arrays to be raised by array powers To performexponentiation on an element-by-element basis use the ^ symbolFor example if x=[3 5 8] then x^3 produces the matrix [33 53
83] = [27 125 512]
A scalar can be raised to an array power For example if p=[2 4 5]then typing 3^p produces the array [32 34 35] = [9 81 243]
With element-by-element array operations it is important to re-member that the dot () is a part of a two-character symbol for aparticular array operator such as () () () (^) Sometimesthis can cause some confusion the dot in 3^p is not a decimalpoint associated with the number 3 The following operations withthe value of p=[2 4 5] are equivalent
3^p
30^p
3^p
(3)^p
3^[2 4 5]
B5 Builtndashin functions
B51 Arithmetic functions
Arithmetic expressions often require computations other than ad-dition subtraction multiplication division and exponentiationFor example many expressions require the use of logarithms ex-ponentials and trigonometric functions Scilab has predefined alarge number of functions that perform these types of computations(see Tables B5 and B6)
238 Appendix B Programming with Scilab
An example of a predefined function is sin which returns the sineof a given angle
m = sin(angle)
where the variable angle contains the angle in radians and theresult is stored in the variable m The variable angle is called theargument or parameter of the function If the angle is stored indegrees it must be converted to radians for example
m = sin(anglepi180)
or
radians =anglepi180
m = sin(radians)
In Scilab the arguments or parameters of the function are con-tained in parentheses following the name of the function This isreferred to as pre-fix notation Many common functions in math-ematics have a syntax using in-fix notation where two argumentsappear to the right and left of the function name as for examplein x+y atimes b nmod r etc Scilab does not support in-fix nota-tion for any but the arithmetic functions Classical mathematicalnotation also has examples of post-fix notationmdash n x squared mdashand other hybrids like |x| while Scilab has ahb to confound thepurist
A function may contain no arguments one argument or many ar-guments depending on its definition If a function contains morethan one argument it is vital to give the arguments in the cor-rect order For example consider the problem of finding θ in thefollowing triangle using the atan command
5
8θ
Now by hand calculation the answer is 05586 radians If we useatan as below we get the answers shown
--gt atan(58)
ans =
05586
--gt atan(85)
ans =
10122
The first answer is the correct one The second answer is incorrect
B5 Builtndashin functions 239
Table B5 Some common mathematical functions defined inScilabFunction Description
abs(x) Computes the absolute value of xsqrt(x) Square root of x The parameter can be nega-
tiveround(x) Round x to the nearest integerfix(x) Round x to the nearest integer toward zerofloor(x) Round x to the nearest integer toward minusinfinceil(x) Round x to the nearest integer toward +infinsign(x) Returns -1 if x is less than zero zero if x equals
zero and 1 otherwisemodulo(xy) Returns the remainder of xy For example
modulo(254) is 1exp(x) Returns ex
log(x) Returns the natural logarithm of x That isthe logarithm to the base e Parameter can benegative but cannot be zero
log10(x) Returns the common logarithm of x That isthe logarithm to the base 10 Parameter can benegative but cannot be zero
since we passed the values to the function in the incorrect order
Some functions also require that the arguments be in specific unitsFor example the trigonometric functions assume that the argu-ments are in radians
In Scilab some functions use the number of arguments to deter-mine the output of the function For example the zeros functioncan have one or two arguments which then determine the output(check this with the command help zeros)
A function reference cannot be made on the left hand side of anassignment since a function call always returns a value(s) whereasan assignment is always to a variable Functions can of courseappear on the right side of an assignment since they return valuesto be assigned to some variable Functions may be applied tothe output of other functions and may be called in the algorithmdefining a function for example
logx = log(abs(x))
When one function is used to compute the argument of anotherfunction be sure to enclose the argument of each function in itsown set of parentheses
Scilab allows recursive functions that is functions that call them-selves in their defining algorithm
Exercise B4 Use examples to show the differences between the
240 Appendix B Programming with Scilab
Table B6 Some common trigonometric functions defined inScilab
Function Description
sin(x) Sine of x where x is in radianscos(x) Cosine of x where x is in radianstan(x) Tangent of x where x is in radiansasin(x) Arcsine or inverse sine of x where x must be
between -1 and 1 The return angle is in radiansbetween minusπ2 and π2
acos(x) Arccosine or inverse cosine of x where x mustbe between minus1 and 1 The return angle is inradians between 0 and π
atan(x) Arctangent or inverse tangent of x The returnangle is in radians between minusπ2 and π2
atan(xy) Arctangent or inverse tangent of xy The re-turn angle is in radians between minusπ and π de-pending on the sign of x and y
sec(x) Secant (1(cos x)) of x where x is in radianscsc(x) Cosecant (1(sin x)) of x where x is in radianscotg(x) Cotangent (1(tan x)) of x where x is in radians
Table B7 Some of Scilabrsquos utility functionsFunction Description
clear Clear variables from memoryhelp Display the first set of comment lines from
scriptswho List the variables in memorywhos List the variables in memory with their sizeerror Terminates the execution of the Sci-file and out-
puts the supplied error message
functions round(x) fix(x) floor(x) and ceil(x)
Exercise B5 The functions listed in Tables B5 B6 are only asmall sample of all of Scilabrsquos mathematical functions
Using Scilabrsquos help command list all of the hyperbolic func-tions available Also list any restrictions on the values of thefunction arguments
B52 Utility functions
In addition to the arithmetic functions Scilab has several utilityfunctions (see Table B7) These functions allow users to trackvariables (whoswho) feedback an error message if an error occursin the code (error) clear the variables form the memory (clear)etc Essential these functions allow users to debug and performbookkeeping operations inside their code
B6 Visualising data in Scilab 241
6
5
4- 04- 8
3
- 02
- 6
00
2- 4 Y- 2
02
Z
10
04
X 02
06
4 - 1
08
6- 28
10
Figure B2 Plot of z = sin(radicx2 + y2)
radicx2 + y2 illustrating the
3d capabilities of Scilab
B6 Visualising data in Scilab
One of the big advantages in using Scilab is the ability to visualisedata and results of complex calculations (see Figure B2 as exampleof the 3d capabilities) While Scilab is capability of complex 3dgraphics (see help pages for mesh and surf) in this section we willmesh amp surf
focus on the 2d plotting command (plot) Hence in this sectionplot
we will focus on
1 plotting mathematical functions from within Scilab
2 changing stylistic elements of the plot
3 adding titles labels and legends to the plot
4 including mathematical notation in titles labels and legendsand
5 plotting data stored in an external ASCII file
B61 Plotting mathematical functions from within Scilab
To plot a mathematical function in Scilab three steps are requiredThese are
1 Define the domain of the function (eg x=linspace(02pi))
2 Calculate the values of function at the selected points in thedomain (eg y=sin(x))
242 Appendix B Programming with Scilab
3 Final step is to plot the function using the plot commandplot
(eg plot(xy))
Using these steps it is possible to produce a basic plot of any math-ematical function (see Figure B3) Note that the plot shown inFigure B3 does not have a title axes labels or a legend all ofwhich are crucial for a good plot The Scilab commands to addthese elements to the plot are given in the following sections
- 10
- 08
- 06
- 04
- 02
00
02
04
06
08
10
0 1 2 3 4 5 6 7
Figure B3 The default plot of y = sin(x) in Scilab Notethat by default axes are not labelled and there is no title orlegend These need to added using the additional commandsxlabelylabeltitle and legend
Changing stylistic elements of the plot
An optional argument can be given to the plot command to cus-tomise each new line aspect This argument is a concatenatedstring containing information about colour line style or markersbased on the specifiers shown in Table B8 To specify the plotbe drawn with a red longdash-dot line with diamond marker thestring that would be passed to plot command would be rsquor-drsquo Thestring which is a concatenation (in any order) of the three typesof properties shown in Table B8 must be unambiguous Further-more the string specification is not case sensitive An exampleof specifying various line styles is shown in Example B6 and Fig-ure B4
Example B6 The following commands illustrate how line stylescan be used to distinguish between the plots of sin(2πt) sin(2πt+π6) and sin(2πt+ π2)
t=linspace(0130)
B6 Visualising data in Scilab 243
Table B8 Plot specifiers required to change line styles coloursand markers in plotsSpecifier Line style Specifier Colour Specifier Marker stylendash Solid line (default) r Red + Plus signndash ndash Dashed line g Green o Circle Dotted line b Blue Asteriskndash Dash-dotted line c Cyan x Cross
m Magenta s Squarey Yellow rsquononersquo No marker(default)k Blackw White
Figure B4 Resulting plotting after issuing the commands givenin Example B6
244 Appendix B Programming with Scilab
Example B6 illustrate a number of key commands relating to thecreation of multiple plots in Scilab The mtlb hold4 commandmtlb hold
holds the current plot and axis properties so that all subsequentplots are added to the existing plot
To distinguish each plot a different line specification is passed eachplot command In the example y = sin(2πt) is plotted using a redsolid line with data points indicated using a red circle as indicatedby the string rsquor-orsquo y = sin(2πt + π6) is plotted using greendashed-dotted line with data points marked by a square (rsquog-srsquo)while y = sin(2πt+ π2) is plotted using a cyan dashed line usingdiamonds to mark the data points (rsquoc--drsquo) Finally to completethe plot axes labels and a legend has been added to plot the com-mands for doing this are discussed in sectB62
B62 Adding titleslabels and legends to the plot
All good plots should have the following properties
1 The style of figure is appropriate to the information beingconveyed
2 The contents are well organised so it is easy for readers tointerpret the information
3 Legends labels etc are clear and concise (eg contain unitsif required do not obstruct parts of the plot etc) and
4 A caption or title that summarises the information presented
Like most mathematical applications Scilab has a wide variety ofdifferent plotting styles that can be used (eg look up the help forbar polarplot etc) This allows you to select the most appropri-barpolarplot
ate style to display and organise your data
The Scilab commands xlabel ylabel title and legend al-xlabel ylabel title and
legend lows you to add axes labels titles and legends to plot As shownin Example B6 the commands xlabel ylabel and title takea string input containing the title or label text For the legend
command a string is required for each separate plot Note theorder of the strings in the legend command reflects the order inwhich the each of plots were made
Including mathematical notation in titles labels and legends
It is possible to include mathematical characters such as π inScilab plots This is done by inserting special commands intoeither the title legend or label string These commands are basedon the abbreviations for mathematical notation used in programssuch as LATEX MathType etc A brief summary of the commands
4 mtlb hold on is equivalent to the Scilab commandset(gca()auto clearoff) while mtlb hold off is equivalentto the Scilab command set(gca()auto clearon)
B6 Visualising data in Scilab 245
Table B9 Basic mathematical symbols available in Scilab forincluding mathematical notation in titles labels and legendscommand symbol command symbol command symbolalpha α upsilon υ sim sim
Im = cup cup wp weierpotimes otimes subseteq sube oslash cap cap in isin supseteq supesupset sup lceil d subset subint
intcdot middot o oslash
rfloor c neg 6= nabla nablalfloor b times times ldots perp perp surd
radicprime prime
wedge and varpi $ O Oslashrceil e rangle 〉 mid |
vee or copyright ccopy langle 〈
available in Scilab for typesetting mathematical notation is givenin Table B9 To insert these commands into a text string simplystart and end the string with $ signs as shown in Example B7
Example B7 Scilab commands to produce the labelled plot ofy = sin
There are a few key concepts to note about the Example B7 Thefirst can be seen in the string being passed to the title commandtitle
In this string sin is prefixed with a backslash () This is doneso that the letters for sin are not typeset in italic which shouldonly be used for variables and not common mathematical func-tions Table B10 shows the command sequences to use for themore common mathematical functions5 To get the fraction in thetitle the command fracab has been used to produce a
b Ifsuperscripts and subscripts are required in a title or label these aresuperscripts amp subscripts
obtained by adding the subscript character ( ) and the superscriptcharacter (^) to the string These characters modify the characteror substring defined in braces immediately following For examplex 2 gives x2 while e^2x gives e2x
Note in Example B7 the commands xlabel and ylabel has beenxlabelylabel
used to set the axes labels in Figure B5 Instead of using thesethree commands the title and the axes labels could have been setusing the single xtitle command (see help xtitle for details)xtitle
In the example we have also customised the font size to make thetext more readable by a) passing an option to the title xlabelylabel commands and b) by using the command xset to set thexset
default font size before executing the xstring command Finallythe xstring command adds the text string to plot at the locationxstring
specified by variables b and c In the example the coordinates wewant to add the text are obtained by executing the xclick functionxclick
and clicking on the point required in the plot window
5 Additional details can be found in books covering Mathematics in LATEX
B6 Visualising data in Scilab 247
Table B10 Command sequences for the common mathematicalfunctions in Scilabarccos arcsin arctan cos cot
csc exp lim ln log
max min sec sin tan
B63 Plotting numeric data stored in an ASCII file
As data can be stored in a wide variety of ways in external filesScilab has a wide variety of functions which can be used to importthe data stored in these files (see help for the commands read
fscanf and xlsread) Since data can be stored in such a wideread fscanf and xlsread
range of formats we will confine our discussions here to the situa-tion where numeric data is stored in a ASCII (text) file where eachrow of the file contains the same number of entries
To illustrate the procedure for plotting data stored in ASCII file wewill consider the data obtained measuring the temperature of a cupof coffee every minute for a total of 100 mins The data recorded isstored in the a file called coffeedatatxt6 which simply containsthe two columns of numbers shown in Table B11 As an examplethe first 10 lines of the text file contains the following numbers7
00000 1000000
10000 950000
20000 903125
30000 859180
40000 817981
50000 779357
60000 743147
70000 709201
80000 677376
90000 647540
Note the current version of Scilab has a builtin text editor whichcan be used to view such text files To access this editor simplytype the command edit at the command prompt Once open youedit
can use the graphical interface to open the required file
The first issue is to get the data into Scilab so it is available forplot command In Scilab the command read can used to importread
the data and assign it to a variable In the current example wewould use the following command to import the data and assignit to a variable called cdata assuming the data is in the currentdirectory8
6 The file coffeedatatxt has been attached and should be available for savingprovided you are using Adobe Reader To save the file right click on thepurple Windows version or the red MacOSLinux version in the margin andselect save file from the options Ensure you save the file as coffeedatatxt
7 normally you will have received the data file in this format8 You can check what directory Scilab is in by typing the command pwd
Figure B6 Temperature of coffee based on the data recorded inthe file coffeedatatxt
cdata=read(rsquocoffeedatatxtrsquo-12)
The -12 following the name of the file tells Scilab that you donrsquotknow how many rows you have but you have two columns of dataSince we know that we have 101 rows of data we could have alsoused the following command to load the data
cdata=read(rsquocoffeedatatxtrsquo1012)
We now have a variable called cdata which has two columns ofdata The first column contains the time while the second containsthe Temperature records Hence we can plot it using the followingcommand
plot(cdata(1)cdata(2))
This gives the plot shown in Figure B6 after appropriate labelshave been added as described above
B7 Script files
Executing commands entered at the keyboard or keypad is themost primitive use of a calculating device It is only when se-quences of operations (to perform complex tasks) are stored awayand re-called for repeated application with different inputs that wehave a programmed operation Complex programs must necessarilyhave an interface with the user The basic interface tool in Scilabcapable of executing sequences of commands that are stored in a fileis the script The file names must have an extension of sce forexample program1sce assignment1sce These files are calledScript Sci-files or simply scripts
A script file is a file of standard keyboard charactersthat define commands to drive a program A script filecan be generated using any text editor or word proces-sor
To invoke a script in Scilab type the commandexec
exec filenamesce
where filename denotes the actual name of the file containing thescript When the script runs it is exactly the same as if you typedthe commands into the command window from the keyboard Thismeans that the Sci-file can use any variables you have defined priorto invoking the script Any variables created in the script are alsoavailable when the script is finished and any variable whose namewas assigned a value in the script will be over-written
If you give a script the same name as any variable that it computesin the running of the script then Scilab will give precedence tothe workspace variable over a script As stated in sectB3 the variablename will hide the script name after the script has run Scilab willtake the name as referring to the variable and not to the definedscript
Scilab also needs to know where to find the script to executedIf Scilab cannot find the script it will report that you have anundefined function or variable
The exact procedure for creating and saving Sci-files depends onthe Scilab version the operating system (MS Windows Macin-tosh or UnixLinux ) and the text editor you use The currentversion of Scilab includes a built-in editor which can be used tocreate and edit Sci-files and text files
To create a new Sci-file under the current version of Scilab simplytype edit newfilenamesce This will open up the Scilab editoredit
with a blank document called newfilename
B71 Script input and output
A script is a named file (with the extension sce) containing asequence of Scilab instructions which when its name is executedin a session performs those instructions in order as though theywere entered at the console
There are two common uses for scripts Pragmatically we maywant to save a particular computational sequence for later use orreference If the tasks involved are to be repeated often thenthe nature of the tasks should be analysed formulated into usefulfunctions (which we will discuss later) and the whole procedurepresented in a single function But if the tasks are only likely toget occasional use then the script may be adequately commentedand stored as a script file It is a quick and dirty way of saving
B7 Script files 251
the instructions for doing a particular job when we do not wantto go to the trouble of writing the functions for the job It is nota highly recommended modus operandi but useful all the same
The second use is for the script to act like a storyboard presentinginformation to a user in a structured and controlled way permit-ting the user to see the steps of a computational procedure andtheir outputs up on the screen At a more sophisticated level thescript takes on the proportions of an application controlling a largeand complex interaction with advanced interface features We willnot take our study of Scilab programming to this level but workmostly at the storyboard level using the simplest of Scilabrsquos in-terface tools
All the variables created in a script become global variables definedin the workspace and available for perusal As an application itis assumed that a script controls the entire workspace and thevariables it creates are lsquothe datarsquo of the application When scriptsare used as storyboards on a small scale then it cannot be assumedthat the variables they create deserve global recognition and it iscourteous to erase these variables when the script is finished unlessthey have some possible further use An unfortunate side-effect isthat if variables with the same name already exist in the workspacebefore a script is run then these variables will be over-written andlost
Used as storyboards scripts allow the user to cut paste and re-trythe various steps of a procedure and to modify the script to suithis own needs Storyboards may be used for demonstrations oftoolbox utilities or extended documentationexplanation of func-tions However if there is any danger that variables defined bythe script may over-write workspace variables then the demo ordocumentation should be re-fashioned into a function that leavesthe workspace variables untouched (more on this later)
Scripts will often read data from pre-prepared files and we will dis-cuss the functions provided by Scilab to perform this task below
Gathering data entered by the user really falls into the realm ofinterface design but we only briefly consider the simplest interfacefunction provided by Scilab Similarly we briefly consider thefunctions for providing the simplest forms of output presentation
Keyboard input
Scilab has a built-in function input that can be used to promptinput
the user to enter the value of a variable from the keyboard Forexample
Tc = input(rsquoPlease enter the temperature in degrees Celsiusrsquo)
will print the message
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
B2 Getting started with Scilab 227
professional looking product but it would take us the whole courseto master these and for the most part we will leave this activity toother courses Ours will be a study of efficient numerical techniquesand display of results will be plain and simple that is why thiscourse is called High Performance Numerical Computing
B2 Getting started with Scilab
Starting Scilab will give a command window similar to the oneshown in Figure B1 Commands are entered at the cursor (--gt)Each command is executed when either the return or enter key ispressed Any command execution can to be interrupted by pressinginterrupt a running command
Ctrl amp C keys together To exit Scilab simply type exit orquit at the command prompt2 Earlier commands executed inquitexit
a session can be recalled using the up (uarr)down (darr) keys on thekeyboard Once recalled the command can either be edited usingthe keyboard or re-executed by pressing the returnenter key Itis also possible to combine multiple commands on the one commandline by separating the commands using either a comma () or asemi-colon () The semi-colon () suppress any output from thesuppressing output from a
command command being executed
Figure B1 The default Scilab Command Window
Help about commands and functions can be obtained by typinghelp at the command plot This will bring up a window where youhelp
2 Note as with all commands in Scilab this will not be executed until eitherthe return or entry is pressed
228 Appendix B Programming with Scilab
can search for help on version topics Note you can short-cut thiswindow if you know the name of the command or function for whichyou require help by typing help command or function name atthe command prompt For example to get help on the log func-tion you would simply type help log at the command promptIn Scilab it is also possible to search for commands andor func-tions which contain certain keywords using the apropos commandapropos
As example the command apropos(rsquologarithmrsquo) will list all thecommands in Scilab relating to logarithms
Example B1 Sample commands in Scilab
--gt3^2-5-632
ans =
0
--gt3^2-5-6(32)
ans =
3
--gt42^2+1
ans =
17
--gtsin(pi2)
ans =
1
--gtsqrt(25)
ans =
5
--gtsqrt(-4)
ans =
2i
--gta1=5b1=3c1=sqrt(a1+b1)
a1 =
5
b1 =
3
c1 =
28284271
--gtans
ans =
2i
--gtt=linspace(0pi26)
t =
0 03141593 06283185 09424778 12566371 15707963
--gty=cos(t)
y =
1 09510565 08090170 05877853 03090170 6123D-17
Example B1 illustrates an important feature of the Scilab com-putational engine If an expression is just typed in and evaluatedthe answer is stored in a special variable called ans This vari-ans
B3 Variables 229
able contains the value of last unassigned (see sectB3 for details onhow variables are assigned) evaluation performed Hence the valuestored in ans changes each time an unassigned calculation is per-formed in Scilab as illustrated in Example B1 Consequentlyif you intend to reuse values of calculations you should store theresults in a variable (see sectB3) to avoid the possible of using theincorrect result
Activity BA To familiarise yourself with some of the basic features of rarrScilab and how to use it for elementary arithmetic andas a sophisticated calculator try some of the examples inExample B1 Make sure you understand what Scilabdid to calculate the answer Also as you work throughthe appendix attempt all the examples and exercises asyou come to them
B3 Variables
Variables are used in a programming language as tags to memorylocations in the computer The data stored in the memory locationis accessed using the variable name in the program The variableis not attached to the data but the memory location This meansthat the value contained within a variable can change
In languages such as C every variable must be declared before it isused specifying both its name and its type The name and type ofvariables do not need to be pre-declared in Scilab In Scilab anyvariable can take integer real complex scalar string or matrixvalues Languages that pre-declare variables have clearly statedat the start of a program or function the type of every variableused The Scilab interpreterrsquos liberal approach to variable typeshas both advantages and disadvantages
The equality sign (=) is used to assign the values to a variableassigning values to
variables This is an unfortunate choice of symbol and means that when wewant to test whether two quantities are equal or not we cannot usethe equal symbol = but must use the more cumbersome == Thus5+2 == 3+4 is true but 5+2 = 3+4 is a syntax error In Scilabprograms the equals sign does not have its mathematical meaningIt is an assignment operator The statement to the right of theequals sign is evaluated and the result is placed in the memorylocation tagged by the variable name on the left For example thestatement x = 3 + 5 means Scilab evaluates the statement onthe right and stores the result in the memory location tagged bythe variable x This means that the statement 3+ 5 = x whichis equivalent to x= 3 + 5 in mathematics will lead to an errormessage in Scilab
The following are examples of variable types in Scilab (it is notcomplete but contains all the variable types that you will encounter
230 Appendix B Programming with Scilab
in this course)
x = 3 Assign the scalar integer value 3 to the variable x
x = 7297e17 Assign the scalar real value 7297 times 1017 to the variable xThe number has been defined using the scientific exponentnotation The number following the e is the power of 10 tomultiply the number in front of the e Either e or E can beused
x = rsquoNorthrsquo Assign the string North to the variable x Strings of char-acters are defined by surrounding the characters with apos-trophes The construct rsquoSouthrsquo defines a constant stringcontaining the characters South
x = 05100 Assign the row vector containing 21 elements from 0 to 100in steps of 5 to the variable x
x = [12 13 25 67] Assign the 2times 2 matrix to the variable x so that
x =
[12 13
25 67
]
x = 51+i63 Assign the complex number 51+ i63 to the variable x
Variable names can be of any length but must be unique withinthe first 31 characters Variable names in Scilab are case sensi-tive that is the variables time and Time are distinct Howeverfunctions and scripts (which we discuss later) are stored in sepa-rate files (with extension sce) outside the interpreter and may ormay not be case sensitive depending on the outside environment(in a dos environment for example they will not be)
Scilab allows overloading of names This means that if you choosea name that has been predefined by Scilab you loose the builtin meaning For example cos = 634 defines a scalar variablecos but it is also the same name for the Scilab cosine functionThe variable assignment means that the cosine function will nolonger be accessible in this session All references to the name coswill assume the variable not the function Take care not to usevariable names that might disable built in functions or constantsthat you may need later When writing scripts for others to usethe names of all variables that will be over-written by the runningof the script must be listed for easy reference (and one shouldread these references before running any script) This is one ofthe reasons why it is nearly always preferable to write functionsrather than scripts Variables defined in the course of runninga functionrsquos algorithm remain local to the function and do notover-write workspace variables
The results of all calculations should be assigned to a variable sothat this value can be used later If this is not done the result
B3 Variables 231
Table B1 Special variables and constants in ScilabName Description
pi The value of π to the precession of the machineis returned
i Recognised as theradicminus1
inf This is Scilabrsquos representation of infin This is arecognised value in a variable and typically ap-pears as a result of division by zero or logarithmof zero
nan Stands for Not a Number This is a recognisedvalue in a variable and appears as a result ofa mathematically undefined operation such asdivision of zero by zero or infinminusinfin
eps Floating point relative accuracy That is epsreturns the smallest difference resolvable be-tween two floating point numbers For examplethe next largest representable number beyond10 is 10+eps Scilab cannot represent anyfloating point number that falls between 10 and10+eps
ans This variable contains the last computed expres-sion that was not stored in a variable
date This variable contains a string representation oftodayrsquos date in the format dd-mmm-yyyy Forexample 01-Apr-2000
232 Appendix B Programming with Scilab
Table B2 Scilab Arithmetic OperatorsOperator Meaning Scilab
^ Exponentiation xy x^y
Multiply xtimes y xy
Right Division xy = xy xy
Left Division yx = xy yx
+ Addition x+ y x+y
- Subtraction xminus y x-y
of a calculation will be stored in a special variable called ans (seeans
Table B1) The value stored in this variable is overridden the nexttime a calculation is done that is not assigned to a variable Thismeans that the result from the previous calculation is then lost
B4 Arithmetic expressions
Table B2 lists the basic arithmetic operators to be found in ScilabWith the exception of the left division operator the arithmetic op-erators in Scilab follow standard notation
B41 Precedence
When more than one operation appears in a mathematical state-ment the programming language needs to define in which order theoperations must be performed or the precedence That is whichparts are evaluated first
If precedence was not defined then the same statement can be eval-uated to a different result for example is 186+3 equal to 2 oris 186+3 equal to 6 The first statement is evaluated from theright using the precedence rule that usually applies in mathemat-ical statements that is work from right to left as for example inlsquosin log tan xrsquo where we start with a value of x compute the tan-gent of that value (tan x) then take the logarithm of the resultingnumber (log(tan x)) and finally take the sine of what this deliv-ers However it is usual (because it is much easier to write poly-nomials with this convention) to give multiplication and divisionprecedence over addition and subtraction The second statementrecognises the precedence of the division over addition and evalu-ates it first Right to left precedence has a much simpler parsingtree than schemes which give precedence to particular operatorsbut the common arithmetic precedences are so widely acceptedand expected that most computer language designers will not riskthe possible lsquoconsumer backlashrsquo that breaking with this traditionwould involve So with every computer language one must searchout and learn the precedence rules that it adopts The usual arith-metic precedences will normally be observed but note that even
B4 Arithmetic expressions 233
Table B3 Precedence of arithmetic expressionsOperator Precedence Order of evaluation
() 1 Left to right innermost first 2 Left to right 3 Left to right 4 Right to left
+ minus 5 Left to right
they are not unambiguous For example which divide has prece-dence in 41632 Working right to left one would get the answerof 8 but Scilab returns (the decimal for) 1128 mdashit works fromleft to right Table B3 lists the operators and their precedencevalue
The rules for evaluation of operators are as follows
bull Operators having high precedence (low numerical value) willbe evaluated before those having a low precedence (high nu-merical value)
bull If an expression contains more than one operator at the samelevel of precedence then the left-to-right rule of evaluationapply
Parentheses () always have the highest precedence Whateveris within the parentheses is evaluated first This means when indoubt about the order of any statement place parentheses aroundthe statement to ensure that the operations are performed in theorder you expect
B42 Matrix arithmetic
One of the strengths of Scilab is its ability to execute matrixarithmetic3 Matrix arithmetic must be implemented in other lan-guages with many lines of code or special functions Scilab isespecially designed to handle matrix arithmetic The representsmatrix multiplication or scalar multiplication of a matrix if one ofthe arguments is a scalar Since scalars are 1times1 matrices ordinaryscalar multiplication can also be performed by the operator Thedot product of two vectors however requires that the first vectorbe a row vector and the second a column vector To multiply thecorresponding elements in two vectors together requires a differentoperator so that
gt a = [2 3 4] b = [6 7 8]
a b
ans =
12 21 32
3 Remember vectors are examples of matrices A column vector of length nis an example of a ntimes 1 matrix
234 Appendix B Programming with Scilab
However because it is a matrix operation
gt a + b
ans =
8 10 40
The standard arithmetic operators described in Table B2 are alsoused with matrices The normal rules of matrix addition subtrac-tion and multiplication apply
The left and right division of matrices are also generalisations ofscalar division Right division BA is roughly the same as Binv(A)Left division AB is roughly the same as inv(A)B The differenceis in how the division is performed Different algorithms are usedwhen A or B are square or not
Scilab also defines a transpose operator for matrices The trans-pose of the matrix A is Arsquo For example if
A =
[25 63
12 09
] then A prime =
[25 12
63 09
]
Accessing matrix elements
Individual elements of a matrix can accessed and used in state-ments by specifying the matrix index of an element The index ofan element is its position in the matrix For example consider thefollowing matrix
A =
6 minus1 4
3 5 7
minus3 6 1
Define this in Scilab with the command
A=[ 6 -1 4 3 5 7 -3 6 7]
to display the contents of matrix A in Scilab just type the variablename A To display or use one element from the matrix specify theelement For example A(13) specifies the element in the first rowand third column that is 4 for the above matrix
To change a specific entry just assign a value to it For example
A(31)=-9
changes the value of the third row first column element from -3
to -9
To display or use a whole row or column use the operator Forexample
gt A(2)
ans =
3 5 7
gt A(2)
B4 Arithmetic expressions 235
ans =
-1
5
6
It is also possible to pick out a sub-matrix by the specifying the rowand column entries required For example To pick out the sub-matrix of A containing the elementsA11 A13 A21 A23 the commandA(12123) could be used as shown below
gt A(12123)
ans =
6 4
3 7
Note 123 means from 1 to 3 in steps of 2 This technique alongwith the assignment operator can be used for values of whole rowscolumns or sub-matrices in one command For example
gt A(12123)=[10111213]
A =
10 -1 11
12 5 13
-3 6 7
Note that the elements A11 A13 A21 A23 now contain the values10 11 12 and 13 respectively instead of their previous values
Index notation is logically an unfortunate phenomenon since itclashes with standard function notation On the face of it A(3)
should be the value of a function called A with input 3 but whenit is interpreted as index notation A is a variable However indexnotation is so familiar and useful to those familiar with classicalmathematical notation that many developers of mathematical soft-ware are not prepared to risk a ldquoconsumer backlashrdquo to providinga functional way of extracting elements from an array such as forexample 3 pick A to pick out the third element of A Similarlywe as teachers feel we would be doing students a disservice if wefailed to introduce them to the index notation that abounds inmathematical texts usually in the form of subscripting as in A3Logic does not always win an argument In the commercial worldof computing an established user base carries more weight thanmathematical logic just as in mathematics tradition may well alsofly in the face of logical consistency
B43 Array arithmetic
Scilab has two forms of arithmetic operations that can be usedwith matrices matrix arithmetic or array arithmetic Matrix arith-metic follows the normal rules of matrix algebra Array arithmeticwhich is also called element-by-element arithmetic is where oper-ations are performed on matrices element by element
236 Appendix B Programming with Scilab
Table B4 Scilab array operatorsOperator Meaning Scilab
^ Exponentiation x^y
Multiply xy
Right Division xy
Left Division yx
+ Addition x+y
- Subtraction x-y
Table B4 lists the Scilab array operators
The array addition and subtraction are identical to matrix additionor subtraction Both array and matrix addition and subtractionoperate element by element Array addition and subtraction re-quire that both arrays have the same size The only exception tothis rule in Scilab occurs when we add or subtract a scalar to orfrom an array In this case the scalar is added or subtracted fromeach element in the array
For example
gt [6 5 6] + 3
ans =
[9 8 9]
gt [6 4 5 8] + [3 1 2 4]
ans =
[9 5 7 12]
gt [4 3] - 7
ans =
[-3 -4]
Array multiplication and division is very different to matrix mul-tiplication and division Array multiplication or division is an ele-ment by element multiplication or division that produces an arraywith the same dimensions as the starting arrays As with addi-tion and subtraction array multiplication and division require thatboth arrays have the same dimensions
Example B2 Consider the two row vectors x=[2 4 7 1] andy=[6 1 3 2] Scilab does not recognise the possibility ofmatrix multiplication of these two vectors They are not com-patible the number of columns of the pre-multiplier must bethe same as the number of rows of the post-multiplier Anattempt to execute x y will bring an error message
Obtain the usual dot product of the two row vectors by trans-posing the post-multiplier as in xyrsquo or yxrsquo The matrixproduct xrsquoy produces a matrix of 4 rows and 4 columns
B5 Builtndashin functions 237
Array multiplication produces the result
gt xy
ans =
[12 4 21 2]
by multiplying corresponding elements of each vector Theresult is a row vector with the same dimensions as x and ythat is a 1times 4 vector
Example B3 Consider the row vectors x=[2 4 7 1] and col-umn vector y=[6 1 3 2]rsquo Array multiplication of thesetwo matrices is not possible because their dimensions do notagree
Matrix multiplication xy produces the result 39 that is theinner product or dot product of the vectors Note vectorsas such do not exist in Scilab there are only single rowedor single columned arrays
Scilab not only allows arrays to be raised to powers but alsoallows scalars and arrays to be raised by array powers To performexponentiation on an element-by-element basis use the ^ symbolFor example if x=[3 5 8] then x^3 produces the matrix [33 53
83] = [27 125 512]
A scalar can be raised to an array power For example if p=[2 4 5]then typing 3^p produces the array [32 34 35] = [9 81 243]
With element-by-element array operations it is important to re-member that the dot () is a part of a two-character symbol for aparticular array operator such as () () () (^) Sometimesthis can cause some confusion the dot in 3^p is not a decimalpoint associated with the number 3 The following operations withthe value of p=[2 4 5] are equivalent
3^p
30^p
3^p
(3)^p
3^[2 4 5]
B5 Builtndashin functions
B51 Arithmetic functions
Arithmetic expressions often require computations other than ad-dition subtraction multiplication division and exponentiationFor example many expressions require the use of logarithms ex-ponentials and trigonometric functions Scilab has predefined alarge number of functions that perform these types of computations(see Tables B5 and B6)
238 Appendix B Programming with Scilab
An example of a predefined function is sin which returns the sineof a given angle
m = sin(angle)
where the variable angle contains the angle in radians and theresult is stored in the variable m The variable angle is called theargument or parameter of the function If the angle is stored indegrees it must be converted to radians for example
m = sin(anglepi180)
or
radians =anglepi180
m = sin(radians)
In Scilab the arguments or parameters of the function are con-tained in parentheses following the name of the function This isreferred to as pre-fix notation Many common functions in math-ematics have a syntax using in-fix notation where two argumentsappear to the right and left of the function name as for examplein x+y atimes b nmod r etc Scilab does not support in-fix nota-tion for any but the arithmetic functions Classical mathematicalnotation also has examples of post-fix notationmdash n x squared mdashand other hybrids like |x| while Scilab has ahb to confound thepurist
A function may contain no arguments one argument or many ar-guments depending on its definition If a function contains morethan one argument it is vital to give the arguments in the cor-rect order For example consider the problem of finding θ in thefollowing triangle using the atan command
5
8θ
Now by hand calculation the answer is 05586 radians If we useatan as below we get the answers shown
--gt atan(58)
ans =
05586
--gt atan(85)
ans =
10122
The first answer is the correct one The second answer is incorrect
B5 Builtndashin functions 239
Table B5 Some common mathematical functions defined inScilabFunction Description
abs(x) Computes the absolute value of xsqrt(x) Square root of x The parameter can be nega-
tiveround(x) Round x to the nearest integerfix(x) Round x to the nearest integer toward zerofloor(x) Round x to the nearest integer toward minusinfinceil(x) Round x to the nearest integer toward +infinsign(x) Returns -1 if x is less than zero zero if x equals
zero and 1 otherwisemodulo(xy) Returns the remainder of xy For example
modulo(254) is 1exp(x) Returns ex
log(x) Returns the natural logarithm of x That isthe logarithm to the base e Parameter can benegative but cannot be zero
log10(x) Returns the common logarithm of x That isthe logarithm to the base 10 Parameter can benegative but cannot be zero
since we passed the values to the function in the incorrect order
Some functions also require that the arguments be in specific unitsFor example the trigonometric functions assume that the argu-ments are in radians
In Scilab some functions use the number of arguments to deter-mine the output of the function For example the zeros functioncan have one or two arguments which then determine the output(check this with the command help zeros)
A function reference cannot be made on the left hand side of anassignment since a function call always returns a value(s) whereasan assignment is always to a variable Functions can of courseappear on the right side of an assignment since they return valuesto be assigned to some variable Functions may be applied tothe output of other functions and may be called in the algorithmdefining a function for example
logx = log(abs(x))
When one function is used to compute the argument of anotherfunction be sure to enclose the argument of each function in itsown set of parentheses
Scilab allows recursive functions that is functions that call them-selves in their defining algorithm
Exercise B4 Use examples to show the differences between the
240 Appendix B Programming with Scilab
Table B6 Some common trigonometric functions defined inScilab
Function Description
sin(x) Sine of x where x is in radianscos(x) Cosine of x where x is in radianstan(x) Tangent of x where x is in radiansasin(x) Arcsine or inverse sine of x where x must be
between -1 and 1 The return angle is in radiansbetween minusπ2 and π2
acos(x) Arccosine or inverse cosine of x where x mustbe between minus1 and 1 The return angle is inradians between 0 and π
atan(x) Arctangent or inverse tangent of x The returnangle is in radians between minusπ2 and π2
atan(xy) Arctangent or inverse tangent of xy The re-turn angle is in radians between minusπ and π de-pending on the sign of x and y
sec(x) Secant (1(cos x)) of x where x is in radianscsc(x) Cosecant (1(sin x)) of x where x is in radianscotg(x) Cotangent (1(tan x)) of x where x is in radians
Table B7 Some of Scilabrsquos utility functionsFunction Description
clear Clear variables from memoryhelp Display the first set of comment lines from
scriptswho List the variables in memorywhos List the variables in memory with their sizeerror Terminates the execution of the Sci-file and out-
puts the supplied error message
functions round(x) fix(x) floor(x) and ceil(x)
Exercise B5 The functions listed in Tables B5 B6 are only asmall sample of all of Scilabrsquos mathematical functions
Using Scilabrsquos help command list all of the hyperbolic func-tions available Also list any restrictions on the values of thefunction arguments
B52 Utility functions
In addition to the arithmetic functions Scilab has several utilityfunctions (see Table B7) These functions allow users to trackvariables (whoswho) feedback an error message if an error occursin the code (error) clear the variables form the memory (clear)etc Essential these functions allow users to debug and performbookkeeping operations inside their code
B6 Visualising data in Scilab 241
6
5
4- 04- 8
3
- 02
- 6
00
2- 4 Y- 2
02
Z
10
04
X 02
06
4 - 1
08
6- 28
10
Figure B2 Plot of z = sin(radicx2 + y2)
radicx2 + y2 illustrating the
3d capabilities of Scilab
B6 Visualising data in Scilab
One of the big advantages in using Scilab is the ability to visualisedata and results of complex calculations (see Figure B2 as exampleof the 3d capabilities) While Scilab is capability of complex 3dgraphics (see help pages for mesh and surf) in this section we willmesh amp surf
focus on the 2d plotting command (plot) Hence in this sectionplot
we will focus on
1 plotting mathematical functions from within Scilab
2 changing stylistic elements of the plot
3 adding titles labels and legends to the plot
4 including mathematical notation in titles labels and legendsand
5 plotting data stored in an external ASCII file
B61 Plotting mathematical functions from within Scilab
To plot a mathematical function in Scilab three steps are requiredThese are
1 Define the domain of the function (eg x=linspace(02pi))
2 Calculate the values of function at the selected points in thedomain (eg y=sin(x))
242 Appendix B Programming with Scilab
3 Final step is to plot the function using the plot commandplot
(eg plot(xy))
Using these steps it is possible to produce a basic plot of any math-ematical function (see Figure B3) Note that the plot shown inFigure B3 does not have a title axes labels or a legend all ofwhich are crucial for a good plot The Scilab commands to addthese elements to the plot are given in the following sections
- 10
- 08
- 06
- 04
- 02
00
02
04
06
08
10
0 1 2 3 4 5 6 7
Figure B3 The default plot of y = sin(x) in Scilab Notethat by default axes are not labelled and there is no title orlegend These need to added using the additional commandsxlabelylabeltitle and legend
Changing stylistic elements of the plot
An optional argument can be given to the plot command to cus-tomise each new line aspect This argument is a concatenatedstring containing information about colour line style or markersbased on the specifiers shown in Table B8 To specify the plotbe drawn with a red longdash-dot line with diamond marker thestring that would be passed to plot command would be rsquor-drsquo Thestring which is a concatenation (in any order) of the three typesof properties shown in Table B8 must be unambiguous Further-more the string specification is not case sensitive An exampleof specifying various line styles is shown in Example B6 and Fig-ure B4
Example B6 The following commands illustrate how line stylescan be used to distinguish between the plots of sin(2πt) sin(2πt+π6) and sin(2πt+ π2)
t=linspace(0130)
B6 Visualising data in Scilab 243
Table B8 Plot specifiers required to change line styles coloursand markers in plotsSpecifier Line style Specifier Colour Specifier Marker stylendash Solid line (default) r Red + Plus signndash ndash Dashed line g Green o Circle Dotted line b Blue Asteriskndash Dash-dotted line c Cyan x Cross
m Magenta s Squarey Yellow rsquononersquo No marker(default)k Blackw White
Figure B4 Resulting plotting after issuing the commands givenin Example B6
244 Appendix B Programming with Scilab
Example B6 illustrate a number of key commands relating to thecreation of multiple plots in Scilab The mtlb hold4 commandmtlb hold
holds the current plot and axis properties so that all subsequentplots are added to the existing plot
To distinguish each plot a different line specification is passed eachplot command In the example y = sin(2πt) is plotted using a redsolid line with data points indicated using a red circle as indicatedby the string rsquor-orsquo y = sin(2πt + π6) is plotted using greendashed-dotted line with data points marked by a square (rsquog-srsquo)while y = sin(2πt+ π2) is plotted using a cyan dashed line usingdiamonds to mark the data points (rsquoc--drsquo) Finally to completethe plot axes labels and a legend has been added to plot the com-mands for doing this are discussed in sectB62
B62 Adding titleslabels and legends to the plot
All good plots should have the following properties
1 The style of figure is appropriate to the information beingconveyed
2 The contents are well organised so it is easy for readers tointerpret the information
3 Legends labels etc are clear and concise (eg contain unitsif required do not obstruct parts of the plot etc) and
4 A caption or title that summarises the information presented
Like most mathematical applications Scilab has a wide variety ofdifferent plotting styles that can be used (eg look up the help forbar polarplot etc) This allows you to select the most appropri-barpolarplot
ate style to display and organise your data
The Scilab commands xlabel ylabel title and legend al-xlabel ylabel title and
legend lows you to add axes labels titles and legends to plot As shownin Example B6 the commands xlabel ylabel and title takea string input containing the title or label text For the legend
command a string is required for each separate plot Note theorder of the strings in the legend command reflects the order inwhich the each of plots were made
Including mathematical notation in titles labels and legends
It is possible to include mathematical characters such as π inScilab plots This is done by inserting special commands intoeither the title legend or label string These commands are basedon the abbreviations for mathematical notation used in programssuch as LATEX MathType etc A brief summary of the commands
4 mtlb hold on is equivalent to the Scilab commandset(gca()auto clearoff) while mtlb hold off is equivalentto the Scilab command set(gca()auto clearon)
B6 Visualising data in Scilab 245
Table B9 Basic mathematical symbols available in Scilab forincluding mathematical notation in titles labels and legendscommand symbol command symbol command symbolalpha α upsilon υ sim sim
Im = cup cup wp weierpotimes otimes subseteq sube oslash cap cap in isin supseteq supesupset sup lceil d subset subint
intcdot middot o oslash
rfloor c neg 6= nabla nablalfloor b times times ldots perp perp surd
radicprime prime
wedge and varpi $ O Oslashrceil e rangle 〉 mid |
vee or copyright ccopy langle 〈
available in Scilab for typesetting mathematical notation is givenin Table B9 To insert these commands into a text string simplystart and end the string with $ signs as shown in Example B7
Example B7 Scilab commands to produce the labelled plot ofy = sin
There are a few key concepts to note about the Example B7 Thefirst can be seen in the string being passed to the title commandtitle
In this string sin is prefixed with a backslash () This is doneso that the letters for sin are not typeset in italic which shouldonly be used for variables and not common mathematical func-tions Table B10 shows the command sequences to use for themore common mathematical functions5 To get the fraction in thetitle the command fracab has been used to produce a
b Ifsuperscripts and subscripts are required in a title or label these aresuperscripts amp subscripts
obtained by adding the subscript character ( ) and the superscriptcharacter (^) to the string These characters modify the characteror substring defined in braces immediately following For examplex 2 gives x2 while e^2x gives e2x
Note in Example B7 the commands xlabel and ylabel has beenxlabelylabel
used to set the axes labels in Figure B5 Instead of using thesethree commands the title and the axes labels could have been setusing the single xtitle command (see help xtitle for details)xtitle
In the example we have also customised the font size to make thetext more readable by a) passing an option to the title xlabelylabel commands and b) by using the command xset to set thexset
default font size before executing the xstring command Finallythe xstring command adds the text string to plot at the locationxstring
specified by variables b and c In the example the coordinates wewant to add the text are obtained by executing the xclick functionxclick
and clicking on the point required in the plot window
5 Additional details can be found in books covering Mathematics in LATEX
B6 Visualising data in Scilab 247
Table B10 Command sequences for the common mathematicalfunctions in Scilabarccos arcsin arctan cos cot
csc exp lim ln log
max min sec sin tan
B63 Plotting numeric data stored in an ASCII file
As data can be stored in a wide variety of ways in external filesScilab has a wide variety of functions which can be used to importthe data stored in these files (see help for the commands read
fscanf and xlsread) Since data can be stored in such a wideread fscanf and xlsread
range of formats we will confine our discussions here to the situa-tion where numeric data is stored in a ASCII (text) file where eachrow of the file contains the same number of entries
To illustrate the procedure for plotting data stored in ASCII file wewill consider the data obtained measuring the temperature of a cupof coffee every minute for a total of 100 mins The data recorded isstored in the a file called coffeedatatxt6 which simply containsthe two columns of numbers shown in Table B11 As an examplethe first 10 lines of the text file contains the following numbers7
00000 1000000
10000 950000
20000 903125
30000 859180
40000 817981
50000 779357
60000 743147
70000 709201
80000 677376
90000 647540
Note the current version of Scilab has a builtin text editor whichcan be used to view such text files To access this editor simplytype the command edit at the command prompt Once open youedit
can use the graphical interface to open the required file
The first issue is to get the data into Scilab so it is available forplot command In Scilab the command read can used to importread
the data and assign it to a variable In the current example wewould use the following command to import the data and assignit to a variable called cdata assuming the data is in the currentdirectory8
6 The file coffeedatatxt has been attached and should be available for savingprovided you are using Adobe Reader To save the file right click on thepurple Windows version or the red MacOSLinux version in the margin andselect save file from the options Ensure you save the file as coffeedatatxt
7 normally you will have received the data file in this format8 You can check what directory Scilab is in by typing the command pwd
Figure B6 Temperature of coffee based on the data recorded inthe file coffeedatatxt
cdata=read(rsquocoffeedatatxtrsquo-12)
The -12 following the name of the file tells Scilab that you donrsquotknow how many rows you have but you have two columns of dataSince we know that we have 101 rows of data we could have alsoused the following command to load the data
cdata=read(rsquocoffeedatatxtrsquo1012)
We now have a variable called cdata which has two columns ofdata The first column contains the time while the second containsthe Temperature records Hence we can plot it using the followingcommand
plot(cdata(1)cdata(2))
This gives the plot shown in Figure B6 after appropriate labelshave been added as described above
B7 Script files
Executing commands entered at the keyboard or keypad is themost primitive use of a calculating device It is only when se-quences of operations (to perform complex tasks) are stored awayand re-called for repeated application with different inputs that wehave a programmed operation Complex programs must necessarilyhave an interface with the user The basic interface tool in Scilabcapable of executing sequences of commands that are stored in a fileis the script The file names must have an extension of sce forexample program1sce assignment1sce These files are calledScript Sci-files or simply scripts
A script file is a file of standard keyboard charactersthat define commands to drive a program A script filecan be generated using any text editor or word proces-sor
To invoke a script in Scilab type the commandexec
exec filenamesce
where filename denotes the actual name of the file containing thescript When the script runs it is exactly the same as if you typedthe commands into the command window from the keyboard Thismeans that the Sci-file can use any variables you have defined priorto invoking the script Any variables created in the script are alsoavailable when the script is finished and any variable whose namewas assigned a value in the script will be over-written
If you give a script the same name as any variable that it computesin the running of the script then Scilab will give precedence tothe workspace variable over a script As stated in sectB3 the variablename will hide the script name after the script has run Scilab willtake the name as referring to the variable and not to the definedscript
Scilab also needs to know where to find the script to executedIf Scilab cannot find the script it will report that you have anundefined function or variable
The exact procedure for creating and saving Sci-files depends onthe Scilab version the operating system (MS Windows Macin-tosh or UnixLinux ) and the text editor you use The currentversion of Scilab includes a built-in editor which can be used tocreate and edit Sci-files and text files
To create a new Sci-file under the current version of Scilab simplytype edit newfilenamesce This will open up the Scilab editoredit
with a blank document called newfilename
B71 Script input and output
A script is a named file (with the extension sce) containing asequence of Scilab instructions which when its name is executedin a session performs those instructions in order as though theywere entered at the console
There are two common uses for scripts Pragmatically we maywant to save a particular computational sequence for later use orreference If the tasks involved are to be repeated often thenthe nature of the tasks should be analysed formulated into usefulfunctions (which we will discuss later) and the whole procedurepresented in a single function But if the tasks are only likely toget occasional use then the script may be adequately commentedand stored as a script file It is a quick and dirty way of saving
B7 Script files 251
the instructions for doing a particular job when we do not wantto go to the trouble of writing the functions for the job It is nota highly recommended modus operandi but useful all the same
The second use is for the script to act like a storyboard presentinginformation to a user in a structured and controlled way permit-ting the user to see the steps of a computational procedure andtheir outputs up on the screen At a more sophisticated level thescript takes on the proportions of an application controlling a largeand complex interaction with advanced interface features We willnot take our study of Scilab programming to this level but workmostly at the storyboard level using the simplest of Scilabrsquos in-terface tools
All the variables created in a script become global variables definedin the workspace and available for perusal As an application itis assumed that a script controls the entire workspace and thevariables it creates are lsquothe datarsquo of the application When scriptsare used as storyboards on a small scale then it cannot be assumedthat the variables they create deserve global recognition and it iscourteous to erase these variables when the script is finished unlessthey have some possible further use An unfortunate side-effect isthat if variables with the same name already exist in the workspacebefore a script is run then these variables will be over-written andlost
Used as storyboards scripts allow the user to cut paste and re-trythe various steps of a procedure and to modify the script to suithis own needs Storyboards may be used for demonstrations oftoolbox utilities or extended documentationexplanation of func-tions However if there is any danger that variables defined bythe script may over-write workspace variables then the demo ordocumentation should be re-fashioned into a function that leavesthe workspace variables untouched (more on this later)
Scripts will often read data from pre-prepared files and we will dis-cuss the functions provided by Scilab to perform this task below
Gathering data entered by the user really falls into the realm ofinterface design but we only briefly consider the simplest interfacefunction provided by Scilab Similarly we briefly consider thefunctions for providing the simplest forms of output presentation
Keyboard input
Scilab has a built-in function input that can be used to promptinput
the user to enter the value of a variable from the keyboard Forexample
Tc = input(rsquoPlease enter the temperature in degrees Celsiusrsquo)
will print the message
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
228 Appendix B Programming with Scilab
can search for help on version topics Note you can short-cut thiswindow if you know the name of the command or function for whichyou require help by typing help command or function name atthe command prompt For example to get help on the log func-tion you would simply type help log at the command promptIn Scilab it is also possible to search for commands andor func-tions which contain certain keywords using the apropos commandapropos
As example the command apropos(rsquologarithmrsquo) will list all thecommands in Scilab relating to logarithms
Example B1 Sample commands in Scilab
--gt3^2-5-632
ans =
0
--gt3^2-5-6(32)
ans =
3
--gt42^2+1
ans =
17
--gtsin(pi2)
ans =
1
--gtsqrt(25)
ans =
5
--gtsqrt(-4)
ans =
2i
--gta1=5b1=3c1=sqrt(a1+b1)
a1 =
5
b1 =
3
c1 =
28284271
--gtans
ans =
2i
--gtt=linspace(0pi26)
t =
0 03141593 06283185 09424778 12566371 15707963
--gty=cos(t)
y =
1 09510565 08090170 05877853 03090170 6123D-17
Example B1 illustrates an important feature of the Scilab com-putational engine If an expression is just typed in and evaluatedthe answer is stored in a special variable called ans This vari-ans
B3 Variables 229
able contains the value of last unassigned (see sectB3 for details onhow variables are assigned) evaluation performed Hence the valuestored in ans changes each time an unassigned calculation is per-formed in Scilab as illustrated in Example B1 Consequentlyif you intend to reuse values of calculations you should store theresults in a variable (see sectB3) to avoid the possible of using theincorrect result
Activity BA To familiarise yourself with some of the basic features of rarrScilab and how to use it for elementary arithmetic andas a sophisticated calculator try some of the examples inExample B1 Make sure you understand what Scilabdid to calculate the answer Also as you work throughthe appendix attempt all the examples and exercises asyou come to them
B3 Variables
Variables are used in a programming language as tags to memorylocations in the computer The data stored in the memory locationis accessed using the variable name in the program The variableis not attached to the data but the memory location This meansthat the value contained within a variable can change
In languages such as C every variable must be declared before it isused specifying both its name and its type The name and type ofvariables do not need to be pre-declared in Scilab In Scilab anyvariable can take integer real complex scalar string or matrixvalues Languages that pre-declare variables have clearly statedat the start of a program or function the type of every variableused The Scilab interpreterrsquos liberal approach to variable typeshas both advantages and disadvantages
The equality sign (=) is used to assign the values to a variableassigning values to
variables This is an unfortunate choice of symbol and means that when wewant to test whether two quantities are equal or not we cannot usethe equal symbol = but must use the more cumbersome == Thus5+2 == 3+4 is true but 5+2 = 3+4 is a syntax error In Scilabprograms the equals sign does not have its mathematical meaningIt is an assignment operator The statement to the right of theequals sign is evaluated and the result is placed in the memorylocation tagged by the variable name on the left For example thestatement x = 3 + 5 means Scilab evaluates the statement onthe right and stores the result in the memory location tagged bythe variable x This means that the statement 3+ 5 = x whichis equivalent to x= 3 + 5 in mathematics will lead to an errormessage in Scilab
The following are examples of variable types in Scilab (it is notcomplete but contains all the variable types that you will encounter
230 Appendix B Programming with Scilab
in this course)
x = 3 Assign the scalar integer value 3 to the variable x
x = 7297e17 Assign the scalar real value 7297 times 1017 to the variable xThe number has been defined using the scientific exponentnotation The number following the e is the power of 10 tomultiply the number in front of the e Either e or E can beused
x = rsquoNorthrsquo Assign the string North to the variable x Strings of char-acters are defined by surrounding the characters with apos-trophes The construct rsquoSouthrsquo defines a constant stringcontaining the characters South
x = 05100 Assign the row vector containing 21 elements from 0 to 100in steps of 5 to the variable x
x = [12 13 25 67] Assign the 2times 2 matrix to the variable x so that
x =
[12 13
25 67
]
x = 51+i63 Assign the complex number 51+ i63 to the variable x
Variable names can be of any length but must be unique withinthe first 31 characters Variable names in Scilab are case sensi-tive that is the variables time and Time are distinct Howeverfunctions and scripts (which we discuss later) are stored in sepa-rate files (with extension sce) outside the interpreter and may ormay not be case sensitive depending on the outside environment(in a dos environment for example they will not be)
Scilab allows overloading of names This means that if you choosea name that has been predefined by Scilab you loose the builtin meaning For example cos = 634 defines a scalar variablecos but it is also the same name for the Scilab cosine functionThe variable assignment means that the cosine function will nolonger be accessible in this session All references to the name coswill assume the variable not the function Take care not to usevariable names that might disable built in functions or constantsthat you may need later When writing scripts for others to usethe names of all variables that will be over-written by the runningof the script must be listed for easy reference (and one shouldread these references before running any script) This is one ofthe reasons why it is nearly always preferable to write functionsrather than scripts Variables defined in the course of runninga functionrsquos algorithm remain local to the function and do notover-write workspace variables
The results of all calculations should be assigned to a variable sothat this value can be used later If this is not done the result
B3 Variables 231
Table B1 Special variables and constants in ScilabName Description
pi The value of π to the precession of the machineis returned
i Recognised as theradicminus1
inf This is Scilabrsquos representation of infin This is arecognised value in a variable and typically ap-pears as a result of division by zero or logarithmof zero
nan Stands for Not a Number This is a recognisedvalue in a variable and appears as a result ofa mathematically undefined operation such asdivision of zero by zero or infinminusinfin
eps Floating point relative accuracy That is epsreturns the smallest difference resolvable be-tween two floating point numbers For examplethe next largest representable number beyond10 is 10+eps Scilab cannot represent anyfloating point number that falls between 10 and10+eps
ans This variable contains the last computed expres-sion that was not stored in a variable
date This variable contains a string representation oftodayrsquos date in the format dd-mmm-yyyy Forexample 01-Apr-2000
232 Appendix B Programming with Scilab
Table B2 Scilab Arithmetic OperatorsOperator Meaning Scilab
^ Exponentiation xy x^y
Multiply xtimes y xy
Right Division xy = xy xy
Left Division yx = xy yx
+ Addition x+ y x+y
- Subtraction xminus y x-y
of a calculation will be stored in a special variable called ans (seeans
Table B1) The value stored in this variable is overridden the nexttime a calculation is done that is not assigned to a variable Thismeans that the result from the previous calculation is then lost
B4 Arithmetic expressions
Table B2 lists the basic arithmetic operators to be found in ScilabWith the exception of the left division operator the arithmetic op-erators in Scilab follow standard notation
B41 Precedence
When more than one operation appears in a mathematical state-ment the programming language needs to define in which order theoperations must be performed or the precedence That is whichparts are evaluated first
If precedence was not defined then the same statement can be eval-uated to a different result for example is 186+3 equal to 2 oris 186+3 equal to 6 The first statement is evaluated from theright using the precedence rule that usually applies in mathemat-ical statements that is work from right to left as for example inlsquosin log tan xrsquo where we start with a value of x compute the tan-gent of that value (tan x) then take the logarithm of the resultingnumber (log(tan x)) and finally take the sine of what this deliv-ers However it is usual (because it is much easier to write poly-nomials with this convention) to give multiplication and divisionprecedence over addition and subtraction The second statementrecognises the precedence of the division over addition and evalu-ates it first Right to left precedence has a much simpler parsingtree than schemes which give precedence to particular operatorsbut the common arithmetic precedences are so widely acceptedand expected that most computer language designers will not riskthe possible lsquoconsumer backlashrsquo that breaking with this traditionwould involve So with every computer language one must searchout and learn the precedence rules that it adopts The usual arith-metic precedences will normally be observed but note that even
B4 Arithmetic expressions 233
Table B3 Precedence of arithmetic expressionsOperator Precedence Order of evaluation
() 1 Left to right innermost first 2 Left to right 3 Left to right 4 Right to left
+ minus 5 Left to right
they are not unambiguous For example which divide has prece-dence in 41632 Working right to left one would get the answerof 8 but Scilab returns (the decimal for) 1128 mdashit works fromleft to right Table B3 lists the operators and their precedencevalue
The rules for evaluation of operators are as follows
bull Operators having high precedence (low numerical value) willbe evaluated before those having a low precedence (high nu-merical value)
bull If an expression contains more than one operator at the samelevel of precedence then the left-to-right rule of evaluationapply
Parentheses () always have the highest precedence Whateveris within the parentheses is evaluated first This means when indoubt about the order of any statement place parentheses aroundthe statement to ensure that the operations are performed in theorder you expect
B42 Matrix arithmetic
One of the strengths of Scilab is its ability to execute matrixarithmetic3 Matrix arithmetic must be implemented in other lan-guages with many lines of code or special functions Scilab isespecially designed to handle matrix arithmetic The representsmatrix multiplication or scalar multiplication of a matrix if one ofthe arguments is a scalar Since scalars are 1times1 matrices ordinaryscalar multiplication can also be performed by the operator Thedot product of two vectors however requires that the first vectorbe a row vector and the second a column vector To multiply thecorresponding elements in two vectors together requires a differentoperator so that
gt a = [2 3 4] b = [6 7 8]
a b
ans =
12 21 32
3 Remember vectors are examples of matrices A column vector of length nis an example of a ntimes 1 matrix
234 Appendix B Programming with Scilab
However because it is a matrix operation
gt a + b
ans =
8 10 40
The standard arithmetic operators described in Table B2 are alsoused with matrices The normal rules of matrix addition subtrac-tion and multiplication apply
The left and right division of matrices are also generalisations ofscalar division Right division BA is roughly the same as Binv(A)Left division AB is roughly the same as inv(A)B The differenceis in how the division is performed Different algorithms are usedwhen A or B are square or not
Scilab also defines a transpose operator for matrices The trans-pose of the matrix A is Arsquo For example if
A =
[25 63
12 09
] then A prime =
[25 12
63 09
]
Accessing matrix elements
Individual elements of a matrix can accessed and used in state-ments by specifying the matrix index of an element The index ofan element is its position in the matrix For example consider thefollowing matrix
A =
6 minus1 4
3 5 7
minus3 6 1
Define this in Scilab with the command
A=[ 6 -1 4 3 5 7 -3 6 7]
to display the contents of matrix A in Scilab just type the variablename A To display or use one element from the matrix specify theelement For example A(13) specifies the element in the first rowand third column that is 4 for the above matrix
To change a specific entry just assign a value to it For example
A(31)=-9
changes the value of the third row first column element from -3
to -9
To display or use a whole row or column use the operator Forexample
gt A(2)
ans =
3 5 7
gt A(2)
B4 Arithmetic expressions 235
ans =
-1
5
6
It is also possible to pick out a sub-matrix by the specifying the rowand column entries required For example To pick out the sub-matrix of A containing the elementsA11 A13 A21 A23 the commandA(12123) could be used as shown below
gt A(12123)
ans =
6 4
3 7
Note 123 means from 1 to 3 in steps of 2 This technique alongwith the assignment operator can be used for values of whole rowscolumns or sub-matrices in one command For example
gt A(12123)=[10111213]
A =
10 -1 11
12 5 13
-3 6 7
Note that the elements A11 A13 A21 A23 now contain the values10 11 12 and 13 respectively instead of their previous values
Index notation is logically an unfortunate phenomenon since itclashes with standard function notation On the face of it A(3)
should be the value of a function called A with input 3 but whenit is interpreted as index notation A is a variable However indexnotation is so familiar and useful to those familiar with classicalmathematical notation that many developers of mathematical soft-ware are not prepared to risk a ldquoconsumer backlashrdquo to providinga functional way of extracting elements from an array such as forexample 3 pick A to pick out the third element of A Similarlywe as teachers feel we would be doing students a disservice if wefailed to introduce them to the index notation that abounds inmathematical texts usually in the form of subscripting as in A3Logic does not always win an argument In the commercial worldof computing an established user base carries more weight thanmathematical logic just as in mathematics tradition may well alsofly in the face of logical consistency
B43 Array arithmetic
Scilab has two forms of arithmetic operations that can be usedwith matrices matrix arithmetic or array arithmetic Matrix arith-metic follows the normal rules of matrix algebra Array arithmeticwhich is also called element-by-element arithmetic is where oper-ations are performed on matrices element by element
236 Appendix B Programming with Scilab
Table B4 Scilab array operatorsOperator Meaning Scilab
^ Exponentiation x^y
Multiply xy
Right Division xy
Left Division yx
+ Addition x+y
- Subtraction x-y
Table B4 lists the Scilab array operators
The array addition and subtraction are identical to matrix additionor subtraction Both array and matrix addition and subtractionoperate element by element Array addition and subtraction re-quire that both arrays have the same size The only exception tothis rule in Scilab occurs when we add or subtract a scalar to orfrom an array In this case the scalar is added or subtracted fromeach element in the array
For example
gt [6 5 6] + 3
ans =
[9 8 9]
gt [6 4 5 8] + [3 1 2 4]
ans =
[9 5 7 12]
gt [4 3] - 7
ans =
[-3 -4]
Array multiplication and division is very different to matrix mul-tiplication and division Array multiplication or division is an ele-ment by element multiplication or division that produces an arraywith the same dimensions as the starting arrays As with addi-tion and subtraction array multiplication and division require thatboth arrays have the same dimensions
Example B2 Consider the two row vectors x=[2 4 7 1] andy=[6 1 3 2] Scilab does not recognise the possibility ofmatrix multiplication of these two vectors They are not com-patible the number of columns of the pre-multiplier must bethe same as the number of rows of the post-multiplier Anattempt to execute x y will bring an error message
Obtain the usual dot product of the two row vectors by trans-posing the post-multiplier as in xyrsquo or yxrsquo The matrixproduct xrsquoy produces a matrix of 4 rows and 4 columns
B5 Builtndashin functions 237
Array multiplication produces the result
gt xy
ans =
[12 4 21 2]
by multiplying corresponding elements of each vector Theresult is a row vector with the same dimensions as x and ythat is a 1times 4 vector
Example B3 Consider the row vectors x=[2 4 7 1] and col-umn vector y=[6 1 3 2]rsquo Array multiplication of thesetwo matrices is not possible because their dimensions do notagree
Matrix multiplication xy produces the result 39 that is theinner product or dot product of the vectors Note vectorsas such do not exist in Scilab there are only single rowedor single columned arrays
Scilab not only allows arrays to be raised to powers but alsoallows scalars and arrays to be raised by array powers To performexponentiation on an element-by-element basis use the ^ symbolFor example if x=[3 5 8] then x^3 produces the matrix [33 53
83] = [27 125 512]
A scalar can be raised to an array power For example if p=[2 4 5]then typing 3^p produces the array [32 34 35] = [9 81 243]
With element-by-element array operations it is important to re-member that the dot () is a part of a two-character symbol for aparticular array operator such as () () () (^) Sometimesthis can cause some confusion the dot in 3^p is not a decimalpoint associated with the number 3 The following operations withthe value of p=[2 4 5] are equivalent
3^p
30^p
3^p
(3)^p
3^[2 4 5]
B5 Builtndashin functions
B51 Arithmetic functions
Arithmetic expressions often require computations other than ad-dition subtraction multiplication division and exponentiationFor example many expressions require the use of logarithms ex-ponentials and trigonometric functions Scilab has predefined alarge number of functions that perform these types of computations(see Tables B5 and B6)
238 Appendix B Programming with Scilab
An example of a predefined function is sin which returns the sineof a given angle
m = sin(angle)
where the variable angle contains the angle in radians and theresult is stored in the variable m The variable angle is called theargument or parameter of the function If the angle is stored indegrees it must be converted to radians for example
m = sin(anglepi180)
or
radians =anglepi180
m = sin(radians)
In Scilab the arguments or parameters of the function are con-tained in parentheses following the name of the function This isreferred to as pre-fix notation Many common functions in math-ematics have a syntax using in-fix notation where two argumentsappear to the right and left of the function name as for examplein x+y atimes b nmod r etc Scilab does not support in-fix nota-tion for any but the arithmetic functions Classical mathematicalnotation also has examples of post-fix notationmdash n x squared mdashand other hybrids like |x| while Scilab has ahb to confound thepurist
A function may contain no arguments one argument or many ar-guments depending on its definition If a function contains morethan one argument it is vital to give the arguments in the cor-rect order For example consider the problem of finding θ in thefollowing triangle using the atan command
5
8θ
Now by hand calculation the answer is 05586 radians If we useatan as below we get the answers shown
--gt atan(58)
ans =
05586
--gt atan(85)
ans =
10122
The first answer is the correct one The second answer is incorrect
B5 Builtndashin functions 239
Table B5 Some common mathematical functions defined inScilabFunction Description
abs(x) Computes the absolute value of xsqrt(x) Square root of x The parameter can be nega-
tiveround(x) Round x to the nearest integerfix(x) Round x to the nearest integer toward zerofloor(x) Round x to the nearest integer toward minusinfinceil(x) Round x to the nearest integer toward +infinsign(x) Returns -1 if x is less than zero zero if x equals
zero and 1 otherwisemodulo(xy) Returns the remainder of xy For example
modulo(254) is 1exp(x) Returns ex
log(x) Returns the natural logarithm of x That isthe logarithm to the base e Parameter can benegative but cannot be zero
log10(x) Returns the common logarithm of x That isthe logarithm to the base 10 Parameter can benegative but cannot be zero
since we passed the values to the function in the incorrect order
Some functions also require that the arguments be in specific unitsFor example the trigonometric functions assume that the argu-ments are in radians
In Scilab some functions use the number of arguments to deter-mine the output of the function For example the zeros functioncan have one or two arguments which then determine the output(check this with the command help zeros)
A function reference cannot be made on the left hand side of anassignment since a function call always returns a value(s) whereasan assignment is always to a variable Functions can of courseappear on the right side of an assignment since they return valuesto be assigned to some variable Functions may be applied tothe output of other functions and may be called in the algorithmdefining a function for example
logx = log(abs(x))
When one function is used to compute the argument of anotherfunction be sure to enclose the argument of each function in itsown set of parentheses
Scilab allows recursive functions that is functions that call them-selves in their defining algorithm
Exercise B4 Use examples to show the differences between the
240 Appendix B Programming with Scilab
Table B6 Some common trigonometric functions defined inScilab
Function Description
sin(x) Sine of x where x is in radianscos(x) Cosine of x where x is in radianstan(x) Tangent of x where x is in radiansasin(x) Arcsine or inverse sine of x where x must be
between -1 and 1 The return angle is in radiansbetween minusπ2 and π2
acos(x) Arccosine or inverse cosine of x where x mustbe between minus1 and 1 The return angle is inradians between 0 and π
atan(x) Arctangent or inverse tangent of x The returnangle is in radians between minusπ2 and π2
atan(xy) Arctangent or inverse tangent of xy The re-turn angle is in radians between minusπ and π de-pending on the sign of x and y
sec(x) Secant (1(cos x)) of x where x is in radianscsc(x) Cosecant (1(sin x)) of x where x is in radianscotg(x) Cotangent (1(tan x)) of x where x is in radians
Table B7 Some of Scilabrsquos utility functionsFunction Description
clear Clear variables from memoryhelp Display the first set of comment lines from
scriptswho List the variables in memorywhos List the variables in memory with their sizeerror Terminates the execution of the Sci-file and out-
puts the supplied error message
functions round(x) fix(x) floor(x) and ceil(x)
Exercise B5 The functions listed in Tables B5 B6 are only asmall sample of all of Scilabrsquos mathematical functions
Using Scilabrsquos help command list all of the hyperbolic func-tions available Also list any restrictions on the values of thefunction arguments
B52 Utility functions
In addition to the arithmetic functions Scilab has several utilityfunctions (see Table B7) These functions allow users to trackvariables (whoswho) feedback an error message if an error occursin the code (error) clear the variables form the memory (clear)etc Essential these functions allow users to debug and performbookkeeping operations inside their code
B6 Visualising data in Scilab 241
6
5
4- 04- 8
3
- 02
- 6
00
2- 4 Y- 2
02
Z
10
04
X 02
06
4 - 1
08
6- 28
10
Figure B2 Plot of z = sin(radicx2 + y2)
radicx2 + y2 illustrating the
3d capabilities of Scilab
B6 Visualising data in Scilab
One of the big advantages in using Scilab is the ability to visualisedata and results of complex calculations (see Figure B2 as exampleof the 3d capabilities) While Scilab is capability of complex 3dgraphics (see help pages for mesh and surf) in this section we willmesh amp surf
focus on the 2d plotting command (plot) Hence in this sectionplot
we will focus on
1 plotting mathematical functions from within Scilab
2 changing stylistic elements of the plot
3 adding titles labels and legends to the plot
4 including mathematical notation in titles labels and legendsand
5 plotting data stored in an external ASCII file
B61 Plotting mathematical functions from within Scilab
To plot a mathematical function in Scilab three steps are requiredThese are
1 Define the domain of the function (eg x=linspace(02pi))
2 Calculate the values of function at the selected points in thedomain (eg y=sin(x))
242 Appendix B Programming with Scilab
3 Final step is to plot the function using the plot commandplot
(eg plot(xy))
Using these steps it is possible to produce a basic plot of any math-ematical function (see Figure B3) Note that the plot shown inFigure B3 does not have a title axes labels or a legend all ofwhich are crucial for a good plot The Scilab commands to addthese elements to the plot are given in the following sections
- 10
- 08
- 06
- 04
- 02
00
02
04
06
08
10
0 1 2 3 4 5 6 7
Figure B3 The default plot of y = sin(x) in Scilab Notethat by default axes are not labelled and there is no title orlegend These need to added using the additional commandsxlabelylabeltitle and legend
Changing stylistic elements of the plot
An optional argument can be given to the plot command to cus-tomise each new line aspect This argument is a concatenatedstring containing information about colour line style or markersbased on the specifiers shown in Table B8 To specify the plotbe drawn with a red longdash-dot line with diamond marker thestring that would be passed to plot command would be rsquor-drsquo Thestring which is a concatenation (in any order) of the three typesof properties shown in Table B8 must be unambiguous Further-more the string specification is not case sensitive An exampleof specifying various line styles is shown in Example B6 and Fig-ure B4
Example B6 The following commands illustrate how line stylescan be used to distinguish between the plots of sin(2πt) sin(2πt+π6) and sin(2πt+ π2)
t=linspace(0130)
B6 Visualising data in Scilab 243
Table B8 Plot specifiers required to change line styles coloursand markers in plotsSpecifier Line style Specifier Colour Specifier Marker stylendash Solid line (default) r Red + Plus signndash ndash Dashed line g Green o Circle Dotted line b Blue Asteriskndash Dash-dotted line c Cyan x Cross
m Magenta s Squarey Yellow rsquononersquo No marker(default)k Blackw White
Figure B4 Resulting plotting after issuing the commands givenin Example B6
244 Appendix B Programming with Scilab
Example B6 illustrate a number of key commands relating to thecreation of multiple plots in Scilab The mtlb hold4 commandmtlb hold
holds the current plot and axis properties so that all subsequentplots are added to the existing plot
To distinguish each plot a different line specification is passed eachplot command In the example y = sin(2πt) is plotted using a redsolid line with data points indicated using a red circle as indicatedby the string rsquor-orsquo y = sin(2πt + π6) is plotted using greendashed-dotted line with data points marked by a square (rsquog-srsquo)while y = sin(2πt+ π2) is plotted using a cyan dashed line usingdiamonds to mark the data points (rsquoc--drsquo) Finally to completethe plot axes labels and a legend has been added to plot the com-mands for doing this are discussed in sectB62
B62 Adding titleslabels and legends to the plot
All good plots should have the following properties
1 The style of figure is appropriate to the information beingconveyed
2 The contents are well organised so it is easy for readers tointerpret the information
3 Legends labels etc are clear and concise (eg contain unitsif required do not obstruct parts of the plot etc) and
4 A caption or title that summarises the information presented
Like most mathematical applications Scilab has a wide variety ofdifferent plotting styles that can be used (eg look up the help forbar polarplot etc) This allows you to select the most appropri-barpolarplot
ate style to display and organise your data
The Scilab commands xlabel ylabel title and legend al-xlabel ylabel title and
legend lows you to add axes labels titles and legends to plot As shownin Example B6 the commands xlabel ylabel and title takea string input containing the title or label text For the legend
command a string is required for each separate plot Note theorder of the strings in the legend command reflects the order inwhich the each of plots were made
Including mathematical notation in titles labels and legends
It is possible to include mathematical characters such as π inScilab plots This is done by inserting special commands intoeither the title legend or label string These commands are basedon the abbreviations for mathematical notation used in programssuch as LATEX MathType etc A brief summary of the commands
4 mtlb hold on is equivalent to the Scilab commandset(gca()auto clearoff) while mtlb hold off is equivalentto the Scilab command set(gca()auto clearon)
B6 Visualising data in Scilab 245
Table B9 Basic mathematical symbols available in Scilab forincluding mathematical notation in titles labels and legendscommand symbol command symbol command symbolalpha α upsilon υ sim sim
Im = cup cup wp weierpotimes otimes subseteq sube oslash cap cap in isin supseteq supesupset sup lceil d subset subint
intcdot middot o oslash
rfloor c neg 6= nabla nablalfloor b times times ldots perp perp surd
radicprime prime
wedge and varpi $ O Oslashrceil e rangle 〉 mid |
vee or copyright ccopy langle 〈
available in Scilab for typesetting mathematical notation is givenin Table B9 To insert these commands into a text string simplystart and end the string with $ signs as shown in Example B7
Example B7 Scilab commands to produce the labelled plot ofy = sin
There are a few key concepts to note about the Example B7 Thefirst can be seen in the string being passed to the title commandtitle
In this string sin is prefixed with a backslash () This is doneso that the letters for sin are not typeset in italic which shouldonly be used for variables and not common mathematical func-tions Table B10 shows the command sequences to use for themore common mathematical functions5 To get the fraction in thetitle the command fracab has been used to produce a
b Ifsuperscripts and subscripts are required in a title or label these aresuperscripts amp subscripts
obtained by adding the subscript character ( ) and the superscriptcharacter (^) to the string These characters modify the characteror substring defined in braces immediately following For examplex 2 gives x2 while e^2x gives e2x
Note in Example B7 the commands xlabel and ylabel has beenxlabelylabel
used to set the axes labels in Figure B5 Instead of using thesethree commands the title and the axes labels could have been setusing the single xtitle command (see help xtitle for details)xtitle
In the example we have also customised the font size to make thetext more readable by a) passing an option to the title xlabelylabel commands and b) by using the command xset to set thexset
default font size before executing the xstring command Finallythe xstring command adds the text string to plot at the locationxstring
specified by variables b and c In the example the coordinates wewant to add the text are obtained by executing the xclick functionxclick
and clicking on the point required in the plot window
5 Additional details can be found in books covering Mathematics in LATEX
B6 Visualising data in Scilab 247
Table B10 Command sequences for the common mathematicalfunctions in Scilabarccos arcsin arctan cos cot
csc exp lim ln log
max min sec sin tan
B63 Plotting numeric data stored in an ASCII file
As data can be stored in a wide variety of ways in external filesScilab has a wide variety of functions which can be used to importthe data stored in these files (see help for the commands read
fscanf and xlsread) Since data can be stored in such a wideread fscanf and xlsread
range of formats we will confine our discussions here to the situa-tion where numeric data is stored in a ASCII (text) file where eachrow of the file contains the same number of entries
To illustrate the procedure for plotting data stored in ASCII file wewill consider the data obtained measuring the temperature of a cupof coffee every minute for a total of 100 mins The data recorded isstored in the a file called coffeedatatxt6 which simply containsthe two columns of numbers shown in Table B11 As an examplethe first 10 lines of the text file contains the following numbers7
00000 1000000
10000 950000
20000 903125
30000 859180
40000 817981
50000 779357
60000 743147
70000 709201
80000 677376
90000 647540
Note the current version of Scilab has a builtin text editor whichcan be used to view such text files To access this editor simplytype the command edit at the command prompt Once open youedit
can use the graphical interface to open the required file
The first issue is to get the data into Scilab so it is available forplot command In Scilab the command read can used to importread
the data and assign it to a variable In the current example wewould use the following command to import the data and assignit to a variable called cdata assuming the data is in the currentdirectory8
6 The file coffeedatatxt has been attached and should be available for savingprovided you are using Adobe Reader To save the file right click on thepurple Windows version or the red MacOSLinux version in the margin andselect save file from the options Ensure you save the file as coffeedatatxt
7 normally you will have received the data file in this format8 You can check what directory Scilab is in by typing the command pwd
Figure B6 Temperature of coffee based on the data recorded inthe file coffeedatatxt
cdata=read(rsquocoffeedatatxtrsquo-12)
The -12 following the name of the file tells Scilab that you donrsquotknow how many rows you have but you have two columns of dataSince we know that we have 101 rows of data we could have alsoused the following command to load the data
cdata=read(rsquocoffeedatatxtrsquo1012)
We now have a variable called cdata which has two columns ofdata The first column contains the time while the second containsthe Temperature records Hence we can plot it using the followingcommand
plot(cdata(1)cdata(2))
This gives the plot shown in Figure B6 after appropriate labelshave been added as described above
B7 Script files
Executing commands entered at the keyboard or keypad is themost primitive use of a calculating device It is only when se-quences of operations (to perform complex tasks) are stored awayand re-called for repeated application with different inputs that wehave a programmed operation Complex programs must necessarilyhave an interface with the user The basic interface tool in Scilabcapable of executing sequences of commands that are stored in a fileis the script The file names must have an extension of sce forexample program1sce assignment1sce These files are calledScript Sci-files or simply scripts
A script file is a file of standard keyboard charactersthat define commands to drive a program A script filecan be generated using any text editor or word proces-sor
To invoke a script in Scilab type the commandexec
exec filenamesce
where filename denotes the actual name of the file containing thescript When the script runs it is exactly the same as if you typedthe commands into the command window from the keyboard Thismeans that the Sci-file can use any variables you have defined priorto invoking the script Any variables created in the script are alsoavailable when the script is finished and any variable whose namewas assigned a value in the script will be over-written
If you give a script the same name as any variable that it computesin the running of the script then Scilab will give precedence tothe workspace variable over a script As stated in sectB3 the variablename will hide the script name after the script has run Scilab willtake the name as referring to the variable and not to the definedscript
Scilab also needs to know where to find the script to executedIf Scilab cannot find the script it will report that you have anundefined function or variable
The exact procedure for creating and saving Sci-files depends onthe Scilab version the operating system (MS Windows Macin-tosh or UnixLinux ) and the text editor you use The currentversion of Scilab includes a built-in editor which can be used tocreate and edit Sci-files and text files
To create a new Sci-file under the current version of Scilab simplytype edit newfilenamesce This will open up the Scilab editoredit
with a blank document called newfilename
B71 Script input and output
A script is a named file (with the extension sce) containing asequence of Scilab instructions which when its name is executedin a session performs those instructions in order as though theywere entered at the console
There are two common uses for scripts Pragmatically we maywant to save a particular computational sequence for later use orreference If the tasks involved are to be repeated often thenthe nature of the tasks should be analysed formulated into usefulfunctions (which we will discuss later) and the whole procedurepresented in a single function But if the tasks are only likely toget occasional use then the script may be adequately commentedand stored as a script file It is a quick and dirty way of saving
B7 Script files 251
the instructions for doing a particular job when we do not wantto go to the trouble of writing the functions for the job It is nota highly recommended modus operandi but useful all the same
The second use is for the script to act like a storyboard presentinginformation to a user in a structured and controlled way permit-ting the user to see the steps of a computational procedure andtheir outputs up on the screen At a more sophisticated level thescript takes on the proportions of an application controlling a largeand complex interaction with advanced interface features We willnot take our study of Scilab programming to this level but workmostly at the storyboard level using the simplest of Scilabrsquos in-terface tools
All the variables created in a script become global variables definedin the workspace and available for perusal As an application itis assumed that a script controls the entire workspace and thevariables it creates are lsquothe datarsquo of the application When scriptsare used as storyboards on a small scale then it cannot be assumedthat the variables they create deserve global recognition and it iscourteous to erase these variables when the script is finished unlessthey have some possible further use An unfortunate side-effect isthat if variables with the same name already exist in the workspacebefore a script is run then these variables will be over-written andlost
Used as storyboards scripts allow the user to cut paste and re-trythe various steps of a procedure and to modify the script to suithis own needs Storyboards may be used for demonstrations oftoolbox utilities or extended documentationexplanation of func-tions However if there is any danger that variables defined bythe script may over-write workspace variables then the demo ordocumentation should be re-fashioned into a function that leavesthe workspace variables untouched (more on this later)
Scripts will often read data from pre-prepared files and we will dis-cuss the functions provided by Scilab to perform this task below
Gathering data entered by the user really falls into the realm ofinterface design but we only briefly consider the simplest interfacefunction provided by Scilab Similarly we briefly consider thefunctions for providing the simplest forms of output presentation
Keyboard input
Scilab has a built-in function input that can be used to promptinput
the user to enter the value of a variable from the keyboard Forexample
Tc = input(rsquoPlease enter the temperature in degrees Celsiusrsquo)
will print the message
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
B3 Variables 229
able contains the value of last unassigned (see sectB3 for details onhow variables are assigned) evaluation performed Hence the valuestored in ans changes each time an unassigned calculation is per-formed in Scilab as illustrated in Example B1 Consequentlyif you intend to reuse values of calculations you should store theresults in a variable (see sectB3) to avoid the possible of using theincorrect result
Activity BA To familiarise yourself with some of the basic features of rarrScilab and how to use it for elementary arithmetic andas a sophisticated calculator try some of the examples inExample B1 Make sure you understand what Scilabdid to calculate the answer Also as you work throughthe appendix attempt all the examples and exercises asyou come to them
B3 Variables
Variables are used in a programming language as tags to memorylocations in the computer The data stored in the memory locationis accessed using the variable name in the program The variableis not attached to the data but the memory location This meansthat the value contained within a variable can change
In languages such as C every variable must be declared before it isused specifying both its name and its type The name and type ofvariables do not need to be pre-declared in Scilab In Scilab anyvariable can take integer real complex scalar string or matrixvalues Languages that pre-declare variables have clearly statedat the start of a program or function the type of every variableused The Scilab interpreterrsquos liberal approach to variable typeshas both advantages and disadvantages
The equality sign (=) is used to assign the values to a variableassigning values to
variables This is an unfortunate choice of symbol and means that when wewant to test whether two quantities are equal or not we cannot usethe equal symbol = but must use the more cumbersome == Thus5+2 == 3+4 is true but 5+2 = 3+4 is a syntax error In Scilabprograms the equals sign does not have its mathematical meaningIt is an assignment operator The statement to the right of theequals sign is evaluated and the result is placed in the memorylocation tagged by the variable name on the left For example thestatement x = 3 + 5 means Scilab evaluates the statement onthe right and stores the result in the memory location tagged bythe variable x This means that the statement 3+ 5 = x whichis equivalent to x= 3 + 5 in mathematics will lead to an errormessage in Scilab
The following are examples of variable types in Scilab (it is notcomplete but contains all the variable types that you will encounter
230 Appendix B Programming with Scilab
in this course)
x = 3 Assign the scalar integer value 3 to the variable x
x = 7297e17 Assign the scalar real value 7297 times 1017 to the variable xThe number has been defined using the scientific exponentnotation The number following the e is the power of 10 tomultiply the number in front of the e Either e or E can beused
x = rsquoNorthrsquo Assign the string North to the variable x Strings of char-acters are defined by surrounding the characters with apos-trophes The construct rsquoSouthrsquo defines a constant stringcontaining the characters South
x = 05100 Assign the row vector containing 21 elements from 0 to 100in steps of 5 to the variable x
x = [12 13 25 67] Assign the 2times 2 matrix to the variable x so that
x =
[12 13
25 67
]
x = 51+i63 Assign the complex number 51+ i63 to the variable x
Variable names can be of any length but must be unique withinthe first 31 characters Variable names in Scilab are case sensi-tive that is the variables time and Time are distinct Howeverfunctions and scripts (which we discuss later) are stored in sepa-rate files (with extension sce) outside the interpreter and may ormay not be case sensitive depending on the outside environment(in a dos environment for example they will not be)
Scilab allows overloading of names This means that if you choosea name that has been predefined by Scilab you loose the builtin meaning For example cos = 634 defines a scalar variablecos but it is also the same name for the Scilab cosine functionThe variable assignment means that the cosine function will nolonger be accessible in this session All references to the name coswill assume the variable not the function Take care not to usevariable names that might disable built in functions or constantsthat you may need later When writing scripts for others to usethe names of all variables that will be over-written by the runningof the script must be listed for easy reference (and one shouldread these references before running any script) This is one ofthe reasons why it is nearly always preferable to write functionsrather than scripts Variables defined in the course of runninga functionrsquos algorithm remain local to the function and do notover-write workspace variables
The results of all calculations should be assigned to a variable sothat this value can be used later If this is not done the result
B3 Variables 231
Table B1 Special variables and constants in ScilabName Description
pi The value of π to the precession of the machineis returned
i Recognised as theradicminus1
inf This is Scilabrsquos representation of infin This is arecognised value in a variable and typically ap-pears as a result of division by zero or logarithmof zero
nan Stands for Not a Number This is a recognisedvalue in a variable and appears as a result ofa mathematically undefined operation such asdivision of zero by zero or infinminusinfin
eps Floating point relative accuracy That is epsreturns the smallest difference resolvable be-tween two floating point numbers For examplethe next largest representable number beyond10 is 10+eps Scilab cannot represent anyfloating point number that falls between 10 and10+eps
ans This variable contains the last computed expres-sion that was not stored in a variable
date This variable contains a string representation oftodayrsquos date in the format dd-mmm-yyyy Forexample 01-Apr-2000
232 Appendix B Programming with Scilab
Table B2 Scilab Arithmetic OperatorsOperator Meaning Scilab
^ Exponentiation xy x^y
Multiply xtimes y xy
Right Division xy = xy xy
Left Division yx = xy yx
+ Addition x+ y x+y
- Subtraction xminus y x-y
of a calculation will be stored in a special variable called ans (seeans
Table B1) The value stored in this variable is overridden the nexttime a calculation is done that is not assigned to a variable Thismeans that the result from the previous calculation is then lost
B4 Arithmetic expressions
Table B2 lists the basic arithmetic operators to be found in ScilabWith the exception of the left division operator the arithmetic op-erators in Scilab follow standard notation
B41 Precedence
When more than one operation appears in a mathematical state-ment the programming language needs to define in which order theoperations must be performed or the precedence That is whichparts are evaluated first
If precedence was not defined then the same statement can be eval-uated to a different result for example is 186+3 equal to 2 oris 186+3 equal to 6 The first statement is evaluated from theright using the precedence rule that usually applies in mathemat-ical statements that is work from right to left as for example inlsquosin log tan xrsquo where we start with a value of x compute the tan-gent of that value (tan x) then take the logarithm of the resultingnumber (log(tan x)) and finally take the sine of what this deliv-ers However it is usual (because it is much easier to write poly-nomials with this convention) to give multiplication and divisionprecedence over addition and subtraction The second statementrecognises the precedence of the division over addition and evalu-ates it first Right to left precedence has a much simpler parsingtree than schemes which give precedence to particular operatorsbut the common arithmetic precedences are so widely acceptedand expected that most computer language designers will not riskthe possible lsquoconsumer backlashrsquo that breaking with this traditionwould involve So with every computer language one must searchout and learn the precedence rules that it adopts The usual arith-metic precedences will normally be observed but note that even
B4 Arithmetic expressions 233
Table B3 Precedence of arithmetic expressionsOperator Precedence Order of evaluation
() 1 Left to right innermost first 2 Left to right 3 Left to right 4 Right to left
+ minus 5 Left to right
they are not unambiguous For example which divide has prece-dence in 41632 Working right to left one would get the answerof 8 but Scilab returns (the decimal for) 1128 mdashit works fromleft to right Table B3 lists the operators and their precedencevalue
The rules for evaluation of operators are as follows
bull Operators having high precedence (low numerical value) willbe evaluated before those having a low precedence (high nu-merical value)
bull If an expression contains more than one operator at the samelevel of precedence then the left-to-right rule of evaluationapply
Parentheses () always have the highest precedence Whateveris within the parentheses is evaluated first This means when indoubt about the order of any statement place parentheses aroundthe statement to ensure that the operations are performed in theorder you expect
B42 Matrix arithmetic
One of the strengths of Scilab is its ability to execute matrixarithmetic3 Matrix arithmetic must be implemented in other lan-guages with many lines of code or special functions Scilab isespecially designed to handle matrix arithmetic The representsmatrix multiplication or scalar multiplication of a matrix if one ofthe arguments is a scalar Since scalars are 1times1 matrices ordinaryscalar multiplication can also be performed by the operator Thedot product of two vectors however requires that the first vectorbe a row vector and the second a column vector To multiply thecorresponding elements in two vectors together requires a differentoperator so that
gt a = [2 3 4] b = [6 7 8]
a b
ans =
12 21 32
3 Remember vectors are examples of matrices A column vector of length nis an example of a ntimes 1 matrix
234 Appendix B Programming with Scilab
However because it is a matrix operation
gt a + b
ans =
8 10 40
The standard arithmetic operators described in Table B2 are alsoused with matrices The normal rules of matrix addition subtrac-tion and multiplication apply
The left and right division of matrices are also generalisations ofscalar division Right division BA is roughly the same as Binv(A)Left division AB is roughly the same as inv(A)B The differenceis in how the division is performed Different algorithms are usedwhen A or B are square or not
Scilab also defines a transpose operator for matrices The trans-pose of the matrix A is Arsquo For example if
A =
[25 63
12 09
] then A prime =
[25 12
63 09
]
Accessing matrix elements
Individual elements of a matrix can accessed and used in state-ments by specifying the matrix index of an element The index ofan element is its position in the matrix For example consider thefollowing matrix
A =
6 minus1 4
3 5 7
minus3 6 1
Define this in Scilab with the command
A=[ 6 -1 4 3 5 7 -3 6 7]
to display the contents of matrix A in Scilab just type the variablename A To display or use one element from the matrix specify theelement For example A(13) specifies the element in the first rowand third column that is 4 for the above matrix
To change a specific entry just assign a value to it For example
A(31)=-9
changes the value of the third row first column element from -3
to -9
To display or use a whole row or column use the operator Forexample
gt A(2)
ans =
3 5 7
gt A(2)
B4 Arithmetic expressions 235
ans =
-1
5
6
It is also possible to pick out a sub-matrix by the specifying the rowand column entries required For example To pick out the sub-matrix of A containing the elementsA11 A13 A21 A23 the commandA(12123) could be used as shown below
gt A(12123)
ans =
6 4
3 7
Note 123 means from 1 to 3 in steps of 2 This technique alongwith the assignment operator can be used for values of whole rowscolumns or sub-matrices in one command For example
gt A(12123)=[10111213]
A =
10 -1 11
12 5 13
-3 6 7
Note that the elements A11 A13 A21 A23 now contain the values10 11 12 and 13 respectively instead of their previous values
Index notation is logically an unfortunate phenomenon since itclashes with standard function notation On the face of it A(3)
should be the value of a function called A with input 3 but whenit is interpreted as index notation A is a variable However indexnotation is so familiar and useful to those familiar with classicalmathematical notation that many developers of mathematical soft-ware are not prepared to risk a ldquoconsumer backlashrdquo to providinga functional way of extracting elements from an array such as forexample 3 pick A to pick out the third element of A Similarlywe as teachers feel we would be doing students a disservice if wefailed to introduce them to the index notation that abounds inmathematical texts usually in the form of subscripting as in A3Logic does not always win an argument In the commercial worldof computing an established user base carries more weight thanmathematical logic just as in mathematics tradition may well alsofly in the face of logical consistency
B43 Array arithmetic
Scilab has two forms of arithmetic operations that can be usedwith matrices matrix arithmetic or array arithmetic Matrix arith-metic follows the normal rules of matrix algebra Array arithmeticwhich is also called element-by-element arithmetic is where oper-ations are performed on matrices element by element
236 Appendix B Programming with Scilab
Table B4 Scilab array operatorsOperator Meaning Scilab
^ Exponentiation x^y
Multiply xy
Right Division xy
Left Division yx
+ Addition x+y
- Subtraction x-y
Table B4 lists the Scilab array operators
The array addition and subtraction are identical to matrix additionor subtraction Both array and matrix addition and subtractionoperate element by element Array addition and subtraction re-quire that both arrays have the same size The only exception tothis rule in Scilab occurs when we add or subtract a scalar to orfrom an array In this case the scalar is added or subtracted fromeach element in the array
For example
gt [6 5 6] + 3
ans =
[9 8 9]
gt [6 4 5 8] + [3 1 2 4]
ans =
[9 5 7 12]
gt [4 3] - 7
ans =
[-3 -4]
Array multiplication and division is very different to matrix mul-tiplication and division Array multiplication or division is an ele-ment by element multiplication or division that produces an arraywith the same dimensions as the starting arrays As with addi-tion and subtraction array multiplication and division require thatboth arrays have the same dimensions
Example B2 Consider the two row vectors x=[2 4 7 1] andy=[6 1 3 2] Scilab does not recognise the possibility ofmatrix multiplication of these two vectors They are not com-patible the number of columns of the pre-multiplier must bethe same as the number of rows of the post-multiplier Anattempt to execute x y will bring an error message
Obtain the usual dot product of the two row vectors by trans-posing the post-multiplier as in xyrsquo or yxrsquo The matrixproduct xrsquoy produces a matrix of 4 rows and 4 columns
B5 Builtndashin functions 237
Array multiplication produces the result
gt xy
ans =
[12 4 21 2]
by multiplying corresponding elements of each vector Theresult is a row vector with the same dimensions as x and ythat is a 1times 4 vector
Example B3 Consider the row vectors x=[2 4 7 1] and col-umn vector y=[6 1 3 2]rsquo Array multiplication of thesetwo matrices is not possible because their dimensions do notagree
Matrix multiplication xy produces the result 39 that is theinner product or dot product of the vectors Note vectorsas such do not exist in Scilab there are only single rowedor single columned arrays
Scilab not only allows arrays to be raised to powers but alsoallows scalars and arrays to be raised by array powers To performexponentiation on an element-by-element basis use the ^ symbolFor example if x=[3 5 8] then x^3 produces the matrix [33 53
83] = [27 125 512]
A scalar can be raised to an array power For example if p=[2 4 5]then typing 3^p produces the array [32 34 35] = [9 81 243]
With element-by-element array operations it is important to re-member that the dot () is a part of a two-character symbol for aparticular array operator such as () () () (^) Sometimesthis can cause some confusion the dot in 3^p is not a decimalpoint associated with the number 3 The following operations withthe value of p=[2 4 5] are equivalent
3^p
30^p
3^p
(3)^p
3^[2 4 5]
B5 Builtndashin functions
B51 Arithmetic functions
Arithmetic expressions often require computations other than ad-dition subtraction multiplication division and exponentiationFor example many expressions require the use of logarithms ex-ponentials and trigonometric functions Scilab has predefined alarge number of functions that perform these types of computations(see Tables B5 and B6)
238 Appendix B Programming with Scilab
An example of a predefined function is sin which returns the sineof a given angle
m = sin(angle)
where the variable angle contains the angle in radians and theresult is stored in the variable m The variable angle is called theargument or parameter of the function If the angle is stored indegrees it must be converted to radians for example
m = sin(anglepi180)
or
radians =anglepi180
m = sin(radians)
In Scilab the arguments or parameters of the function are con-tained in parentheses following the name of the function This isreferred to as pre-fix notation Many common functions in math-ematics have a syntax using in-fix notation where two argumentsappear to the right and left of the function name as for examplein x+y atimes b nmod r etc Scilab does not support in-fix nota-tion for any but the arithmetic functions Classical mathematicalnotation also has examples of post-fix notationmdash n x squared mdashand other hybrids like |x| while Scilab has ahb to confound thepurist
A function may contain no arguments one argument or many ar-guments depending on its definition If a function contains morethan one argument it is vital to give the arguments in the cor-rect order For example consider the problem of finding θ in thefollowing triangle using the atan command
5
8θ
Now by hand calculation the answer is 05586 radians If we useatan as below we get the answers shown
--gt atan(58)
ans =
05586
--gt atan(85)
ans =
10122
The first answer is the correct one The second answer is incorrect
B5 Builtndashin functions 239
Table B5 Some common mathematical functions defined inScilabFunction Description
abs(x) Computes the absolute value of xsqrt(x) Square root of x The parameter can be nega-
tiveround(x) Round x to the nearest integerfix(x) Round x to the nearest integer toward zerofloor(x) Round x to the nearest integer toward minusinfinceil(x) Round x to the nearest integer toward +infinsign(x) Returns -1 if x is less than zero zero if x equals
zero and 1 otherwisemodulo(xy) Returns the remainder of xy For example
modulo(254) is 1exp(x) Returns ex
log(x) Returns the natural logarithm of x That isthe logarithm to the base e Parameter can benegative but cannot be zero
log10(x) Returns the common logarithm of x That isthe logarithm to the base 10 Parameter can benegative but cannot be zero
since we passed the values to the function in the incorrect order
Some functions also require that the arguments be in specific unitsFor example the trigonometric functions assume that the argu-ments are in radians
In Scilab some functions use the number of arguments to deter-mine the output of the function For example the zeros functioncan have one or two arguments which then determine the output(check this with the command help zeros)
A function reference cannot be made on the left hand side of anassignment since a function call always returns a value(s) whereasan assignment is always to a variable Functions can of courseappear on the right side of an assignment since they return valuesto be assigned to some variable Functions may be applied tothe output of other functions and may be called in the algorithmdefining a function for example
logx = log(abs(x))
When one function is used to compute the argument of anotherfunction be sure to enclose the argument of each function in itsown set of parentheses
Scilab allows recursive functions that is functions that call them-selves in their defining algorithm
Exercise B4 Use examples to show the differences between the
240 Appendix B Programming with Scilab
Table B6 Some common trigonometric functions defined inScilab
Function Description
sin(x) Sine of x where x is in radianscos(x) Cosine of x where x is in radianstan(x) Tangent of x where x is in radiansasin(x) Arcsine or inverse sine of x where x must be
between -1 and 1 The return angle is in radiansbetween minusπ2 and π2
acos(x) Arccosine or inverse cosine of x where x mustbe between minus1 and 1 The return angle is inradians between 0 and π
atan(x) Arctangent or inverse tangent of x The returnangle is in radians between minusπ2 and π2
atan(xy) Arctangent or inverse tangent of xy The re-turn angle is in radians between minusπ and π de-pending on the sign of x and y
sec(x) Secant (1(cos x)) of x where x is in radianscsc(x) Cosecant (1(sin x)) of x where x is in radianscotg(x) Cotangent (1(tan x)) of x where x is in radians
Table B7 Some of Scilabrsquos utility functionsFunction Description
clear Clear variables from memoryhelp Display the first set of comment lines from
scriptswho List the variables in memorywhos List the variables in memory with their sizeerror Terminates the execution of the Sci-file and out-
puts the supplied error message
functions round(x) fix(x) floor(x) and ceil(x)
Exercise B5 The functions listed in Tables B5 B6 are only asmall sample of all of Scilabrsquos mathematical functions
Using Scilabrsquos help command list all of the hyperbolic func-tions available Also list any restrictions on the values of thefunction arguments
B52 Utility functions
In addition to the arithmetic functions Scilab has several utilityfunctions (see Table B7) These functions allow users to trackvariables (whoswho) feedback an error message if an error occursin the code (error) clear the variables form the memory (clear)etc Essential these functions allow users to debug and performbookkeeping operations inside their code
B6 Visualising data in Scilab 241
6
5
4- 04- 8
3
- 02
- 6
00
2- 4 Y- 2
02
Z
10
04
X 02
06
4 - 1
08
6- 28
10
Figure B2 Plot of z = sin(radicx2 + y2)
radicx2 + y2 illustrating the
3d capabilities of Scilab
B6 Visualising data in Scilab
One of the big advantages in using Scilab is the ability to visualisedata and results of complex calculations (see Figure B2 as exampleof the 3d capabilities) While Scilab is capability of complex 3dgraphics (see help pages for mesh and surf) in this section we willmesh amp surf
focus on the 2d plotting command (plot) Hence in this sectionplot
we will focus on
1 plotting mathematical functions from within Scilab
2 changing stylistic elements of the plot
3 adding titles labels and legends to the plot
4 including mathematical notation in titles labels and legendsand
5 plotting data stored in an external ASCII file
B61 Plotting mathematical functions from within Scilab
To plot a mathematical function in Scilab three steps are requiredThese are
1 Define the domain of the function (eg x=linspace(02pi))
2 Calculate the values of function at the selected points in thedomain (eg y=sin(x))
242 Appendix B Programming with Scilab
3 Final step is to plot the function using the plot commandplot
(eg plot(xy))
Using these steps it is possible to produce a basic plot of any math-ematical function (see Figure B3) Note that the plot shown inFigure B3 does not have a title axes labels or a legend all ofwhich are crucial for a good plot The Scilab commands to addthese elements to the plot are given in the following sections
- 10
- 08
- 06
- 04
- 02
00
02
04
06
08
10
0 1 2 3 4 5 6 7
Figure B3 The default plot of y = sin(x) in Scilab Notethat by default axes are not labelled and there is no title orlegend These need to added using the additional commandsxlabelylabeltitle and legend
Changing stylistic elements of the plot
An optional argument can be given to the plot command to cus-tomise each new line aspect This argument is a concatenatedstring containing information about colour line style or markersbased on the specifiers shown in Table B8 To specify the plotbe drawn with a red longdash-dot line with diamond marker thestring that would be passed to plot command would be rsquor-drsquo Thestring which is a concatenation (in any order) of the three typesof properties shown in Table B8 must be unambiguous Further-more the string specification is not case sensitive An exampleof specifying various line styles is shown in Example B6 and Fig-ure B4
Example B6 The following commands illustrate how line stylescan be used to distinguish between the plots of sin(2πt) sin(2πt+π6) and sin(2πt+ π2)
t=linspace(0130)
B6 Visualising data in Scilab 243
Table B8 Plot specifiers required to change line styles coloursand markers in plotsSpecifier Line style Specifier Colour Specifier Marker stylendash Solid line (default) r Red + Plus signndash ndash Dashed line g Green o Circle Dotted line b Blue Asteriskndash Dash-dotted line c Cyan x Cross
m Magenta s Squarey Yellow rsquononersquo No marker(default)k Blackw White
Figure B4 Resulting plotting after issuing the commands givenin Example B6
244 Appendix B Programming with Scilab
Example B6 illustrate a number of key commands relating to thecreation of multiple plots in Scilab The mtlb hold4 commandmtlb hold
holds the current plot and axis properties so that all subsequentplots are added to the existing plot
To distinguish each plot a different line specification is passed eachplot command In the example y = sin(2πt) is plotted using a redsolid line with data points indicated using a red circle as indicatedby the string rsquor-orsquo y = sin(2πt + π6) is plotted using greendashed-dotted line with data points marked by a square (rsquog-srsquo)while y = sin(2πt+ π2) is plotted using a cyan dashed line usingdiamonds to mark the data points (rsquoc--drsquo) Finally to completethe plot axes labels and a legend has been added to plot the com-mands for doing this are discussed in sectB62
B62 Adding titleslabels and legends to the plot
All good plots should have the following properties
1 The style of figure is appropriate to the information beingconveyed
2 The contents are well organised so it is easy for readers tointerpret the information
3 Legends labels etc are clear and concise (eg contain unitsif required do not obstruct parts of the plot etc) and
4 A caption or title that summarises the information presented
Like most mathematical applications Scilab has a wide variety ofdifferent plotting styles that can be used (eg look up the help forbar polarplot etc) This allows you to select the most appropri-barpolarplot
ate style to display and organise your data
The Scilab commands xlabel ylabel title and legend al-xlabel ylabel title and
legend lows you to add axes labels titles and legends to plot As shownin Example B6 the commands xlabel ylabel and title takea string input containing the title or label text For the legend
command a string is required for each separate plot Note theorder of the strings in the legend command reflects the order inwhich the each of plots were made
Including mathematical notation in titles labels and legends
It is possible to include mathematical characters such as π inScilab plots This is done by inserting special commands intoeither the title legend or label string These commands are basedon the abbreviations for mathematical notation used in programssuch as LATEX MathType etc A brief summary of the commands
4 mtlb hold on is equivalent to the Scilab commandset(gca()auto clearoff) while mtlb hold off is equivalentto the Scilab command set(gca()auto clearon)
B6 Visualising data in Scilab 245
Table B9 Basic mathematical symbols available in Scilab forincluding mathematical notation in titles labels and legendscommand symbol command symbol command symbolalpha α upsilon υ sim sim
Im = cup cup wp weierpotimes otimes subseteq sube oslash cap cap in isin supseteq supesupset sup lceil d subset subint
intcdot middot o oslash
rfloor c neg 6= nabla nablalfloor b times times ldots perp perp surd
radicprime prime
wedge and varpi $ O Oslashrceil e rangle 〉 mid |
vee or copyright ccopy langle 〈
available in Scilab for typesetting mathematical notation is givenin Table B9 To insert these commands into a text string simplystart and end the string with $ signs as shown in Example B7
Example B7 Scilab commands to produce the labelled plot ofy = sin
There are a few key concepts to note about the Example B7 Thefirst can be seen in the string being passed to the title commandtitle
In this string sin is prefixed with a backslash () This is doneso that the letters for sin are not typeset in italic which shouldonly be used for variables and not common mathematical func-tions Table B10 shows the command sequences to use for themore common mathematical functions5 To get the fraction in thetitle the command fracab has been used to produce a
b Ifsuperscripts and subscripts are required in a title or label these aresuperscripts amp subscripts
obtained by adding the subscript character ( ) and the superscriptcharacter (^) to the string These characters modify the characteror substring defined in braces immediately following For examplex 2 gives x2 while e^2x gives e2x
Note in Example B7 the commands xlabel and ylabel has beenxlabelylabel
used to set the axes labels in Figure B5 Instead of using thesethree commands the title and the axes labels could have been setusing the single xtitle command (see help xtitle for details)xtitle
In the example we have also customised the font size to make thetext more readable by a) passing an option to the title xlabelylabel commands and b) by using the command xset to set thexset
default font size before executing the xstring command Finallythe xstring command adds the text string to plot at the locationxstring
specified by variables b and c In the example the coordinates wewant to add the text are obtained by executing the xclick functionxclick
and clicking on the point required in the plot window
5 Additional details can be found in books covering Mathematics in LATEX
B6 Visualising data in Scilab 247
Table B10 Command sequences for the common mathematicalfunctions in Scilabarccos arcsin arctan cos cot
csc exp lim ln log
max min sec sin tan
B63 Plotting numeric data stored in an ASCII file
As data can be stored in a wide variety of ways in external filesScilab has a wide variety of functions which can be used to importthe data stored in these files (see help for the commands read
fscanf and xlsread) Since data can be stored in such a wideread fscanf and xlsread
range of formats we will confine our discussions here to the situa-tion where numeric data is stored in a ASCII (text) file where eachrow of the file contains the same number of entries
To illustrate the procedure for plotting data stored in ASCII file wewill consider the data obtained measuring the temperature of a cupof coffee every minute for a total of 100 mins The data recorded isstored in the a file called coffeedatatxt6 which simply containsthe two columns of numbers shown in Table B11 As an examplethe first 10 lines of the text file contains the following numbers7
00000 1000000
10000 950000
20000 903125
30000 859180
40000 817981
50000 779357
60000 743147
70000 709201
80000 677376
90000 647540
Note the current version of Scilab has a builtin text editor whichcan be used to view such text files To access this editor simplytype the command edit at the command prompt Once open youedit
can use the graphical interface to open the required file
The first issue is to get the data into Scilab so it is available forplot command In Scilab the command read can used to importread
the data and assign it to a variable In the current example wewould use the following command to import the data and assignit to a variable called cdata assuming the data is in the currentdirectory8
6 The file coffeedatatxt has been attached and should be available for savingprovided you are using Adobe Reader To save the file right click on thepurple Windows version or the red MacOSLinux version in the margin andselect save file from the options Ensure you save the file as coffeedatatxt
7 normally you will have received the data file in this format8 You can check what directory Scilab is in by typing the command pwd
Figure B6 Temperature of coffee based on the data recorded inthe file coffeedatatxt
cdata=read(rsquocoffeedatatxtrsquo-12)
The -12 following the name of the file tells Scilab that you donrsquotknow how many rows you have but you have two columns of dataSince we know that we have 101 rows of data we could have alsoused the following command to load the data
cdata=read(rsquocoffeedatatxtrsquo1012)
We now have a variable called cdata which has two columns ofdata The first column contains the time while the second containsthe Temperature records Hence we can plot it using the followingcommand
plot(cdata(1)cdata(2))
This gives the plot shown in Figure B6 after appropriate labelshave been added as described above
B7 Script files
Executing commands entered at the keyboard or keypad is themost primitive use of a calculating device It is only when se-quences of operations (to perform complex tasks) are stored awayand re-called for repeated application with different inputs that wehave a programmed operation Complex programs must necessarilyhave an interface with the user The basic interface tool in Scilabcapable of executing sequences of commands that are stored in a fileis the script The file names must have an extension of sce forexample program1sce assignment1sce These files are calledScript Sci-files or simply scripts
A script file is a file of standard keyboard charactersthat define commands to drive a program A script filecan be generated using any text editor or word proces-sor
To invoke a script in Scilab type the commandexec
exec filenamesce
where filename denotes the actual name of the file containing thescript When the script runs it is exactly the same as if you typedthe commands into the command window from the keyboard Thismeans that the Sci-file can use any variables you have defined priorto invoking the script Any variables created in the script are alsoavailable when the script is finished and any variable whose namewas assigned a value in the script will be over-written
If you give a script the same name as any variable that it computesin the running of the script then Scilab will give precedence tothe workspace variable over a script As stated in sectB3 the variablename will hide the script name after the script has run Scilab willtake the name as referring to the variable and not to the definedscript
Scilab also needs to know where to find the script to executedIf Scilab cannot find the script it will report that you have anundefined function or variable
The exact procedure for creating and saving Sci-files depends onthe Scilab version the operating system (MS Windows Macin-tosh or UnixLinux ) and the text editor you use The currentversion of Scilab includes a built-in editor which can be used tocreate and edit Sci-files and text files
To create a new Sci-file under the current version of Scilab simplytype edit newfilenamesce This will open up the Scilab editoredit
with a blank document called newfilename
B71 Script input and output
A script is a named file (with the extension sce) containing asequence of Scilab instructions which when its name is executedin a session performs those instructions in order as though theywere entered at the console
There are two common uses for scripts Pragmatically we maywant to save a particular computational sequence for later use orreference If the tasks involved are to be repeated often thenthe nature of the tasks should be analysed formulated into usefulfunctions (which we will discuss later) and the whole procedurepresented in a single function But if the tasks are only likely toget occasional use then the script may be adequately commentedand stored as a script file It is a quick and dirty way of saving
B7 Script files 251
the instructions for doing a particular job when we do not wantto go to the trouble of writing the functions for the job It is nota highly recommended modus operandi but useful all the same
The second use is for the script to act like a storyboard presentinginformation to a user in a structured and controlled way permit-ting the user to see the steps of a computational procedure andtheir outputs up on the screen At a more sophisticated level thescript takes on the proportions of an application controlling a largeand complex interaction with advanced interface features We willnot take our study of Scilab programming to this level but workmostly at the storyboard level using the simplest of Scilabrsquos in-terface tools
All the variables created in a script become global variables definedin the workspace and available for perusal As an application itis assumed that a script controls the entire workspace and thevariables it creates are lsquothe datarsquo of the application When scriptsare used as storyboards on a small scale then it cannot be assumedthat the variables they create deserve global recognition and it iscourteous to erase these variables when the script is finished unlessthey have some possible further use An unfortunate side-effect isthat if variables with the same name already exist in the workspacebefore a script is run then these variables will be over-written andlost
Used as storyboards scripts allow the user to cut paste and re-trythe various steps of a procedure and to modify the script to suithis own needs Storyboards may be used for demonstrations oftoolbox utilities or extended documentationexplanation of func-tions However if there is any danger that variables defined bythe script may over-write workspace variables then the demo ordocumentation should be re-fashioned into a function that leavesthe workspace variables untouched (more on this later)
Scripts will often read data from pre-prepared files and we will dis-cuss the functions provided by Scilab to perform this task below
Gathering data entered by the user really falls into the realm ofinterface design but we only briefly consider the simplest interfacefunction provided by Scilab Similarly we briefly consider thefunctions for providing the simplest forms of output presentation
Keyboard input
Scilab has a built-in function input that can be used to promptinput
the user to enter the value of a variable from the keyboard Forexample
Tc = input(rsquoPlease enter the temperature in degrees Celsiusrsquo)
will print the message
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
230 Appendix B Programming with Scilab
in this course)
x = 3 Assign the scalar integer value 3 to the variable x
x = 7297e17 Assign the scalar real value 7297 times 1017 to the variable xThe number has been defined using the scientific exponentnotation The number following the e is the power of 10 tomultiply the number in front of the e Either e or E can beused
x = rsquoNorthrsquo Assign the string North to the variable x Strings of char-acters are defined by surrounding the characters with apos-trophes The construct rsquoSouthrsquo defines a constant stringcontaining the characters South
x = 05100 Assign the row vector containing 21 elements from 0 to 100in steps of 5 to the variable x
x = [12 13 25 67] Assign the 2times 2 matrix to the variable x so that
x =
[12 13
25 67
]
x = 51+i63 Assign the complex number 51+ i63 to the variable x
Variable names can be of any length but must be unique withinthe first 31 characters Variable names in Scilab are case sensi-tive that is the variables time and Time are distinct Howeverfunctions and scripts (which we discuss later) are stored in sepa-rate files (with extension sce) outside the interpreter and may ormay not be case sensitive depending on the outside environment(in a dos environment for example they will not be)
Scilab allows overloading of names This means that if you choosea name that has been predefined by Scilab you loose the builtin meaning For example cos = 634 defines a scalar variablecos but it is also the same name for the Scilab cosine functionThe variable assignment means that the cosine function will nolonger be accessible in this session All references to the name coswill assume the variable not the function Take care not to usevariable names that might disable built in functions or constantsthat you may need later When writing scripts for others to usethe names of all variables that will be over-written by the runningof the script must be listed for easy reference (and one shouldread these references before running any script) This is one ofthe reasons why it is nearly always preferable to write functionsrather than scripts Variables defined in the course of runninga functionrsquos algorithm remain local to the function and do notover-write workspace variables
The results of all calculations should be assigned to a variable sothat this value can be used later If this is not done the result
B3 Variables 231
Table B1 Special variables and constants in ScilabName Description
pi The value of π to the precession of the machineis returned
i Recognised as theradicminus1
inf This is Scilabrsquos representation of infin This is arecognised value in a variable and typically ap-pears as a result of division by zero or logarithmof zero
nan Stands for Not a Number This is a recognisedvalue in a variable and appears as a result ofa mathematically undefined operation such asdivision of zero by zero or infinminusinfin
eps Floating point relative accuracy That is epsreturns the smallest difference resolvable be-tween two floating point numbers For examplethe next largest representable number beyond10 is 10+eps Scilab cannot represent anyfloating point number that falls between 10 and10+eps
ans This variable contains the last computed expres-sion that was not stored in a variable
date This variable contains a string representation oftodayrsquos date in the format dd-mmm-yyyy Forexample 01-Apr-2000
232 Appendix B Programming with Scilab
Table B2 Scilab Arithmetic OperatorsOperator Meaning Scilab
^ Exponentiation xy x^y
Multiply xtimes y xy
Right Division xy = xy xy
Left Division yx = xy yx
+ Addition x+ y x+y
- Subtraction xminus y x-y
of a calculation will be stored in a special variable called ans (seeans
Table B1) The value stored in this variable is overridden the nexttime a calculation is done that is not assigned to a variable Thismeans that the result from the previous calculation is then lost
B4 Arithmetic expressions
Table B2 lists the basic arithmetic operators to be found in ScilabWith the exception of the left division operator the arithmetic op-erators in Scilab follow standard notation
B41 Precedence
When more than one operation appears in a mathematical state-ment the programming language needs to define in which order theoperations must be performed or the precedence That is whichparts are evaluated first
If precedence was not defined then the same statement can be eval-uated to a different result for example is 186+3 equal to 2 oris 186+3 equal to 6 The first statement is evaluated from theright using the precedence rule that usually applies in mathemat-ical statements that is work from right to left as for example inlsquosin log tan xrsquo where we start with a value of x compute the tan-gent of that value (tan x) then take the logarithm of the resultingnumber (log(tan x)) and finally take the sine of what this deliv-ers However it is usual (because it is much easier to write poly-nomials with this convention) to give multiplication and divisionprecedence over addition and subtraction The second statementrecognises the precedence of the division over addition and evalu-ates it first Right to left precedence has a much simpler parsingtree than schemes which give precedence to particular operatorsbut the common arithmetic precedences are so widely acceptedand expected that most computer language designers will not riskthe possible lsquoconsumer backlashrsquo that breaking with this traditionwould involve So with every computer language one must searchout and learn the precedence rules that it adopts The usual arith-metic precedences will normally be observed but note that even
B4 Arithmetic expressions 233
Table B3 Precedence of arithmetic expressionsOperator Precedence Order of evaluation
() 1 Left to right innermost first 2 Left to right 3 Left to right 4 Right to left
+ minus 5 Left to right
they are not unambiguous For example which divide has prece-dence in 41632 Working right to left one would get the answerof 8 but Scilab returns (the decimal for) 1128 mdashit works fromleft to right Table B3 lists the operators and their precedencevalue
The rules for evaluation of operators are as follows
bull Operators having high precedence (low numerical value) willbe evaluated before those having a low precedence (high nu-merical value)
bull If an expression contains more than one operator at the samelevel of precedence then the left-to-right rule of evaluationapply
Parentheses () always have the highest precedence Whateveris within the parentheses is evaluated first This means when indoubt about the order of any statement place parentheses aroundthe statement to ensure that the operations are performed in theorder you expect
B42 Matrix arithmetic
One of the strengths of Scilab is its ability to execute matrixarithmetic3 Matrix arithmetic must be implemented in other lan-guages with many lines of code or special functions Scilab isespecially designed to handle matrix arithmetic The representsmatrix multiplication or scalar multiplication of a matrix if one ofthe arguments is a scalar Since scalars are 1times1 matrices ordinaryscalar multiplication can also be performed by the operator Thedot product of two vectors however requires that the first vectorbe a row vector and the second a column vector To multiply thecorresponding elements in two vectors together requires a differentoperator so that
gt a = [2 3 4] b = [6 7 8]
a b
ans =
12 21 32
3 Remember vectors are examples of matrices A column vector of length nis an example of a ntimes 1 matrix
234 Appendix B Programming with Scilab
However because it is a matrix operation
gt a + b
ans =
8 10 40
The standard arithmetic operators described in Table B2 are alsoused with matrices The normal rules of matrix addition subtrac-tion and multiplication apply
The left and right division of matrices are also generalisations ofscalar division Right division BA is roughly the same as Binv(A)Left division AB is roughly the same as inv(A)B The differenceis in how the division is performed Different algorithms are usedwhen A or B are square or not
Scilab also defines a transpose operator for matrices The trans-pose of the matrix A is Arsquo For example if
A =
[25 63
12 09
] then A prime =
[25 12
63 09
]
Accessing matrix elements
Individual elements of a matrix can accessed and used in state-ments by specifying the matrix index of an element The index ofan element is its position in the matrix For example consider thefollowing matrix
A =
6 minus1 4
3 5 7
minus3 6 1
Define this in Scilab with the command
A=[ 6 -1 4 3 5 7 -3 6 7]
to display the contents of matrix A in Scilab just type the variablename A To display or use one element from the matrix specify theelement For example A(13) specifies the element in the first rowand third column that is 4 for the above matrix
To change a specific entry just assign a value to it For example
A(31)=-9
changes the value of the third row first column element from -3
to -9
To display or use a whole row or column use the operator Forexample
gt A(2)
ans =
3 5 7
gt A(2)
B4 Arithmetic expressions 235
ans =
-1
5
6
It is also possible to pick out a sub-matrix by the specifying the rowand column entries required For example To pick out the sub-matrix of A containing the elementsA11 A13 A21 A23 the commandA(12123) could be used as shown below
gt A(12123)
ans =
6 4
3 7
Note 123 means from 1 to 3 in steps of 2 This technique alongwith the assignment operator can be used for values of whole rowscolumns or sub-matrices in one command For example
gt A(12123)=[10111213]
A =
10 -1 11
12 5 13
-3 6 7
Note that the elements A11 A13 A21 A23 now contain the values10 11 12 and 13 respectively instead of their previous values
Index notation is logically an unfortunate phenomenon since itclashes with standard function notation On the face of it A(3)
should be the value of a function called A with input 3 but whenit is interpreted as index notation A is a variable However indexnotation is so familiar and useful to those familiar with classicalmathematical notation that many developers of mathematical soft-ware are not prepared to risk a ldquoconsumer backlashrdquo to providinga functional way of extracting elements from an array such as forexample 3 pick A to pick out the third element of A Similarlywe as teachers feel we would be doing students a disservice if wefailed to introduce them to the index notation that abounds inmathematical texts usually in the form of subscripting as in A3Logic does not always win an argument In the commercial worldof computing an established user base carries more weight thanmathematical logic just as in mathematics tradition may well alsofly in the face of logical consistency
B43 Array arithmetic
Scilab has two forms of arithmetic operations that can be usedwith matrices matrix arithmetic or array arithmetic Matrix arith-metic follows the normal rules of matrix algebra Array arithmeticwhich is also called element-by-element arithmetic is where oper-ations are performed on matrices element by element
236 Appendix B Programming with Scilab
Table B4 Scilab array operatorsOperator Meaning Scilab
^ Exponentiation x^y
Multiply xy
Right Division xy
Left Division yx
+ Addition x+y
- Subtraction x-y
Table B4 lists the Scilab array operators
The array addition and subtraction are identical to matrix additionor subtraction Both array and matrix addition and subtractionoperate element by element Array addition and subtraction re-quire that both arrays have the same size The only exception tothis rule in Scilab occurs when we add or subtract a scalar to orfrom an array In this case the scalar is added or subtracted fromeach element in the array
For example
gt [6 5 6] + 3
ans =
[9 8 9]
gt [6 4 5 8] + [3 1 2 4]
ans =
[9 5 7 12]
gt [4 3] - 7
ans =
[-3 -4]
Array multiplication and division is very different to matrix mul-tiplication and division Array multiplication or division is an ele-ment by element multiplication or division that produces an arraywith the same dimensions as the starting arrays As with addi-tion and subtraction array multiplication and division require thatboth arrays have the same dimensions
Example B2 Consider the two row vectors x=[2 4 7 1] andy=[6 1 3 2] Scilab does not recognise the possibility ofmatrix multiplication of these two vectors They are not com-patible the number of columns of the pre-multiplier must bethe same as the number of rows of the post-multiplier Anattempt to execute x y will bring an error message
Obtain the usual dot product of the two row vectors by trans-posing the post-multiplier as in xyrsquo or yxrsquo The matrixproduct xrsquoy produces a matrix of 4 rows and 4 columns
B5 Builtndashin functions 237
Array multiplication produces the result
gt xy
ans =
[12 4 21 2]
by multiplying corresponding elements of each vector Theresult is a row vector with the same dimensions as x and ythat is a 1times 4 vector
Example B3 Consider the row vectors x=[2 4 7 1] and col-umn vector y=[6 1 3 2]rsquo Array multiplication of thesetwo matrices is not possible because their dimensions do notagree
Matrix multiplication xy produces the result 39 that is theinner product or dot product of the vectors Note vectorsas such do not exist in Scilab there are only single rowedor single columned arrays
Scilab not only allows arrays to be raised to powers but alsoallows scalars and arrays to be raised by array powers To performexponentiation on an element-by-element basis use the ^ symbolFor example if x=[3 5 8] then x^3 produces the matrix [33 53
83] = [27 125 512]
A scalar can be raised to an array power For example if p=[2 4 5]then typing 3^p produces the array [32 34 35] = [9 81 243]
With element-by-element array operations it is important to re-member that the dot () is a part of a two-character symbol for aparticular array operator such as () () () (^) Sometimesthis can cause some confusion the dot in 3^p is not a decimalpoint associated with the number 3 The following operations withthe value of p=[2 4 5] are equivalent
3^p
30^p
3^p
(3)^p
3^[2 4 5]
B5 Builtndashin functions
B51 Arithmetic functions
Arithmetic expressions often require computations other than ad-dition subtraction multiplication division and exponentiationFor example many expressions require the use of logarithms ex-ponentials and trigonometric functions Scilab has predefined alarge number of functions that perform these types of computations(see Tables B5 and B6)
238 Appendix B Programming with Scilab
An example of a predefined function is sin which returns the sineof a given angle
m = sin(angle)
where the variable angle contains the angle in radians and theresult is stored in the variable m The variable angle is called theargument or parameter of the function If the angle is stored indegrees it must be converted to radians for example
m = sin(anglepi180)
or
radians =anglepi180
m = sin(radians)
In Scilab the arguments or parameters of the function are con-tained in parentheses following the name of the function This isreferred to as pre-fix notation Many common functions in math-ematics have a syntax using in-fix notation where two argumentsappear to the right and left of the function name as for examplein x+y atimes b nmod r etc Scilab does not support in-fix nota-tion for any but the arithmetic functions Classical mathematicalnotation also has examples of post-fix notationmdash n x squared mdashand other hybrids like |x| while Scilab has ahb to confound thepurist
A function may contain no arguments one argument or many ar-guments depending on its definition If a function contains morethan one argument it is vital to give the arguments in the cor-rect order For example consider the problem of finding θ in thefollowing triangle using the atan command
5
8θ
Now by hand calculation the answer is 05586 radians If we useatan as below we get the answers shown
--gt atan(58)
ans =
05586
--gt atan(85)
ans =
10122
The first answer is the correct one The second answer is incorrect
B5 Builtndashin functions 239
Table B5 Some common mathematical functions defined inScilabFunction Description
abs(x) Computes the absolute value of xsqrt(x) Square root of x The parameter can be nega-
tiveround(x) Round x to the nearest integerfix(x) Round x to the nearest integer toward zerofloor(x) Round x to the nearest integer toward minusinfinceil(x) Round x to the nearest integer toward +infinsign(x) Returns -1 if x is less than zero zero if x equals
zero and 1 otherwisemodulo(xy) Returns the remainder of xy For example
modulo(254) is 1exp(x) Returns ex
log(x) Returns the natural logarithm of x That isthe logarithm to the base e Parameter can benegative but cannot be zero
log10(x) Returns the common logarithm of x That isthe logarithm to the base 10 Parameter can benegative but cannot be zero
since we passed the values to the function in the incorrect order
Some functions also require that the arguments be in specific unitsFor example the trigonometric functions assume that the argu-ments are in radians
In Scilab some functions use the number of arguments to deter-mine the output of the function For example the zeros functioncan have one or two arguments which then determine the output(check this with the command help zeros)
A function reference cannot be made on the left hand side of anassignment since a function call always returns a value(s) whereasan assignment is always to a variable Functions can of courseappear on the right side of an assignment since they return valuesto be assigned to some variable Functions may be applied tothe output of other functions and may be called in the algorithmdefining a function for example
logx = log(abs(x))
When one function is used to compute the argument of anotherfunction be sure to enclose the argument of each function in itsown set of parentheses
Scilab allows recursive functions that is functions that call them-selves in their defining algorithm
Exercise B4 Use examples to show the differences between the
240 Appendix B Programming with Scilab
Table B6 Some common trigonometric functions defined inScilab
Function Description
sin(x) Sine of x where x is in radianscos(x) Cosine of x where x is in radianstan(x) Tangent of x where x is in radiansasin(x) Arcsine or inverse sine of x where x must be
between -1 and 1 The return angle is in radiansbetween minusπ2 and π2
acos(x) Arccosine or inverse cosine of x where x mustbe between minus1 and 1 The return angle is inradians between 0 and π
atan(x) Arctangent or inverse tangent of x The returnangle is in radians between minusπ2 and π2
atan(xy) Arctangent or inverse tangent of xy The re-turn angle is in radians between minusπ and π de-pending on the sign of x and y
sec(x) Secant (1(cos x)) of x where x is in radianscsc(x) Cosecant (1(sin x)) of x where x is in radianscotg(x) Cotangent (1(tan x)) of x where x is in radians
Table B7 Some of Scilabrsquos utility functionsFunction Description
clear Clear variables from memoryhelp Display the first set of comment lines from
scriptswho List the variables in memorywhos List the variables in memory with their sizeerror Terminates the execution of the Sci-file and out-
puts the supplied error message
functions round(x) fix(x) floor(x) and ceil(x)
Exercise B5 The functions listed in Tables B5 B6 are only asmall sample of all of Scilabrsquos mathematical functions
Using Scilabrsquos help command list all of the hyperbolic func-tions available Also list any restrictions on the values of thefunction arguments
B52 Utility functions
In addition to the arithmetic functions Scilab has several utilityfunctions (see Table B7) These functions allow users to trackvariables (whoswho) feedback an error message if an error occursin the code (error) clear the variables form the memory (clear)etc Essential these functions allow users to debug and performbookkeeping operations inside their code
B6 Visualising data in Scilab 241
6
5
4- 04- 8
3
- 02
- 6
00
2- 4 Y- 2
02
Z
10
04
X 02
06
4 - 1
08
6- 28
10
Figure B2 Plot of z = sin(radicx2 + y2)
radicx2 + y2 illustrating the
3d capabilities of Scilab
B6 Visualising data in Scilab
One of the big advantages in using Scilab is the ability to visualisedata and results of complex calculations (see Figure B2 as exampleof the 3d capabilities) While Scilab is capability of complex 3dgraphics (see help pages for mesh and surf) in this section we willmesh amp surf
focus on the 2d plotting command (plot) Hence in this sectionplot
we will focus on
1 plotting mathematical functions from within Scilab
2 changing stylistic elements of the plot
3 adding titles labels and legends to the plot
4 including mathematical notation in titles labels and legendsand
5 plotting data stored in an external ASCII file
B61 Plotting mathematical functions from within Scilab
To plot a mathematical function in Scilab three steps are requiredThese are
1 Define the domain of the function (eg x=linspace(02pi))
2 Calculate the values of function at the selected points in thedomain (eg y=sin(x))
242 Appendix B Programming with Scilab
3 Final step is to plot the function using the plot commandplot
(eg plot(xy))
Using these steps it is possible to produce a basic plot of any math-ematical function (see Figure B3) Note that the plot shown inFigure B3 does not have a title axes labels or a legend all ofwhich are crucial for a good plot The Scilab commands to addthese elements to the plot are given in the following sections
- 10
- 08
- 06
- 04
- 02
00
02
04
06
08
10
0 1 2 3 4 5 6 7
Figure B3 The default plot of y = sin(x) in Scilab Notethat by default axes are not labelled and there is no title orlegend These need to added using the additional commandsxlabelylabeltitle and legend
Changing stylistic elements of the plot
An optional argument can be given to the plot command to cus-tomise each new line aspect This argument is a concatenatedstring containing information about colour line style or markersbased on the specifiers shown in Table B8 To specify the plotbe drawn with a red longdash-dot line with diamond marker thestring that would be passed to plot command would be rsquor-drsquo Thestring which is a concatenation (in any order) of the three typesof properties shown in Table B8 must be unambiguous Further-more the string specification is not case sensitive An exampleof specifying various line styles is shown in Example B6 and Fig-ure B4
Example B6 The following commands illustrate how line stylescan be used to distinguish between the plots of sin(2πt) sin(2πt+π6) and sin(2πt+ π2)
t=linspace(0130)
B6 Visualising data in Scilab 243
Table B8 Plot specifiers required to change line styles coloursand markers in plotsSpecifier Line style Specifier Colour Specifier Marker stylendash Solid line (default) r Red + Plus signndash ndash Dashed line g Green o Circle Dotted line b Blue Asteriskndash Dash-dotted line c Cyan x Cross
m Magenta s Squarey Yellow rsquononersquo No marker(default)k Blackw White
Figure B4 Resulting plotting after issuing the commands givenin Example B6
244 Appendix B Programming with Scilab
Example B6 illustrate a number of key commands relating to thecreation of multiple plots in Scilab The mtlb hold4 commandmtlb hold
holds the current plot and axis properties so that all subsequentplots are added to the existing plot
To distinguish each plot a different line specification is passed eachplot command In the example y = sin(2πt) is plotted using a redsolid line with data points indicated using a red circle as indicatedby the string rsquor-orsquo y = sin(2πt + π6) is plotted using greendashed-dotted line with data points marked by a square (rsquog-srsquo)while y = sin(2πt+ π2) is plotted using a cyan dashed line usingdiamonds to mark the data points (rsquoc--drsquo) Finally to completethe plot axes labels and a legend has been added to plot the com-mands for doing this are discussed in sectB62
B62 Adding titleslabels and legends to the plot
All good plots should have the following properties
1 The style of figure is appropriate to the information beingconveyed
2 The contents are well organised so it is easy for readers tointerpret the information
3 Legends labels etc are clear and concise (eg contain unitsif required do not obstruct parts of the plot etc) and
4 A caption or title that summarises the information presented
Like most mathematical applications Scilab has a wide variety ofdifferent plotting styles that can be used (eg look up the help forbar polarplot etc) This allows you to select the most appropri-barpolarplot
ate style to display and organise your data
The Scilab commands xlabel ylabel title and legend al-xlabel ylabel title and
legend lows you to add axes labels titles and legends to plot As shownin Example B6 the commands xlabel ylabel and title takea string input containing the title or label text For the legend
command a string is required for each separate plot Note theorder of the strings in the legend command reflects the order inwhich the each of plots were made
Including mathematical notation in titles labels and legends
It is possible to include mathematical characters such as π inScilab plots This is done by inserting special commands intoeither the title legend or label string These commands are basedon the abbreviations for mathematical notation used in programssuch as LATEX MathType etc A brief summary of the commands
4 mtlb hold on is equivalent to the Scilab commandset(gca()auto clearoff) while mtlb hold off is equivalentto the Scilab command set(gca()auto clearon)
B6 Visualising data in Scilab 245
Table B9 Basic mathematical symbols available in Scilab forincluding mathematical notation in titles labels and legendscommand symbol command symbol command symbolalpha α upsilon υ sim sim
Im = cup cup wp weierpotimes otimes subseteq sube oslash cap cap in isin supseteq supesupset sup lceil d subset subint
intcdot middot o oslash
rfloor c neg 6= nabla nablalfloor b times times ldots perp perp surd
radicprime prime
wedge and varpi $ O Oslashrceil e rangle 〉 mid |
vee or copyright ccopy langle 〈
available in Scilab for typesetting mathematical notation is givenin Table B9 To insert these commands into a text string simplystart and end the string with $ signs as shown in Example B7
Example B7 Scilab commands to produce the labelled plot ofy = sin
There are a few key concepts to note about the Example B7 Thefirst can be seen in the string being passed to the title commandtitle
In this string sin is prefixed with a backslash () This is doneso that the letters for sin are not typeset in italic which shouldonly be used for variables and not common mathematical func-tions Table B10 shows the command sequences to use for themore common mathematical functions5 To get the fraction in thetitle the command fracab has been used to produce a
b Ifsuperscripts and subscripts are required in a title or label these aresuperscripts amp subscripts
obtained by adding the subscript character ( ) and the superscriptcharacter (^) to the string These characters modify the characteror substring defined in braces immediately following For examplex 2 gives x2 while e^2x gives e2x
Note in Example B7 the commands xlabel and ylabel has beenxlabelylabel
used to set the axes labels in Figure B5 Instead of using thesethree commands the title and the axes labels could have been setusing the single xtitle command (see help xtitle for details)xtitle
In the example we have also customised the font size to make thetext more readable by a) passing an option to the title xlabelylabel commands and b) by using the command xset to set thexset
default font size before executing the xstring command Finallythe xstring command adds the text string to plot at the locationxstring
specified by variables b and c In the example the coordinates wewant to add the text are obtained by executing the xclick functionxclick
and clicking on the point required in the plot window
5 Additional details can be found in books covering Mathematics in LATEX
B6 Visualising data in Scilab 247
Table B10 Command sequences for the common mathematicalfunctions in Scilabarccos arcsin arctan cos cot
csc exp lim ln log
max min sec sin tan
B63 Plotting numeric data stored in an ASCII file
As data can be stored in a wide variety of ways in external filesScilab has a wide variety of functions which can be used to importthe data stored in these files (see help for the commands read
fscanf and xlsread) Since data can be stored in such a wideread fscanf and xlsread
range of formats we will confine our discussions here to the situa-tion where numeric data is stored in a ASCII (text) file where eachrow of the file contains the same number of entries
To illustrate the procedure for plotting data stored in ASCII file wewill consider the data obtained measuring the temperature of a cupof coffee every minute for a total of 100 mins The data recorded isstored in the a file called coffeedatatxt6 which simply containsthe two columns of numbers shown in Table B11 As an examplethe first 10 lines of the text file contains the following numbers7
00000 1000000
10000 950000
20000 903125
30000 859180
40000 817981
50000 779357
60000 743147
70000 709201
80000 677376
90000 647540
Note the current version of Scilab has a builtin text editor whichcan be used to view such text files To access this editor simplytype the command edit at the command prompt Once open youedit
can use the graphical interface to open the required file
The first issue is to get the data into Scilab so it is available forplot command In Scilab the command read can used to importread
the data and assign it to a variable In the current example wewould use the following command to import the data and assignit to a variable called cdata assuming the data is in the currentdirectory8
6 The file coffeedatatxt has been attached and should be available for savingprovided you are using Adobe Reader To save the file right click on thepurple Windows version or the red MacOSLinux version in the margin andselect save file from the options Ensure you save the file as coffeedatatxt
7 normally you will have received the data file in this format8 You can check what directory Scilab is in by typing the command pwd
Figure B6 Temperature of coffee based on the data recorded inthe file coffeedatatxt
cdata=read(rsquocoffeedatatxtrsquo-12)
The -12 following the name of the file tells Scilab that you donrsquotknow how many rows you have but you have two columns of dataSince we know that we have 101 rows of data we could have alsoused the following command to load the data
cdata=read(rsquocoffeedatatxtrsquo1012)
We now have a variable called cdata which has two columns ofdata The first column contains the time while the second containsthe Temperature records Hence we can plot it using the followingcommand
plot(cdata(1)cdata(2))
This gives the plot shown in Figure B6 after appropriate labelshave been added as described above
B7 Script files
Executing commands entered at the keyboard or keypad is themost primitive use of a calculating device It is only when se-quences of operations (to perform complex tasks) are stored awayand re-called for repeated application with different inputs that wehave a programmed operation Complex programs must necessarilyhave an interface with the user The basic interface tool in Scilabcapable of executing sequences of commands that are stored in a fileis the script The file names must have an extension of sce forexample program1sce assignment1sce These files are calledScript Sci-files or simply scripts
A script file is a file of standard keyboard charactersthat define commands to drive a program A script filecan be generated using any text editor or word proces-sor
To invoke a script in Scilab type the commandexec
exec filenamesce
where filename denotes the actual name of the file containing thescript When the script runs it is exactly the same as if you typedthe commands into the command window from the keyboard Thismeans that the Sci-file can use any variables you have defined priorto invoking the script Any variables created in the script are alsoavailable when the script is finished and any variable whose namewas assigned a value in the script will be over-written
If you give a script the same name as any variable that it computesin the running of the script then Scilab will give precedence tothe workspace variable over a script As stated in sectB3 the variablename will hide the script name after the script has run Scilab willtake the name as referring to the variable and not to the definedscript
Scilab also needs to know where to find the script to executedIf Scilab cannot find the script it will report that you have anundefined function or variable
The exact procedure for creating and saving Sci-files depends onthe Scilab version the operating system (MS Windows Macin-tosh or UnixLinux ) and the text editor you use The currentversion of Scilab includes a built-in editor which can be used tocreate and edit Sci-files and text files
To create a new Sci-file under the current version of Scilab simplytype edit newfilenamesce This will open up the Scilab editoredit
with a blank document called newfilename
B71 Script input and output
A script is a named file (with the extension sce) containing asequence of Scilab instructions which when its name is executedin a session performs those instructions in order as though theywere entered at the console
There are two common uses for scripts Pragmatically we maywant to save a particular computational sequence for later use orreference If the tasks involved are to be repeated often thenthe nature of the tasks should be analysed formulated into usefulfunctions (which we will discuss later) and the whole procedurepresented in a single function But if the tasks are only likely toget occasional use then the script may be adequately commentedand stored as a script file It is a quick and dirty way of saving
B7 Script files 251
the instructions for doing a particular job when we do not wantto go to the trouble of writing the functions for the job It is nota highly recommended modus operandi but useful all the same
The second use is for the script to act like a storyboard presentinginformation to a user in a structured and controlled way permit-ting the user to see the steps of a computational procedure andtheir outputs up on the screen At a more sophisticated level thescript takes on the proportions of an application controlling a largeand complex interaction with advanced interface features We willnot take our study of Scilab programming to this level but workmostly at the storyboard level using the simplest of Scilabrsquos in-terface tools
All the variables created in a script become global variables definedin the workspace and available for perusal As an application itis assumed that a script controls the entire workspace and thevariables it creates are lsquothe datarsquo of the application When scriptsare used as storyboards on a small scale then it cannot be assumedthat the variables they create deserve global recognition and it iscourteous to erase these variables when the script is finished unlessthey have some possible further use An unfortunate side-effect isthat if variables with the same name already exist in the workspacebefore a script is run then these variables will be over-written andlost
Used as storyboards scripts allow the user to cut paste and re-trythe various steps of a procedure and to modify the script to suithis own needs Storyboards may be used for demonstrations oftoolbox utilities or extended documentationexplanation of func-tions However if there is any danger that variables defined bythe script may over-write workspace variables then the demo ordocumentation should be re-fashioned into a function that leavesthe workspace variables untouched (more on this later)
Scripts will often read data from pre-prepared files and we will dis-cuss the functions provided by Scilab to perform this task below
Gathering data entered by the user really falls into the realm ofinterface design but we only briefly consider the simplest interfacefunction provided by Scilab Similarly we briefly consider thefunctions for providing the simplest forms of output presentation
Keyboard input
Scilab has a built-in function input that can be used to promptinput
the user to enter the value of a variable from the keyboard Forexample
Tc = input(rsquoPlease enter the temperature in degrees Celsiusrsquo)
will print the message
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
B3 Variables 231
Table B1 Special variables and constants in ScilabName Description
pi The value of π to the precession of the machineis returned
i Recognised as theradicminus1
inf This is Scilabrsquos representation of infin This is arecognised value in a variable and typically ap-pears as a result of division by zero or logarithmof zero
nan Stands for Not a Number This is a recognisedvalue in a variable and appears as a result ofa mathematically undefined operation such asdivision of zero by zero or infinminusinfin
eps Floating point relative accuracy That is epsreturns the smallest difference resolvable be-tween two floating point numbers For examplethe next largest representable number beyond10 is 10+eps Scilab cannot represent anyfloating point number that falls between 10 and10+eps
ans This variable contains the last computed expres-sion that was not stored in a variable
date This variable contains a string representation oftodayrsquos date in the format dd-mmm-yyyy Forexample 01-Apr-2000
232 Appendix B Programming with Scilab
Table B2 Scilab Arithmetic OperatorsOperator Meaning Scilab
^ Exponentiation xy x^y
Multiply xtimes y xy
Right Division xy = xy xy
Left Division yx = xy yx
+ Addition x+ y x+y
- Subtraction xminus y x-y
of a calculation will be stored in a special variable called ans (seeans
Table B1) The value stored in this variable is overridden the nexttime a calculation is done that is not assigned to a variable Thismeans that the result from the previous calculation is then lost
B4 Arithmetic expressions
Table B2 lists the basic arithmetic operators to be found in ScilabWith the exception of the left division operator the arithmetic op-erators in Scilab follow standard notation
B41 Precedence
When more than one operation appears in a mathematical state-ment the programming language needs to define in which order theoperations must be performed or the precedence That is whichparts are evaluated first
If precedence was not defined then the same statement can be eval-uated to a different result for example is 186+3 equal to 2 oris 186+3 equal to 6 The first statement is evaluated from theright using the precedence rule that usually applies in mathemat-ical statements that is work from right to left as for example inlsquosin log tan xrsquo where we start with a value of x compute the tan-gent of that value (tan x) then take the logarithm of the resultingnumber (log(tan x)) and finally take the sine of what this deliv-ers However it is usual (because it is much easier to write poly-nomials with this convention) to give multiplication and divisionprecedence over addition and subtraction The second statementrecognises the precedence of the division over addition and evalu-ates it first Right to left precedence has a much simpler parsingtree than schemes which give precedence to particular operatorsbut the common arithmetic precedences are so widely acceptedand expected that most computer language designers will not riskthe possible lsquoconsumer backlashrsquo that breaking with this traditionwould involve So with every computer language one must searchout and learn the precedence rules that it adopts The usual arith-metic precedences will normally be observed but note that even
B4 Arithmetic expressions 233
Table B3 Precedence of arithmetic expressionsOperator Precedence Order of evaluation
() 1 Left to right innermost first 2 Left to right 3 Left to right 4 Right to left
+ minus 5 Left to right
they are not unambiguous For example which divide has prece-dence in 41632 Working right to left one would get the answerof 8 but Scilab returns (the decimal for) 1128 mdashit works fromleft to right Table B3 lists the operators and their precedencevalue
The rules for evaluation of operators are as follows
bull Operators having high precedence (low numerical value) willbe evaluated before those having a low precedence (high nu-merical value)
bull If an expression contains more than one operator at the samelevel of precedence then the left-to-right rule of evaluationapply
Parentheses () always have the highest precedence Whateveris within the parentheses is evaluated first This means when indoubt about the order of any statement place parentheses aroundthe statement to ensure that the operations are performed in theorder you expect
B42 Matrix arithmetic
One of the strengths of Scilab is its ability to execute matrixarithmetic3 Matrix arithmetic must be implemented in other lan-guages with many lines of code or special functions Scilab isespecially designed to handle matrix arithmetic The representsmatrix multiplication or scalar multiplication of a matrix if one ofthe arguments is a scalar Since scalars are 1times1 matrices ordinaryscalar multiplication can also be performed by the operator Thedot product of two vectors however requires that the first vectorbe a row vector and the second a column vector To multiply thecorresponding elements in two vectors together requires a differentoperator so that
gt a = [2 3 4] b = [6 7 8]
a b
ans =
12 21 32
3 Remember vectors are examples of matrices A column vector of length nis an example of a ntimes 1 matrix
234 Appendix B Programming with Scilab
However because it is a matrix operation
gt a + b
ans =
8 10 40
The standard arithmetic operators described in Table B2 are alsoused with matrices The normal rules of matrix addition subtrac-tion and multiplication apply
The left and right division of matrices are also generalisations ofscalar division Right division BA is roughly the same as Binv(A)Left division AB is roughly the same as inv(A)B The differenceis in how the division is performed Different algorithms are usedwhen A or B are square or not
Scilab also defines a transpose operator for matrices The trans-pose of the matrix A is Arsquo For example if
A =
[25 63
12 09
] then A prime =
[25 12
63 09
]
Accessing matrix elements
Individual elements of a matrix can accessed and used in state-ments by specifying the matrix index of an element The index ofan element is its position in the matrix For example consider thefollowing matrix
A =
6 minus1 4
3 5 7
minus3 6 1
Define this in Scilab with the command
A=[ 6 -1 4 3 5 7 -3 6 7]
to display the contents of matrix A in Scilab just type the variablename A To display or use one element from the matrix specify theelement For example A(13) specifies the element in the first rowand third column that is 4 for the above matrix
To change a specific entry just assign a value to it For example
A(31)=-9
changes the value of the third row first column element from -3
to -9
To display or use a whole row or column use the operator Forexample
gt A(2)
ans =
3 5 7
gt A(2)
B4 Arithmetic expressions 235
ans =
-1
5
6
It is also possible to pick out a sub-matrix by the specifying the rowand column entries required For example To pick out the sub-matrix of A containing the elementsA11 A13 A21 A23 the commandA(12123) could be used as shown below
gt A(12123)
ans =
6 4
3 7
Note 123 means from 1 to 3 in steps of 2 This technique alongwith the assignment operator can be used for values of whole rowscolumns or sub-matrices in one command For example
gt A(12123)=[10111213]
A =
10 -1 11
12 5 13
-3 6 7
Note that the elements A11 A13 A21 A23 now contain the values10 11 12 and 13 respectively instead of their previous values
Index notation is logically an unfortunate phenomenon since itclashes with standard function notation On the face of it A(3)
should be the value of a function called A with input 3 but whenit is interpreted as index notation A is a variable However indexnotation is so familiar and useful to those familiar with classicalmathematical notation that many developers of mathematical soft-ware are not prepared to risk a ldquoconsumer backlashrdquo to providinga functional way of extracting elements from an array such as forexample 3 pick A to pick out the third element of A Similarlywe as teachers feel we would be doing students a disservice if wefailed to introduce them to the index notation that abounds inmathematical texts usually in the form of subscripting as in A3Logic does not always win an argument In the commercial worldof computing an established user base carries more weight thanmathematical logic just as in mathematics tradition may well alsofly in the face of logical consistency
B43 Array arithmetic
Scilab has two forms of arithmetic operations that can be usedwith matrices matrix arithmetic or array arithmetic Matrix arith-metic follows the normal rules of matrix algebra Array arithmeticwhich is also called element-by-element arithmetic is where oper-ations are performed on matrices element by element
236 Appendix B Programming with Scilab
Table B4 Scilab array operatorsOperator Meaning Scilab
^ Exponentiation x^y
Multiply xy
Right Division xy
Left Division yx
+ Addition x+y
- Subtraction x-y
Table B4 lists the Scilab array operators
The array addition and subtraction are identical to matrix additionor subtraction Both array and matrix addition and subtractionoperate element by element Array addition and subtraction re-quire that both arrays have the same size The only exception tothis rule in Scilab occurs when we add or subtract a scalar to orfrom an array In this case the scalar is added or subtracted fromeach element in the array
For example
gt [6 5 6] + 3
ans =
[9 8 9]
gt [6 4 5 8] + [3 1 2 4]
ans =
[9 5 7 12]
gt [4 3] - 7
ans =
[-3 -4]
Array multiplication and division is very different to matrix mul-tiplication and division Array multiplication or division is an ele-ment by element multiplication or division that produces an arraywith the same dimensions as the starting arrays As with addi-tion and subtraction array multiplication and division require thatboth arrays have the same dimensions
Example B2 Consider the two row vectors x=[2 4 7 1] andy=[6 1 3 2] Scilab does not recognise the possibility ofmatrix multiplication of these two vectors They are not com-patible the number of columns of the pre-multiplier must bethe same as the number of rows of the post-multiplier Anattempt to execute x y will bring an error message
Obtain the usual dot product of the two row vectors by trans-posing the post-multiplier as in xyrsquo or yxrsquo The matrixproduct xrsquoy produces a matrix of 4 rows and 4 columns
B5 Builtndashin functions 237
Array multiplication produces the result
gt xy
ans =
[12 4 21 2]
by multiplying corresponding elements of each vector Theresult is a row vector with the same dimensions as x and ythat is a 1times 4 vector
Example B3 Consider the row vectors x=[2 4 7 1] and col-umn vector y=[6 1 3 2]rsquo Array multiplication of thesetwo matrices is not possible because their dimensions do notagree
Matrix multiplication xy produces the result 39 that is theinner product or dot product of the vectors Note vectorsas such do not exist in Scilab there are only single rowedor single columned arrays
Scilab not only allows arrays to be raised to powers but alsoallows scalars and arrays to be raised by array powers To performexponentiation on an element-by-element basis use the ^ symbolFor example if x=[3 5 8] then x^3 produces the matrix [33 53
83] = [27 125 512]
A scalar can be raised to an array power For example if p=[2 4 5]then typing 3^p produces the array [32 34 35] = [9 81 243]
With element-by-element array operations it is important to re-member that the dot () is a part of a two-character symbol for aparticular array operator such as () () () (^) Sometimesthis can cause some confusion the dot in 3^p is not a decimalpoint associated with the number 3 The following operations withthe value of p=[2 4 5] are equivalent
3^p
30^p
3^p
(3)^p
3^[2 4 5]
B5 Builtndashin functions
B51 Arithmetic functions
Arithmetic expressions often require computations other than ad-dition subtraction multiplication division and exponentiationFor example many expressions require the use of logarithms ex-ponentials and trigonometric functions Scilab has predefined alarge number of functions that perform these types of computations(see Tables B5 and B6)
238 Appendix B Programming with Scilab
An example of a predefined function is sin which returns the sineof a given angle
m = sin(angle)
where the variable angle contains the angle in radians and theresult is stored in the variable m The variable angle is called theargument or parameter of the function If the angle is stored indegrees it must be converted to radians for example
m = sin(anglepi180)
or
radians =anglepi180
m = sin(radians)
In Scilab the arguments or parameters of the function are con-tained in parentheses following the name of the function This isreferred to as pre-fix notation Many common functions in math-ematics have a syntax using in-fix notation where two argumentsappear to the right and left of the function name as for examplein x+y atimes b nmod r etc Scilab does not support in-fix nota-tion for any but the arithmetic functions Classical mathematicalnotation also has examples of post-fix notationmdash n x squared mdashand other hybrids like |x| while Scilab has ahb to confound thepurist
A function may contain no arguments one argument or many ar-guments depending on its definition If a function contains morethan one argument it is vital to give the arguments in the cor-rect order For example consider the problem of finding θ in thefollowing triangle using the atan command
5
8θ
Now by hand calculation the answer is 05586 radians If we useatan as below we get the answers shown
--gt atan(58)
ans =
05586
--gt atan(85)
ans =
10122
The first answer is the correct one The second answer is incorrect
B5 Builtndashin functions 239
Table B5 Some common mathematical functions defined inScilabFunction Description
abs(x) Computes the absolute value of xsqrt(x) Square root of x The parameter can be nega-
tiveround(x) Round x to the nearest integerfix(x) Round x to the nearest integer toward zerofloor(x) Round x to the nearest integer toward minusinfinceil(x) Round x to the nearest integer toward +infinsign(x) Returns -1 if x is less than zero zero if x equals
zero and 1 otherwisemodulo(xy) Returns the remainder of xy For example
modulo(254) is 1exp(x) Returns ex
log(x) Returns the natural logarithm of x That isthe logarithm to the base e Parameter can benegative but cannot be zero
log10(x) Returns the common logarithm of x That isthe logarithm to the base 10 Parameter can benegative but cannot be zero
since we passed the values to the function in the incorrect order
Some functions also require that the arguments be in specific unitsFor example the trigonometric functions assume that the argu-ments are in radians
In Scilab some functions use the number of arguments to deter-mine the output of the function For example the zeros functioncan have one or two arguments which then determine the output(check this with the command help zeros)
A function reference cannot be made on the left hand side of anassignment since a function call always returns a value(s) whereasan assignment is always to a variable Functions can of courseappear on the right side of an assignment since they return valuesto be assigned to some variable Functions may be applied tothe output of other functions and may be called in the algorithmdefining a function for example
logx = log(abs(x))
When one function is used to compute the argument of anotherfunction be sure to enclose the argument of each function in itsown set of parentheses
Scilab allows recursive functions that is functions that call them-selves in their defining algorithm
Exercise B4 Use examples to show the differences between the
240 Appendix B Programming with Scilab
Table B6 Some common trigonometric functions defined inScilab
Function Description
sin(x) Sine of x where x is in radianscos(x) Cosine of x where x is in radianstan(x) Tangent of x where x is in radiansasin(x) Arcsine or inverse sine of x where x must be
between -1 and 1 The return angle is in radiansbetween minusπ2 and π2
acos(x) Arccosine or inverse cosine of x where x mustbe between minus1 and 1 The return angle is inradians between 0 and π
atan(x) Arctangent or inverse tangent of x The returnangle is in radians between minusπ2 and π2
atan(xy) Arctangent or inverse tangent of xy The re-turn angle is in radians between minusπ and π de-pending on the sign of x and y
sec(x) Secant (1(cos x)) of x where x is in radianscsc(x) Cosecant (1(sin x)) of x where x is in radianscotg(x) Cotangent (1(tan x)) of x where x is in radians
Table B7 Some of Scilabrsquos utility functionsFunction Description
clear Clear variables from memoryhelp Display the first set of comment lines from
scriptswho List the variables in memorywhos List the variables in memory with their sizeerror Terminates the execution of the Sci-file and out-
puts the supplied error message
functions round(x) fix(x) floor(x) and ceil(x)
Exercise B5 The functions listed in Tables B5 B6 are only asmall sample of all of Scilabrsquos mathematical functions
Using Scilabrsquos help command list all of the hyperbolic func-tions available Also list any restrictions on the values of thefunction arguments
B52 Utility functions
In addition to the arithmetic functions Scilab has several utilityfunctions (see Table B7) These functions allow users to trackvariables (whoswho) feedback an error message if an error occursin the code (error) clear the variables form the memory (clear)etc Essential these functions allow users to debug and performbookkeeping operations inside their code
B6 Visualising data in Scilab 241
6
5
4- 04- 8
3
- 02
- 6
00
2- 4 Y- 2
02
Z
10
04
X 02
06
4 - 1
08
6- 28
10
Figure B2 Plot of z = sin(radicx2 + y2)
radicx2 + y2 illustrating the
3d capabilities of Scilab
B6 Visualising data in Scilab
One of the big advantages in using Scilab is the ability to visualisedata and results of complex calculations (see Figure B2 as exampleof the 3d capabilities) While Scilab is capability of complex 3dgraphics (see help pages for mesh and surf) in this section we willmesh amp surf
focus on the 2d plotting command (plot) Hence in this sectionplot
we will focus on
1 plotting mathematical functions from within Scilab
2 changing stylistic elements of the plot
3 adding titles labels and legends to the plot
4 including mathematical notation in titles labels and legendsand
5 plotting data stored in an external ASCII file
B61 Plotting mathematical functions from within Scilab
To plot a mathematical function in Scilab three steps are requiredThese are
1 Define the domain of the function (eg x=linspace(02pi))
2 Calculate the values of function at the selected points in thedomain (eg y=sin(x))
242 Appendix B Programming with Scilab
3 Final step is to plot the function using the plot commandplot
(eg plot(xy))
Using these steps it is possible to produce a basic plot of any math-ematical function (see Figure B3) Note that the plot shown inFigure B3 does not have a title axes labels or a legend all ofwhich are crucial for a good plot The Scilab commands to addthese elements to the plot are given in the following sections
- 10
- 08
- 06
- 04
- 02
00
02
04
06
08
10
0 1 2 3 4 5 6 7
Figure B3 The default plot of y = sin(x) in Scilab Notethat by default axes are not labelled and there is no title orlegend These need to added using the additional commandsxlabelylabeltitle and legend
Changing stylistic elements of the plot
An optional argument can be given to the plot command to cus-tomise each new line aspect This argument is a concatenatedstring containing information about colour line style or markersbased on the specifiers shown in Table B8 To specify the plotbe drawn with a red longdash-dot line with diamond marker thestring that would be passed to plot command would be rsquor-drsquo Thestring which is a concatenation (in any order) of the three typesof properties shown in Table B8 must be unambiguous Further-more the string specification is not case sensitive An exampleof specifying various line styles is shown in Example B6 and Fig-ure B4
Example B6 The following commands illustrate how line stylescan be used to distinguish between the plots of sin(2πt) sin(2πt+π6) and sin(2πt+ π2)
t=linspace(0130)
B6 Visualising data in Scilab 243
Table B8 Plot specifiers required to change line styles coloursand markers in plotsSpecifier Line style Specifier Colour Specifier Marker stylendash Solid line (default) r Red + Plus signndash ndash Dashed line g Green o Circle Dotted line b Blue Asteriskndash Dash-dotted line c Cyan x Cross
m Magenta s Squarey Yellow rsquononersquo No marker(default)k Blackw White
Figure B4 Resulting plotting after issuing the commands givenin Example B6
244 Appendix B Programming with Scilab
Example B6 illustrate a number of key commands relating to thecreation of multiple plots in Scilab The mtlb hold4 commandmtlb hold
holds the current plot and axis properties so that all subsequentplots are added to the existing plot
To distinguish each plot a different line specification is passed eachplot command In the example y = sin(2πt) is plotted using a redsolid line with data points indicated using a red circle as indicatedby the string rsquor-orsquo y = sin(2πt + π6) is plotted using greendashed-dotted line with data points marked by a square (rsquog-srsquo)while y = sin(2πt+ π2) is plotted using a cyan dashed line usingdiamonds to mark the data points (rsquoc--drsquo) Finally to completethe plot axes labels and a legend has been added to plot the com-mands for doing this are discussed in sectB62
B62 Adding titleslabels and legends to the plot
All good plots should have the following properties
1 The style of figure is appropriate to the information beingconveyed
2 The contents are well organised so it is easy for readers tointerpret the information
3 Legends labels etc are clear and concise (eg contain unitsif required do not obstruct parts of the plot etc) and
4 A caption or title that summarises the information presented
Like most mathematical applications Scilab has a wide variety ofdifferent plotting styles that can be used (eg look up the help forbar polarplot etc) This allows you to select the most appropri-barpolarplot
ate style to display and organise your data
The Scilab commands xlabel ylabel title and legend al-xlabel ylabel title and
legend lows you to add axes labels titles and legends to plot As shownin Example B6 the commands xlabel ylabel and title takea string input containing the title or label text For the legend
command a string is required for each separate plot Note theorder of the strings in the legend command reflects the order inwhich the each of plots were made
Including mathematical notation in titles labels and legends
It is possible to include mathematical characters such as π inScilab plots This is done by inserting special commands intoeither the title legend or label string These commands are basedon the abbreviations for mathematical notation used in programssuch as LATEX MathType etc A brief summary of the commands
4 mtlb hold on is equivalent to the Scilab commandset(gca()auto clearoff) while mtlb hold off is equivalentto the Scilab command set(gca()auto clearon)
B6 Visualising data in Scilab 245
Table B9 Basic mathematical symbols available in Scilab forincluding mathematical notation in titles labels and legendscommand symbol command symbol command symbolalpha α upsilon υ sim sim
Im = cup cup wp weierpotimes otimes subseteq sube oslash cap cap in isin supseteq supesupset sup lceil d subset subint
intcdot middot o oslash
rfloor c neg 6= nabla nablalfloor b times times ldots perp perp surd
radicprime prime
wedge and varpi $ O Oslashrceil e rangle 〉 mid |
vee or copyright ccopy langle 〈
available in Scilab for typesetting mathematical notation is givenin Table B9 To insert these commands into a text string simplystart and end the string with $ signs as shown in Example B7
Example B7 Scilab commands to produce the labelled plot ofy = sin
There are a few key concepts to note about the Example B7 Thefirst can be seen in the string being passed to the title commandtitle
In this string sin is prefixed with a backslash () This is doneso that the letters for sin are not typeset in italic which shouldonly be used for variables and not common mathematical func-tions Table B10 shows the command sequences to use for themore common mathematical functions5 To get the fraction in thetitle the command fracab has been used to produce a
b Ifsuperscripts and subscripts are required in a title or label these aresuperscripts amp subscripts
obtained by adding the subscript character ( ) and the superscriptcharacter (^) to the string These characters modify the characteror substring defined in braces immediately following For examplex 2 gives x2 while e^2x gives e2x
Note in Example B7 the commands xlabel and ylabel has beenxlabelylabel
used to set the axes labels in Figure B5 Instead of using thesethree commands the title and the axes labels could have been setusing the single xtitle command (see help xtitle for details)xtitle
In the example we have also customised the font size to make thetext more readable by a) passing an option to the title xlabelylabel commands and b) by using the command xset to set thexset
default font size before executing the xstring command Finallythe xstring command adds the text string to plot at the locationxstring
specified by variables b and c In the example the coordinates wewant to add the text are obtained by executing the xclick functionxclick
and clicking on the point required in the plot window
5 Additional details can be found in books covering Mathematics in LATEX
B6 Visualising data in Scilab 247
Table B10 Command sequences for the common mathematicalfunctions in Scilabarccos arcsin arctan cos cot
csc exp lim ln log
max min sec sin tan
B63 Plotting numeric data stored in an ASCII file
As data can be stored in a wide variety of ways in external filesScilab has a wide variety of functions which can be used to importthe data stored in these files (see help for the commands read
fscanf and xlsread) Since data can be stored in such a wideread fscanf and xlsread
range of formats we will confine our discussions here to the situa-tion where numeric data is stored in a ASCII (text) file where eachrow of the file contains the same number of entries
To illustrate the procedure for plotting data stored in ASCII file wewill consider the data obtained measuring the temperature of a cupof coffee every minute for a total of 100 mins The data recorded isstored in the a file called coffeedatatxt6 which simply containsthe two columns of numbers shown in Table B11 As an examplethe first 10 lines of the text file contains the following numbers7
00000 1000000
10000 950000
20000 903125
30000 859180
40000 817981
50000 779357
60000 743147
70000 709201
80000 677376
90000 647540
Note the current version of Scilab has a builtin text editor whichcan be used to view such text files To access this editor simplytype the command edit at the command prompt Once open youedit
can use the graphical interface to open the required file
The first issue is to get the data into Scilab so it is available forplot command In Scilab the command read can used to importread
the data and assign it to a variable In the current example wewould use the following command to import the data and assignit to a variable called cdata assuming the data is in the currentdirectory8
6 The file coffeedatatxt has been attached and should be available for savingprovided you are using Adobe Reader To save the file right click on thepurple Windows version or the red MacOSLinux version in the margin andselect save file from the options Ensure you save the file as coffeedatatxt
7 normally you will have received the data file in this format8 You can check what directory Scilab is in by typing the command pwd
Figure B6 Temperature of coffee based on the data recorded inthe file coffeedatatxt
cdata=read(rsquocoffeedatatxtrsquo-12)
The -12 following the name of the file tells Scilab that you donrsquotknow how many rows you have but you have two columns of dataSince we know that we have 101 rows of data we could have alsoused the following command to load the data
cdata=read(rsquocoffeedatatxtrsquo1012)
We now have a variable called cdata which has two columns ofdata The first column contains the time while the second containsthe Temperature records Hence we can plot it using the followingcommand
plot(cdata(1)cdata(2))
This gives the plot shown in Figure B6 after appropriate labelshave been added as described above
B7 Script files
Executing commands entered at the keyboard or keypad is themost primitive use of a calculating device It is only when se-quences of operations (to perform complex tasks) are stored awayand re-called for repeated application with different inputs that wehave a programmed operation Complex programs must necessarilyhave an interface with the user The basic interface tool in Scilabcapable of executing sequences of commands that are stored in a fileis the script The file names must have an extension of sce forexample program1sce assignment1sce These files are calledScript Sci-files or simply scripts
A script file is a file of standard keyboard charactersthat define commands to drive a program A script filecan be generated using any text editor or word proces-sor
To invoke a script in Scilab type the commandexec
exec filenamesce
where filename denotes the actual name of the file containing thescript When the script runs it is exactly the same as if you typedthe commands into the command window from the keyboard Thismeans that the Sci-file can use any variables you have defined priorto invoking the script Any variables created in the script are alsoavailable when the script is finished and any variable whose namewas assigned a value in the script will be over-written
If you give a script the same name as any variable that it computesin the running of the script then Scilab will give precedence tothe workspace variable over a script As stated in sectB3 the variablename will hide the script name after the script has run Scilab willtake the name as referring to the variable and not to the definedscript
Scilab also needs to know where to find the script to executedIf Scilab cannot find the script it will report that you have anundefined function or variable
The exact procedure for creating and saving Sci-files depends onthe Scilab version the operating system (MS Windows Macin-tosh or UnixLinux ) and the text editor you use The currentversion of Scilab includes a built-in editor which can be used tocreate and edit Sci-files and text files
To create a new Sci-file under the current version of Scilab simplytype edit newfilenamesce This will open up the Scilab editoredit
with a blank document called newfilename
B71 Script input and output
A script is a named file (with the extension sce) containing asequence of Scilab instructions which when its name is executedin a session performs those instructions in order as though theywere entered at the console
There are two common uses for scripts Pragmatically we maywant to save a particular computational sequence for later use orreference If the tasks involved are to be repeated often thenthe nature of the tasks should be analysed formulated into usefulfunctions (which we will discuss later) and the whole procedurepresented in a single function But if the tasks are only likely toget occasional use then the script may be adequately commentedand stored as a script file It is a quick and dirty way of saving
B7 Script files 251
the instructions for doing a particular job when we do not wantto go to the trouble of writing the functions for the job It is nota highly recommended modus operandi but useful all the same
The second use is for the script to act like a storyboard presentinginformation to a user in a structured and controlled way permit-ting the user to see the steps of a computational procedure andtheir outputs up on the screen At a more sophisticated level thescript takes on the proportions of an application controlling a largeand complex interaction with advanced interface features We willnot take our study of Scilab programming to this level but workmostly at the storyboard level using the simplest of Scilabrsquos in-terface tools
All the variables created in a script become global variables definedin the workspace and available for perusal As an application itis assumed that a script controls the entire workspace and thevariables it creates are lsquothe datarsquo of the application When scriptsare used as storyboards on a small scale then it cannot be assumedthat the variables they create deserve global recognition and it iscourteous to erase these variables when the script is finished unlessthey have some possible further use An unfortunate side-effect isthat if variables with the same name already exist in the workspacebefore a script is run then these variables will be over-written andlost
Used as storyboards scripts allow the user to cut paste and re-trythe various steps of a procedure and to modify the script to suithis own needs Storyboards may be used for demonstrations oftoolbox utilities or extended documentationexplanation of func-tions However if there is any danger that variables defined bythe script may over-write workspace variables then the demo ordocumentation should be re-fashioned into a function that leavesthe workspace variables untouched (more on this later)
Scripts will often read data from pre-prepared files and we will dis-cuss the functions provided by Scilab to perform this task below
Gathering data entered by the user really falls into the realm ofinterface design but we only briefly consider the simplest interfacefunction provided by Scilab Similarly we briefly consider thefunctions for providing the simplest forms of output presentation
Keyboard input
Scilab has a built-in function input that can be used to promptinput
the user to enter the value of a variable from the keyboard Forexample
Tc = input(rsquoPlease enter the temperature in degrees Celsiusrsquo)
will print the message
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
232 Appendix B Programming with Scilab
Table B2 Scilab Arithmetic OperatorsOperator Meaning Scilab
^ Exponentiation xy x^y
Multiply xtimes y xy
Right Division xy = xy xy
Left Division yx = xy yx
+ Addition x+ y x+y
- Subtraction xminus y x-y
of a calculation will be stored in a special variable called ans (seeans
Table B1) The value stored in this variable is overridden the nexttime a calculation is done that is not assigned to a variable Thismeans that the result from the previous calculation is then lost
B4 Arithmetic expressions
Table B2 lists the basic arithmetic operators to be found in ScilabWith the exception of the left division operator the arithmetic op-erators in Scilab follow standard notation
B41 Precedence
When more than one operation appears in a mathematical state-ment the programming language needs to define in which order theoperations must be performed or the precedence That is whichparts are evaluated first
If precedence was not defined then the same statement can be eval-uated to a different result for example is 186+3 equal to 2 oris 186+3 equal to 6 The first statement is evaluated from theright using the precedence rule that usually applies in mathemat-ical statements that is work from right to left as for example inlsquosin log tan xrsquo where we start with a value of x compute the tan-gent of that value (tan x) then take the logarithm of the resultingnumber (log(tan x)) and finally take the sine of what this deliv-ers However it is usual (because it is much easier to write poly-nomials with this convention) to give multiplication and divisionprecedence over addition and subtraction The second statementrecognises the precedence of the division over addition and evalu-ates it first Right to left precedence has a much simpler parsingtree than schemes which give precedence to particular operatorsbut the common arithmetic precedences are so widely acceptedand expected that most computer language designers will not riskthe possible lsquoconsumer backlashrsquo that breaking with this traditionwould involve So with every computer language one must searchout and learn the precedence rules that it adopts The usual arith-metic precedences will normally be observed but note that even
B4 Arithmetic expressions 233
Table B3 Precedence of arithmetic expressionsOperator Precedence Order of evaluation
() 1 Left to right innermost first 2 Left to right 3 Left to right 4 Right to left
+ minus 5 Left to right
they are not unambiguous For example which divide has prece-dence in 41632 Working right to left one would get the answerof 8 but Scilab returns (the decimal for) 1128 mdashit works fromleft to right Table B3 lists the operators and their precedencevalue
The rules for evaluation of operators are as follows
bull Operators having high precedence (low numerical value) willbe evaluated before those having a low precedence (high nu-merical value)
bull If an expression contains more than one operator at the samelevel of precedence then the left-to-right rule of evaluationapply
Parentheses () always have the highest precedence Whateveris within the parentheses is evaluated first This means when indoubt about the order of any statement place parentheses aroundthe statement to ensure that the operations are performed in theorder you expect
B42 Matrix arithmetic
One of the strengths of Scilab is its ability to execute matrixarithmetic3 Matrix arithmetic must be implemented in other lan-guages with many lines of code or special functions Scilab isespecially designed to handle matrix arithmetic The representsmatrix multiplication or scalar multiplication of a matrix if one ofthe arguments is a scalar Since scalars are 1times1 matrices ordinaryscalar multiplication can also be performed by the operator Thedot product of two vectors however requires that the first vectorbe a row vector and the second a column vector To multiply thecorresponding elements in two vectors together requires a differentoperator so that
gt a = [2 3 4] b = [6 7 8]
a b
ans =
12 21 32
3 Remember vectors are examples of matrices A column vector of length nis an example of a ntimes 1 matrix
234 Appendix B Programming with Scilab
However because it is a matrix operation
gt a + b
ans =
8 10 40
The standard arithmetic operators described in Table B2 are alsoused with matrices The normal rules of matrix addition subtrac-tion and multiplication apply
The left and right division of matrices are also generalisations ofscalar division Right division BA is roughly the same as Binv(A)Left division AB is roughly the same as inv(A)B The differenceis in how the division is performed Different algorithms are usedwhen A or B are square or not
Scilab also defines a transpose operator for matrices The trans-pose of the matrix A is Arsquo For example if
A =
[25 63
12 09
] then A prime =
[25 12
63 09
]
Accessing matrix elements
Individual elements of a matrix can accessed and used in state-ments by specifying the matrix index of an element The index ofan element is its position in the matrix For example consider thefollowing matrix
A =
6 minus1 4
3 5 7
minus3 6 1
Define this in Scilab with the command
A=[ 6 -1 4 3 5 7 -3 6 7]
to display the contents of matrix A in Scilab just type the variablename A To display or use one element from the matrix specify theelement For example A(13) specifies the element in the first rowand third column that is 4 for the above matrix
To change a specific entry just assign a value to it For example
A(31)=-9
changes the value of the third row first column element from -3
to -9
To display or use a whole row or column use the operator Forexample
gt A(2)
ans =
3 5 7
gt A(2)
B4 Arithmetic expressions 235
ans =
-1
5
6
It is also possible to pick out a sub-matrix by the specifying the rowand column entries required For example To pick out the sub-matrix of A containing the elementsA11 A13 A21 A23 the commandA(12123) could be used as shown below
gt A(12123)
ans =
6 4
3 7
Note 123 means from 1 to 3 in steps of 2 This technique alongwith the assignment operator can be used for values of whole rowscolumns or sub-matrices in one command For example
gt A(12123)=[10111213]
A =
10 -1 11
12 5 13
-3 6 7
Note that the elements A11 A13 A21 A23 now contain the values10 11 12 and 13 respectively instead of their previous values
Index notation is logically an unfortunate phenomenon since itclashes with standard function notation On the face of it A(3)
should be the value of a function called A with input 3 but whenit is interpreted as index notation A is a variable However indexnotation is so familiar and useful to those familiar with classicalmathematical notation that many developers of mathematical soft-ware are not prepared to risk a ldquoconsumer backlashrdquo to providinga functional way of extracting elements from an array such as forexample 3 pick A to pick out the third element of A Similarlywe as teachers feel we would be doing students a disservice if wefailed to introduce them to the index notation that abounds inmathematical texts usually in the form of subscripting as in A3Logic does not always win an argument In the commercial worldof computing an established user base carries more weight thanmathematical logic just as in mathematics tradition may well alsofly in the face of logical consistency
B43 Array arithmetic
Scilab has two forms of arithmetic operations that can be usedwith matrices matrix arithmetic or array arithmetic Matrix arith-metic follows the normal rules of matrix algebra Array arithmeticwhich is also called element-by-element arithmetic is where oper-ations are performed on matrices element by element
236 Appendix B Programming with Scilab
Table B4 Scilab array operatorsOperator Meaning Scilab
^ Exponentiation x^y
Multiply xy
Right Division xy
Left Division yx
+ Addition x+y
- Subtraction x-y
Table B4 lists the Scilab array operators
The array addition and subtraction are identical to matrix additionor subtraction Both array and matrix addition and subtractionoperate element by element Array addition and subtraction re-quire that both arrays have the same size The only exception tothis rule in Scilab occurs when we add or subtract a scalar to orfrom an array In this case the scalar is added or subtracted fromeach element in the array
For example
gt [6 5 6] + 3
ans =
[9 8 9]
gt [6 4 5 8] + [3 1 2 4]
ans =
[9 5 7 12]
gt [4 3] - 7
ans =
[-3 -4]
Array multiplication and division is very different to matrix mul-tiplication and division Array multiplication or division is an ele-ment by element multiplication or division that produces an arraywith the same dimensions as the starting arrays As with addi-tion and subtraction array multiplication and division require thatboth arrays have the same dimensions
Example B2 Consider the two row vectors x=[2 4 7 1] andy=[6 1 3 2] Scilab does not recognise the possibility ofmatrix multiplication of these two vectors They are not com-patible the number of columns of the pre-multiplier must bethe same as the number of rows of the post-multiplier Anattempt to execute x y will bring an error message
Obtain the usual dot product of the two row vectors by trans-posing the post-multiplier as in xyrsquo or yxrsquo The matrixproduct xrsquoy produces a matrix of 4 rows and 4 columns
B5 Builtndashin functions 237
Array multiplication produces the result
gt xy
ans =
[12 4 21 2]
by multiplying corresponding elements of each vector Theresult is a row vector with the same dimensions as x and ythat is a 1times 4 vector
Example B3 Consider the row vectors x=[2 4 7 1] and col-umn vector y=[6 1 3 2]rsquo Array multiplication of thesetwo matrices is not possible because their dimensions do notagree
Matrix multiplication xy produces the result 39 that is theinner product or dot product of the vectors Note vectorsas such do not exist in Scilab there are only single rowedor single columned arrays
Scilab not only allows arrays to be raised to powers but alsoallows scalars and arrays to be raised by array powers To performexponentiation on an element-by-element basis use the ^ symbolFor example if x=[3 5 8] then x^3 produces the matrix [33 53
83] = [27 125 512]
A scalar can be raised to an array power For example if p=[2 4 5]then typing 3^p produces the array [32 34 35] = [9 81 243]
With element-by-element array operations it is important to re-member that the dot () is a part of a two-character symbol for aparticular array operator such as () () () (^) Sometimesthis can cause some confusion the dot in 3^p is not a decimalpoint associated with the number 3 The following operations withthe value of p=[2 4 5] are equivalent
3^p
30^p
3^p
(3)^p
3^[2 4 5]
B5 Builtndashin functions
B51 Arithmetic functions
Arithmetic expressions often require computations other than ad-dition subtraction multiplication division and exponentiationFor example many expressions require the use of logarithms ex-ponentials and trigonometric functions Scilab has predefined alarge number of functions that perform these types of computations(see Tables B5 and B6)
238 Appendix B Programming with Scilab
An example of a predefined function is sin which returns the sineof a given angle
m = sin(angle)
where the variable angle contains the angle in radians and theresult is stored in the variable m The variable angle is called theargument or parameter of the function If the angle is stored indegrees it must be converted to radians for example
m = sin(anglepi180)
or
radians =anglepi180
m = sin(radians)
In Scilab the arguments or parameters of the function are con-tained in parentheses following the name of the function This isreferred to as pre-fix notation Many common functions in math-ematics have a syntax using in-fix notation where two argumentsappear to the right and left of the function name as for examplein x+y atimes b nmod r etc Scilab does not support in-fix nota-tion for any but the arithmetic functions Classical mathematicalnotation also has examples of post-fix notationmdash n x squared mdashand other hybrids like |x| while Scilab has ahb to confound thepurist
A function may contain no arguments one argument or many ar-guments depending on its definition If a function contains morethan one argument it is vital to give the arguments in the cor-rect order For example consider the problem of finding θ in thefollowing triangle using the atan command
5
8θ
Now by hand calculation the answer is 05586 radians If we useatan as below we get the answers shown
--gt atan(58)
ans =
05586
--gt atan(85)
ans =
10122
The first answer is the correct one The second answer is incorrect
B5 Builtndashin functions 239
Table B5 Some common mathematical functions defined inScilabFunction Description
abs(x) Computes the absolute value of xsqrt(x) Square root of x The parameter can be nega-
tiveround(x) Round x to the nearest integerfix(x) Round x to the nearest integer toward zerofloor(x) Round x to the nearest integer toward minusinfinceil(x) Round x to the nearest integer toward +infinsign(x) Returns -1 if x is less than zero zero if x equals
zero and 1 otherwisemodulo(xy) Returns the remainder of xy For example
modulo(254) is 1exp(x) Returns ex
log(x) Returns the natural logarithm of x That isthe logarithm to the base e Parameter can benegative but cannot be zero
log10(x) Returns the common logarithm of x That isthe logarithm to the base 10 Parameter can benegative but cannot be zero
since we passed the values to the function in the incorrect order
Some functions also require that the arguments be in specific unitsFor example the trigonometric functions assume that the argu-ments are in radians
In Scilab some functions use the number of arguments to deter-mine the output of the function For example the zeros functioncan have one or two arguments which then determine the output(check this with the command help zeros)
A function reference cannot be made on the left hand side of anassignment since a function call always returns a value(s) whereasan assignment is always to a variable Functions can of courseappear on the right side of an assignment since they return valuesto be assigned to some variable Functions may be applied tothe output of other functions and may be called in the algorithmdefining a function for example
logx = log(abs(x))
When one function is used to compute the argument of anotherfunction be sure to enclose the argument of each function in itsown set of parentheses
Scilab allows recursive functions that is functions that call them-selves in their defining algorithm
Exercise B4 Use examples to show the differences between the
240 Appendix B Programming with Scilab
Table B6 Some common trigonometric functions defined inScilab
Function Description
sin(x) Sine of x where x is in radianscos(x) Cosine of x where x is in radianstan(x) Tangent of x where x is in radiansasin(x) Arcsine or inverse sine of x where x must be
between -1 and 1 The return angle is in radiansbetween minusπ2 and π2
acos(x) Arccosine or inverse cosine of x where x mustbe between minus1 and 1 The return angle is inradians between 0 and π
atan(x) Arctangent or inverse tangent of x The returnangle is in radians between minusπ2 and π2
atan(xy) Arctangent or inverse tangent of xy The re-turn angle is in radians between minusπ and π de-pending on the sign of x and y
sec(x) Secant (1(cos x)) of x where x is in radianscsc(x) Cosecant (1(sin x)) of x where x is in radianscotg(x) Cotangent (1(tan x)) of x where x is in radians
Table B7 Some of Scilabrsquos utility functionsFunction Description
clear Clear variables from memoryhelp Display the first set of comment lines from
scriptswho List the variables in memorywhos List the variables in memory with their sizeerror Terminates the execution of the Sci-file and out-
puts the supplied error message
functions round(x) fix(x) floor(x) and ceil(x)
Exercise B5 The functions listed in Tables B5 B6 are only asmall sample of all of Scilabrsquos mathematical functions
Using Scilabrsquos help command list all of the hyperbolic func-tions available Also list any restrictions on the values of thefunction arguments
B52 Utility functions
In addition to the arithmetic functions Scilab has several utilityfunctions (see Table B7) These functions allow users to trackvariables (whoswho) feedback an error message if an error occursin the code (error) clear the variables form the memory (clear)etc Essential these functions allow users to debug and performbookkeeping operations inside their code
B6 Visualising data in Scilab 241
6
5
4- 04- 8
3
- 02
- 6
00
2- 4 Y- 2
02
Z
10
04
X 02
06
4 - 1
08
6- 28
10
Figure B2 Plot of z = sin(radicx2 + y2)
radicx2 + y2 illustrating the
3d capabilities of Scilab
B6 Visualising data in Scilab
One of the big advantages in using Scilab is the ability to visualisedata and results of complex calculations (see Figure B2 as exampleof the 3d capabilities) While Scilab is capability of complex 3dgraphics (see help pages for mesh and surf) in this section we willmesh amp surf
focus on the 2d plotting command (plot) Hence in this sectionplot
we will focus on
1 plotting mathematical functions from within Scilab
2 changing stylistic elements of the plot
3 adding titles labels and legends to the plot
4 including mathematical notation in titles labels and legendsand
5 plotting data stored in an external ASCII file
B61 Plotting mathematical functions from within Scilab
To plot a mathematical function in Scilab three steps are requiredThese are
1 Define the domain of the function (eg x=linspace(02pi))
2 Calculate the values of function at the selected points in thedomain (eg y=sin(x))
242 Appendix B Programming with Scilab
3 Final step is to plot the function using the plot commandplot
(eg plot(xy))
Using these steps it is possible to produce a basic plot of any math-ematical function (see Figure B3) Note that the plot shown inFigure B3 does not have a title axes labels or a legend all ofwhich are crucial for a good plot The Scilab commands to addthese elements to the plot are given in the following sections
- 10
- 08
- 06
- 04
- 02
00
02
04
06
08
10
0 1 2 3 4 5 6 7
Figure B3 The default plot of y = sin(x) in Scilab Notethat by default axes are not labelled and there is no title orlegend These need to added using the additional commandsxlabelylabeltitle and legend
Changing stylistic elements of the plot
An optional argument can be given to the plot command to cus-tomise each new line aspect This argument is a concatenatedstring containing information about colour line style or markersbased on the specifiers shown in Table B8 To specify the plotbe drawn with a red longdash-dot line with diamond marker thestring that would be passed to plot command would be rsquor-drsquo Thestring which is a concatenation (in any order) of the three typesof properties shown in Table B8 must be unambiguous Further-more the string specification is not case sensitive An exampleof specifying various line styles is shown in Example B6 and Fig-ure B4
Example B6 The following commands illustrate how line stylescan be used to distinguish between the plots of sin(2πt) sin(2πt+π6) and sin(2πt+ π2)
t=linspace(0130)
B6 Visualising data in Scilab 243
Table B8 Plot specifiers required to change line styles coloursand markers in plotsSpecifier Line style Specifier Colour Specifier Marker stylendash Solid line (default) r Red + Plus signndash ndash Dashed line g Green o Circle Dotted line b Blue Asteriskndash Dash-dotted line c Cyan x Cross
m Magenta s Squarey Yellow rsquononersquo No marker(default)k Blackw White
Figure B4 Resulting plotting after issuing the commands givenin Example B6
244 Appendix B Programming with Scilab
Example B6 illustrate a number of key commands relating to thecreation of multiple plots in Scilab The mtlb hold4 commandmtlb hold
holds the current plot and axis properties so that all subsequentplots are added to the existing plot
To distinguish each plot a different line specification is passed eachplot command In the example y = sin(2πt) is plotted using a redsolid line with data points indicated using a red circle as indicatedby the string rsquor-orsquo y = sin(2πt + π6) is plotted using greendashed-dotted line with data points marked by a square (rsquog-srsquo)while y = sin(2πt+ π2) is plotted using a cyan dashed line usingdiamonds to mark the data points (rsquoc--drsquo) Finally to completethe plot axes labels and a legend has been added to plot the com-mands for doing this are discussed in sectB62
B62 Adding titleslabels and legends to the plot
All good plots should have the following properties
1 The style of figure is appropriate to the information beingconveyed
2 The contents are well organised so it is easy for readers tointerpret the information
3 Legends labels etc are clear and concise (eg contain unitsif required do not obstruct parts of the plot etc) and
4 A caption or title that summarises the information presented
Like most mathematical applications Scilab has a wide variety ofdifferent plotting styles that can be used (eg look up the help forbar polarplot etc) This allows you to select the most appropri-barpolarplot
ate style to display and organise your data
The Scilab commands xlabel ylabel title and legend al-xlabel ylabel title and
legend lows you to add axes labels titles and legends to plot As shownin Example B6 the commands xlabel ylabel and title takea string input containing the title or label text For the legend
command a string is required for each separate plot Note theorder of the strings in the legend command reflects the order inwhich the each of plots were made
Including mathematical notation in titles labels and legends
It is possible to include mathematical characters such as π inScilab plots This is done by inserting special commands intoeither the title legend or label string These commands are basedon the abbreviations for mathematical notation used in programssuch as LATEX MathType etc A brief summary of the commands
4 mtlb hold on is equivalent to the Scilab commandset(gca()auto clearoff) while mtlb hold off is equivalentto the Scilab command set(gca()auto clearon)
B6 Visualising data in Scilab 245
Table B9 Basic mathematical symbols available in Scilab forincluding mathematical notation in titles labels and legendscommand symbol command symbol command symbolalpha α upsilon υ sim sim
Im = cup cup wp weierpotimes otimes subseteq sube oslash cap cap in isin supseteq supesupset sup lceil d subset subint
intcdot middot o oslash
rfloor c neg 6= nabla nablalfloor b times times ldots perp perp surd
radicprime prime
wedge and varpi $ O Oslashrceil e rangle 〉 mid |
vee or copyright ccopy langle 〈
available in Scilab for typesetting mathematical notation is givenin Table B9 To insert these commands into a text string simplystart and end the string with $ signs as shown in Example B7
Example B7 Scilab commands to produce the labelled plot ofy = sin
There are a few key concepts to note about the Example B7 Thefirst can be seen in the string being passed to the title commandtitle
In this string sin is prefixed with a backslash () This is doneso that the letters for sin are not typeset in italic which shouldonly be used for variables and not common mathematical func-tions Table B10 shows the command sequences to use for themore common mathematical functions5 To get the fraction in thetitle the command fracab has been used to produce a
b Ifsuperscripts and subscripts are required in a title or label these aresuperscripts amp subscripts
obtained by adding the subscript character ( ) and the superscriptcharacter (^) to the string These characters modify the characteror substring defined in braces immediately following For examplex 2 gives x2 while e^2x gives e2x
Note in Example B7 the commands xlabel and ylabel has beenxlabelylabel
used to set the axes labels in Figure B5 Instead of using thesethree commands the title and the axes labels could have been setusing the single xtitle command (see help xtitle for details)xtitle
In the example we have also customised the font size to make thetext more readable by a) passing an option to the title xlabelylabel commands and b) by using the command xset to set thexset
default font size before executing the xstring command Finallythe xstring command adds the text string to plot at the locationxstring
specified by variables b and c In the example the coordinates wewant to add the text are obtained by executing the xclick functionxclick
and clicking on the point required in the plot window
5 Additional details can be found in books covering Mathematics in LATEX
B6 Visualising data in Scilab 247
Table B10 Command sequences for the common mathematicalfunctions in Scilabarccos arcsin arctan cos cot
csc exp lim ln log
max min sec sin tan
B63 Plotting numeric data stored in an ASCII file
As data can be stored in a wide variety of ways in external filesScilab has a wide variety of functions which can be used to importthe data stored in these files (see help for the commands read
fscanf and xlsread) Since data can be stored in such a wideread fscanf and xlsread
range of formats we will confine our discussions here to the situa-tion where numeric data is stored in a ASCII (text) file where eachrow of the file contains the same number of entries
To illustrate the procedure for plotting data stored in ASCII file wewill consider the data obtained measuring the temperature of a cupof coffee every minute for a total of 100 mins The data recorded isstored in the a file called coffeedatatxt6 which simply containsthe two columns of numbers shown in Table B11 As an examplethe first 10 lines of the text file contains the following numbers7
00000 1000000
10000 950000
20000 903125
30000 859180
40000 817981
50000 779357
60000 743147
70000 709201
80000 677376
90000 647540
Note the current version of Scilab has a builtin text editor whichcan be used to view such text files To access this editor simplytype the command edit at the command prompt Once open youedit
can use the graphical interface to open the required file
The first issue is to get the data into Scilab so it is available forplot command In Scilab the command read can used to importread
the data and assign it to a variable In the current example wewould use the following command to import the data and assignit to a variable called cdata assuming the data is in the currentdirectory8
6 The file coffeedatatxt has been attached and should be available for savingprovided you are using Adobe Reader To save the file right click on thepurple Windows version or the red MacOSLinux version in the margin andselect save file from the options Ensure you save the file as coffeedatatxt
7 normally you will have received the data file in this format8 You can check what directory Scilab is in by typing the command pwd
Figure B6 Temperature of coffee based on the data recorded inthe file coffeedatatxt
cdata=read(rsquocoffeedatatxtrsquo-12)
The -12 following the name of the file tells Scilab that you donrsquotknow how many rows you have but you have two columns of dataSince we know that we have 101 rows of data we could have alsoused the following command to load the data
cdata=read(rsquocoffeedatatxtrsquo1012)
We now have a variable called cdata which has two columns ofdata The first column contains the time while the second containsthe Temperature records Hence we can plot it using the followingcommand
plot(cdata(1)cdata(2))
This gives the plot shown in Figure B6 after appropriate labelshave been added as described above
B7 Script files
Executing commands entered at the keyboard or keypad is themost primitive use of a calculating device It is only when se-quences of operations (to perform complex tasks) are stored awayand re-called for repeated application with different inputs that wehave a programmed operation Complex programs must necessarilyhave an interface with the user The basic interface tool in Scilabcapable of executing sequences of commands that are stored in a fileis the script The file names must have an extension of sce forexample program1sce assignment1sce These files are calledScript Sci-files or simply scripts
A script file is a file of standard keyboard charactersthat define commands to drive a program A script filecan be generated using any text editor or word proces-sor
To invoke a script in Scilab type the commandexec
exec filenamesce
where filename denotes the actual name of the file containing thescript When the script runs it is exactly the same as if you typedthe commands into the command window from the keyboard Thismeans that the Sci-file can use any variables you have defined priorto invoking the script Any variables created in the script are alsoavailable when the script is finished and any variable whose namewas assigned a value in the script will be over-written
If you give a script the same name as any variable that it computesin the running of the script then Scilab will give precedence tothe workspace variable over a script As stated in sectB3 the variablename will hide the script name after the script has run Scilab willtake the name as referring to the variable and not to the definedscript
Scilab also needs to know where to find the script to executedIf Scilab cannot find the script it will report that you have anundefined function or variable
The exact procedure for creating and saving Sci-files depends onthe Scilab version the operating system (MS Windows Macin-tosh or UnixLinux ) and the text editor you use The currentversion of Scilab includes a built-in editor which can be used tocreate and edit Sci-files and text files
To create a new Sci-file under the current version of Scilab simplytype edit newfilenamesce This will open up the Scilab editoredit
with a blank document called newfilename
B71 Script input and output
A script is a named file (with the extension sce) containing asequence of Scilab instructions which when its name is executedin a session performs those instructions in order as though theywere entered at the console
There are two common uses for scripts Pragmatically we maywant to save a particular computational sequence for later use orreference If the tasks involved are to be repeated often thenthe nature of the tasks should be analysed formulated into usefulfunctions (which we will discuss later) and the whole procedurepresented in a single function But if the tasks are only likely toget occasional use then the script may be adequately commentedand stored as a script file It is a quick and dirty way of saving
B7 Script files 251
the instructions for doing a particular job when we do not wantto go to the trouble of writing the functions for the job It is nota highly recommended modus operandi but useful all the same
The second use is for the script to act like a storyboard presentinginformation to a user in a structured and controlled way permit-ting the user to see the steps of a computational procedure andtheir outputs up on the screen At a more sophisticated level thescript takes on the proportions of an application controlling a largeand complex interaction with advanced interface features We willnot take our study of Scilab programming to this level but workmostly at the storyboard level using the simplest of Scilabrsquos in-terface tools
All the variables created in a script become global variables definedin the workspace and available for perusal As an application itis assumed that a script controls the entire workspace and thevariables it creates are lsquothe datarsquo of the application When scriptsare used as storyboards on a small scale then it cannot be assumedthat the variables they create deserve global recognition and it iscourteous to erase these variables when the script is finished unlessthey have some possible further use An unfortunate side-effect isthat if variables with the same name already exist in the workspacebefore a script is run then these variables will be over-written andlost
Used as storyboards scripts allow the user to cut paste and re-trythe various steps of a procedure and to modify the script to suithis own needs Storyboards may be used for demonstrations oftoolbox utilities or extended documentationexplanation of func-tions However if there is any danger that variables defined bythe script may over-write workspace variables then the demo ordocumentation should be re-fashioned into a function that leavesthe workspace variables untouched (more on this later)
Scripts will often read data from pre-prepared files and we will dis-cuss the functions provided by Scilab to perform this task below
Gathering data entered by the user really falls into the realm ofinterface design but we only briefly consider the simplest interfacefunction provided by Scilab Similarly we briefly consider thefunctions for providing the simplest forms of output presentation
Keyboard input
Scilab has a built-in function input that can be used to promptinput
the user to enter the value of a variable from the keyboard Forexample
Tc = input(rsquoPlease enter the temperature in degrees Celsiusrsquo)
will print the message
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
B4 Arithmetic expressions 233
Table B3 Precedence of arithmetic expressionsOperator Precedence Order of evaluation
() 1 Left to right innermost first 2 Left to right 3 Left to right 4 Right to left
+ minus 5 Left to right
they are not unambiguous For example which divide has prece-dence in 41632 Working right to left one would get the answerof 8 but Scilab returns (the decimal for) 1128 mdashit works fromleft to right Table B3 lists the operators and their precedencevalue
The rules for evaluation of operators are as follows
bull Operators having high precedence (low numerical value) willbe evaluated before those having a low precedence (high nu-merical value)
bull If an expression contains more than one operator at the samelevel of precedence then the left-to-right rule of evaluationapply
Parentheses () always have the highest precedence Whateveris within the parentheses is evaluated first This means when indoubt about the order of any statement place parentheses aroundthe statement to ensure that the operations are performed in theorder you expect
B42 Matrix arithmetic
One of the strengths of Scilab is its ability to execute matrixarithmetic3 Matrix arithmetic must be implemented in other lan-guages with many lines of code or special functions Scilab isespecially designed to handle matrix arithmetic The representsmatrix multiplication or scalar multiplication of a matrix if one ofthe arguments is a scalar Since scalars are 1times1 matrices ordinaryscalar multiplication can also be performed by the operator Thedot product of two vectors however requires that the first vectorbe a row vector and the second a column vector To multiply thecorresponding elements in two vectors together requires a differentoperator so that
gt a = [2 3 4] b = [6 7 8]
a b
ans =
12 21 32
3 Remember vectors are examples of matrices A column vector of length nis an example of a ntimes 1 matrix
234 Appendix B Programming with Scilab
However because it is a matrix operation
gt a + b
ans =
8 10 40
The standard arithmetic operators described in Table B2 are alsoused with matrices The normal rules of matrix addition subtrac-tion and multiplication apply
The left and right division of matrices are also generalisations ofscalar division Right division BA is roughly the same as Binv(A)Left division AB is roughly the same as inv(A)B The differenceis in how the division is performed Different algorithms are usedwhen A or B are square or not
Scilab also defines a transpose operator for matrices The trans-pose of the matrix A is Arsquo For example if
A =
[25 63
12 09
] then A prime =
[25 12
63 09
]
Accessing matrix elements
Individual elements of a matrix can accessed and used in state-ments by specifying the matrix index of an element The index ofan element is its position in the matrix For example consider thefollowing matrix
A =
6 minus1 4
3 5 7
minus3 6 1
Define this in Scilab with the command
A=[ 6 -1 4 3 5 7 -3 6 7]
to display the contents of matrix A in Scilab just type the variablename A To display or use one element from the matrix specify theelement For example A(13) specifies the element in the first rowand third column that is 4 for the above matrix
To change a specific entry just assign a value to it For example
A(31)=-9
changes the value of the third row first column element from -3
to -9
To display or use a whole row or column use the operator Forexample
gt A(2)
ans =
3 5 7
gt A(2)
B4 Arithmetic expressions 235
ans =
-1
5
6
It is also possible to pick out a sub-matrix by the specifying the rowand column entries required For example To pick out the sub-matrix of A containing the elementsA11 A13 A21 A23 the commandA(12123) could be used as shown below
gt A(12123)
ans =
6 4
3 7
Note 123 means from 1 to 3 in steps of 2 This technique alongwith the assignment operator can be used for values of whole rowscolumns or sub-matrices in one command For example
gt A(12123)=[10111213]
A =
10 -1 11
12 5 13
-3 6 7
Note that the elements A11 A13 A21 A23 now contain the values10 11 12 and 13 respectively instead of their previous values
Index notation is logically an unfortunate phenomenon since itclashes with standard function notation On the face of it A(3)
should be the value of a function called A with input 3 but whenit is interpreted as index notation A is a variable However indexnotation is so familiar and useful to those familiar with classicalmathematical notation that many developers of mathematical soft-ware are not prepared to risk a ldquoconsumer backlashrdquo to providinga functional way of extracting elements from an array such as forexample 3 pick A to pick out the third element of A Similarlywe as teachers feel we would be doing students a disservice if wefailed to introduce them to the index notation that abounds inmathematical texts usually in the form of subscripting as in A3Logic does not always win an argument In the commercial worldof computing an established user base carries more weight thanmathematical logic just as in mathematics tradition may well alsofly in the face of logical consistency
B43 Array arithmetic
Scilab has two forms of arithmetic operations that can be usedwith matrices matrix arithmetic or array arithmetic Matrix arith-metic follows the normal rules of matrix algebra Array arithmeticwhich is also called element-by-element arithmetic is where oper-ations are performed on matrices element by element
236 Appendix B Programming with Scilab
Table B4 Scilab array operatorsOperator Meaning Scilab
^ Exponentiation x^y
Multiply xy
Right Division xy
Left Division yx
+ Addition x+y
- Subtraction x-y
Table B4 lists the Scilab array operators
The array addition and subtraction are identical to matrix additionor subtraction Both array and matrix addition and subtractionoperate element by element Array addition and subtraction re-quire that both arrays have the same size The only exception tothis rule in Scilab occurs when we add or subtract a scalar to orfrom an array In this case the scalar is added or subtracted fromeach element in the array
For example
gt [6 5 6] + 3
ans =
[9 8 9]
gt [6 4 5 8] + [3 1 2 4]
ans =
[9 5 7 12]
gt [4 3] - 7
ans =
[-3 -4]
Array multiplication and division is very different to matrix mul-tiplication and division Array multiplication or division is an ele-ment by element multiplication or division that produces an arraywith the same dimensions as the starting arrays As with addi-tion and subtraction array multiplication and division require thatboth arrays have the same dimensions
Example B2 Consider the two row vectors x=[2 4 7 1] andy=[6 1 3 2] Scilab does not recognise the possibility ofmatrix multiplication of these two vectors They are not com-patible the number of columns of the pre-multiplier must bethe same as the number of rows of the post-multiplier Anattempt to execute x y will bring an error message
Obtain the usual dot product of the two row vectors by trans-posing the post-multiplier as in xyrsquo or yxrsquo The matrixproduct xrsquoy produces a matrix of 4 rows and 4 columns
B5 Builtndashin functions 237
Array multiplication produces the result
gt xy
ans =
[12 4 21 2]
by multiplying corresponding elements of each vector Theresult is a row vector with the same dimensions as x and ythat is a 1times 4 vector
Example B3 Consider the row vectors x=[2 4 7 1] and col-umn vector y=[6 1 3 2]rsquo Array multiplication of thesetwo matrices is not possible because their dimensions do notagree
Matrix multiplication xy produces the result 39 that is theinner product or dot product of the vectors Note vectorsas such do not exist in Scilab there are only single rowedor single columned arrays
Scilab not only allows arrays to be raised to powers but alsoallows scalars and arrays to be raised by array powers To performexponentiation on an element-by-element basis use the ^ symbolFor example if x=[3 5 8] then x^3 produces the matrix [33 53
83] = [27 125 512]
A scalar can be raised to an array power For example if p=[2 4 5]then typing 3^p produces the array [32 34 35] = [9 81 243]
With element-by-element array operations it is important to re-member that the dot () is a part of a two-character symbol for aparticular array operator such as () () () (^) Sometimesthis can cause some confusion the dot in 3^p is not a decimalpoint associated with the number 3 The following operations withthe value of p=[2 4 5] are equivalent
3^p
30^p
3^p
(3)^p
3^[2 4 5]
B5 Builtndashin functions
B51 Arithmetic functions
Arithmetic expressions often require computations other than ad-dition subtraction multiplication division and exponentiationFor example many expressions require the use of logarithms ex-ponentials and trigonometric functions Scilab has predefined alarge number of functions that perform these types of computations(see Tables B5 and B6)
238 Appendix B Programming with Scilab
An example of a predefined function is sin which returns the sineof a given angle
m = sin(angle)
where the variable angle contains the angle in radians and theresult is stored in the variable m The variable angle is called theargument or parameter of the function If the angle is stored indegrees it must be converted to radians for example
m = sin(anglepi180)
or
radians =anglepi180
m = sin(radians)
In Scilab the arguments or parameters of the function are con-tained in parentheses following the name of the function This isreferred to as pre-fix notation Many common functions in math-ematics have a syntax using in-fix notation where two argumentsappear to the right and left of the function name as for examplein x+y atimes b nmod r etc Scilab does not support in-fix nota-tion for any but the arithmetic functions Classical mathematicalnotation also has examples of post-fix notationmdash n x squared mdashand other hybrids like |x| while Scilab has ahb to confound thepurist
A function may contain no arguments one argument or many ar-guments depending on its definition If a function contains morethan one argument it is vital to give the arguments in the cor-rect order For example consider the problem of finding θ in thefollowing triangle using the atan command
5
8θ
Now by hand calculation the answer is 05586 radians If we useatan as below we get the answers shown
--gt atan(58)
ans =
05586
--gt atan(85)
ans =
10122
The first answer is the correct one The second answer is incorrect
B5 Builtndashin functions 239
Table B5 Some common mathematical functions defined inScilabFunction Description
abs(x) Computes the absolute value of xsqrt(x) Square root of x The parameter can be nega-
tiveround(x) Round x to the nearest integerfix(x) Round x to the nearest integer toward zerofloor(x) Round x to the nearest integer toward minusinfinceil(x) Round x to the nearest integer toward +infinsign(x) Returns -1 if x is less than zero zero if x equals
zero and 1 otherwisemodulo(xy) Returns the remainder of xy For example
modulo(254) is 1exp(x) Returns ex
log(x) Returns the natural logarithm of x That isthe logarithm to the base e Parameter can benegative but cannot be zero
log10(x) Returns the common logarithm of x That isthe logarithm to the base 10 Parameter can benegative but cannot be zero
since we passed the values to the function in the incorrect order
Some functions also require that the arguments be in specific unitsFor example the trigonometric functions assume that the argu-ments are in radians
In Scilab some functions use the number of arguments to deter-mine the output of the function For example the zeros functioncan have one or two arguments which then determine the output(check this with the command help zeros)
A function reference cannot be made on the left hand side of anassignment since a function call always returns a value(s) whereasan assignment is always to a variable Functions can of courseappear on the right side of an assignment since they return valuesto be assigned to some variable Functions may be applied tothe output of other functions and may be called in the algorithmdefining a function for example
logx = log(abs(x))
When one function is used to compute the argument of anotherfunction be sure to enclose the argument of each function in itsown set of parentheses
Scilab allows recursive functions that is functions that call them-selves in their defining algorithm
Exercise B4 Use examples to show the differences between the
240 Appendix B Programming with Scilab
Table B6 Some common trigonometric functions defined inScilab
Function Description
sin(x) Sine of x where x is in radianscos(x) Cosine of x where x is in radianstan(x) Tangent of x where x is in radiansasin(x) Arcsine or inverse sine of x where x must be
between -1 and 1 The return angle is in radiansbetween minusπ2 and π2
acos(x) Arccosine or inverse cosine of x where x mustbe between minus1 and 1 The return angle is inradians between 0 and π
atan(x) Arctangent or inverse tangent of x The returnangle is in radians between minusπ2 and π2
atan(xy) Arctangent or inverse tangent of xy The re-turn angle is in radians between minusπ and π de-pending on the sign of x and y
sec(x) Secant (1(cos x)) of x where x is in radianscsc(x) Cosecant (1(sin x)) of x where x is in radianscotg(x) Cotangent (1(tan x)) of x where x is in radians
Table B7 Some of Scilabrsquos utility functionsFunction Description
clear Clear variables from memoryhelp Display the first set of comment lines from
scriptswho List the variables in memorywhos List the variables in memory with their sizeerror Terminates the execution of the Sci-file and out-
puts the supplied error message
functions round(x) fix(x) floor(x) and ceil(x)
Exercise B5 The functions listed in Tables B5 B6 are only asmall sample of all of Scilabrsquos mathematical functions
Using Scilabrsquos help command list all of the hyperbolic func-tions available Also list any restrictions on the values of thefunction arguments
B52 Utility functions
In addition to the arithmetic functions Scilab has several utilityfunctions (see Table B7) These functions allow users to trackvariables (whoswho) feedback an error message if an error occursin the code (error) clear the variables form the memory (clear)etc Essential these functions allow users to debug and performbookkeeping operations inside their code
B6 Visualising data in Scilab 241
6
5
4- 04- 8
3
- 02
- 6
00
2- 4 Y- 2
02
Z
10
04
X 02
06
4 - 1
08
6- 28
10
Figure B2 Plot of z = sin(radicx2 + y2)
radicx2 + y2 illustrating the
3d capabilities of Scilab
B6 Visualising data in Scilab
One of the big advantages in using Scilab is the ability to visualisedata and results of complex calculations (see Figure B2 as exampleof the 3d capabilities) While Scilab is capability of complex 3dgraphics (see help pages for mesh and surf) in this section we willmesh amp surf
focus on the 2d plotting command (plot) Hence in this sectionplot
we will focus on
1 plotting mathematical functions from within Scilab
2 changing stylistic elements of the plot
3 adding titles labels and legends to the plot
4 including mathematical notation in titles labels and legendsand
5 plotting data stored in an external ASCII file
B61 Plotting mathematical functions from within Scilab
To plot a mathematical function in Scilab three steps are requiredThese are
1 Define the domain of the function (eg x=linspace(02pi))
2 Calculate the values of function at the selected points in thedomain (eg y=sin(x))
242 Appendix B Programming with Scilab
3 Final step is to plot the function using the plot commandplot
(eg plot(xy))
Using these steps it is possible to produce a basic plot of any math-ematical function (see Figure B3) Note that the plot shown inFigure B3 does not have a title axes labels or a legend all ofwhich are crucial for a good plot The Scilab commands to addthese elements to the plot are given in the following sections
- 10
- 08
- 06
- 04
- 02
00
02
04
06
08
10
0 1 2 3 4 5 6 7
Figure B3 The default plot of y = sin(x) in Scilab Notethat by default axes are not labelled and there is no title orlegend These need to added using the additional commandsxlabelylabeltitle and legend
Changing stylistic elements of the plot
An optional argument can be given to the plot command to cus-tomise each new line aspect This argument is a concatenatedstring containing information about colour line style or markersbased on the specifiers shown in Table B8 To specify the plotbe drawn with a red longdash-dot line with diamond marker thestring that would be passed to plot command would be rsquor-drsquo Thestring which is a concatenation (in any order) of the three typesof properties shown in Table B8 must be unambiguous Further-more the string specification is not case sensitive An exampleof specifying various line styles is shown in Example B6 and Fig-ure B4
Example B6 The following commands illustrate how line stylescan be used to distinguish between the plots of sin(2πt) sin(2πt+π6) and sin(2πt+ π2)
t=linspace(0130)
B6 Visualising data in Scilab 243
Table B8 Plot specifiers required to change line styles coloursand markers in plotsSpecifier Line style Specifier Colour Specifier Marker stylendash Solid line (default) r Red + Plus signndash ndash Dashed line g Green o Circle Dotted line b Blue Asteriskndash Dash-dotted line c Cyan x Cross
m Magenta s Squarey Yellow rsquononersquo No marker(default)k Blackw White
Figure B4 Resulting plotting after issuing the commands givenin Example B6
244 Appendix B Programming with Scilab
Example B6 illustrate a number of key commands relating to thecreation of multiple plots in Scilab The mtlb hold4 commandmtlb hold
holds the current plot and axis properties so that all subsequentplots are added to the existing plot
To distinguish each plot a different line specification is passed eachplot command In the example y = sin(2πt) is plotted using a redsolid line with data points indicated using a red circle as indicatedby the string rsquor-orsquo y = sin(2πt + π6) is plotted using greendashed-dotted line with data points marked by a square (rsquog-srsquo)while y = sin(2πt+ π2) is plotted using a cyan dashed line usingdiamonds to mark the data points (rsquoc--drsquo) Finally to completethe plot axes labels and a legend has been added to plot the com-mands for doing this are discussed in sectB62
B62 Adding titleslabels and legends to the plot
All good plots should have the following properties
1 The style of figure is appropriate to the information beingconveyed
2 The contents are well organised so it is easy for readers tointerpret the information
3 Legends labels etc are clear and concise (eg contain unitsif required do not obstruct parts of the plot etc) and
4 A caption or title that summarises the information presented
Like most mathematical applications Scilab has a wide variety ofdifferent plotting styles that can be used (eg look up the help forbar polarplot etc) This allows you to select the most appropri-barpolarplot
ate style to display and organise your data
The Scilab commands xlabel ylabel title and legend al-xlabel ylabel title and
legend lows you to add axes labels titles and legends to plot As shownin Example B6 the commands xlabel ylabel and title takea string input containing the title or label text For the legend
command a string is required for each separate plot Note theorder of the strings in the legend command reflects the order inwhich the each of plots were made
Including mathematical notation in titles labels and legends
It is possible to include mathematical characters such as π inScilab plots This is done by inserting special commands intoeither the title legend or label string These commands are basedon the abbreviations for mathematical notation used in programssuch as LATEX MathType etc A brief summary of the commands
4 mtlb hold on is equivalent to the Scilab commandset(gca()auto clearoff) while mtlb hold off is equivalentto the Scilab command set(gca()auto clearon)
B6 Visualising data in Scilab 245
Table B9 Basic mathematical symbols available in Scilab forincluding mathematical notation in titles labels and legendscommand symbol command symbol command symbolalpha α upsilon υ sim sim
Im = cup cup wp weierpotimes otimes subseteq sube oslash cap cap in isin supseteq supesupset sup lceil d subset subint
intcdot middot o oslash
rfloor c neg 6= nabla nablalfloor b times times ldots perp perp surd
radicprime prime
wedge and varpi $ O Oslashrceil e rangle 〉 mid |
vee or copyright ccopy langle 〈
available in Scilab for typesetting mathematical notation is givenin Table B9 To insert these commands into a text string simplystart and end the string with $ signs as shown in Example B7
Example B7 Scilab commands to produce the labelled plot ofy = sin
There are a few key concepts to note about the Example B7 Thefirst can be seen in the string being passed to the title commandtitle
In this string sin is prefixed with a backslash () This is doneso that the letters for sin are not typeset in italic which shouldonly be used for variables and not common mathematical func-tions Table B10 shows the command sequences to use for themore common mathematical functions5 To get the fraction in thetitle the command fracab has been used to produce a
b Ifsuperscripts and subscripts are required in a title or label these aresuperscripts amp subscripts
obtained by adding the subscript character ( ) and the superscriptcharacter (^) to the string These characters modify the characteror substring defined in braces immediately following For examplex 2 gives x2 while e^2x gives e2x
Note in Example B7 the commands xlabel and ylabel has beenxlabelylabel
used to set the axes labels in Figure B5 Instead of using thesethree commands the title and the axes labels could have been setusing the single xtitle command (see help xtitle for details)xtitle
In the example we have also customised the font size to make thetext more readable by a) passing an option to the title xlabelylabel commands and b) by using the command xset to set thexset
default font size before executing the xstring command Finallythe xstring command adds the text string to plot at the locationxstring
specified by variables b and c In the example the coordinates wewant to add the text are obtained by executing the xclick functionxclick
and clicking on the point required in the plot window
5 Additional details can be found in books covering Mathematics in LATEX
B6 Visualising data in Scilab 247
Table B10 Command sequences for the common mathematicalfunctions in Scilabarccos arcsin arctan cos cot
csc exp lim ln log
max min sec sin tan
B63 Plotting numeric data stored in an ASCII file
As data can be stored in a wide variety of ways in external filesScilab has a wide variety of functions which can be used to importthe data stored in these files (see help for the commands read
fscanf and xlsread) Since data can be stored in such a wideread fscanf and xlsread
range of formats we will confine our discussions here to the situa-tion where numeric data is stored in a ASCII (text) file where eachrow of the file contains the same number of entries
To illustrate the procedure for plotting data stored in ASCII file wewill consider the data obtained measuring the temperature of a cupof coffee every minute for a total of 100 mins The data recorded isstored in the a file called coffeedatatxt6 which simply containsthe two columns of numbers shown in Table B11 As an examplethe first 10 lines of the text file contains the following numbers7
00000 1000000
10000 950000
20000 903125
30000 859180
40000 817981
50000 779357
60000 743147
70000 709201
80000 677376
90000 647540
Note the current version of Scilab has a builtin text editor whichcan be used to view such text files To access this editor simplytype the command edit at the command prompt Once open youedit
can use the graphical interface to open the required file
The first issue is to get the data into Scilab so it is available forplot command In Scilab the command read can used to importread
the data and assign it to a variable In the current example wewould use the following command to import the data and assignit to a variable called cdata assuming the data is in the currentdirectory8
6 The file coffeedatatxt has been attached and should be available for savingprovided you are using Adobe Reader To save the file right click on thepurple Windows version or the red MacOSLinux version in the margin andselect save file from the options Ensure you save the file as coffeedatatxt
7 normally you will have received the data file in this format8 You can check what directory Scilab is in by typing the command pwd
Figure B6 Temperature of coffee based on the data recorded inthe file coffeedatatxt
cdata=read(rsquocoffeedatatxtrsquo-12)
The -12 following the name of the file tells Scilab that you donrsquotknow how many rows you have but you have two columns of dataSince we know that we have 101 rows of data we could have alsoused the following command to load the data
cdata=read(rsquocoffeedatatxtrsquo1012)
We now have a variable called cdata which has two columns ofdata The first column contains the time while the second containsthe Temperature records Hence we can plot it using the followingcommand
plot(cdata(1)cdata(2))
This gives the plot shown in Figure B6 after appropriate labelshave been added as described above
B7 Script files
Executing commands entered at the keyboard or keypad is themost primitive use of a calculating device It is only when se-quences of operations (to perform complex tasks) are stored awayand re-called for repeated application with different inputs that wehave a programmed operation Complex programs must necessarilyhave an interface with the user The basic interface tool in Scilabcapable of executing sequences of commands that are stored in a fileis the script The file names must have an extension of sce forexample program1sce assignment1sce These files are calledScript Sci-files or simply scripts
A script file is a file of standard keyboard charactersthat define commands to drive a program A script filecan be generated using any text editor or word proces-sor
To invoke a script in Scilab type the commandexec
exec filenamesce
where filename denotes the actual name of the file containing thescript When the script runs it is exactly the same as if you typedthe commands into the command window from the keyboard Thismeans that the Sci-file can use any variables you have defined priorto invoking the script Any variables created in the script are alsoavailable when the script is finished and any variable whose namewas assigned a value in the script will be over-written
If you give a script the same name as any variable that it computesin the running of the script then Scilab will give precedence tothe workspace variable over a script As stated in sectB3 the variablename will hide the script name after the script has run Scilab willtake the name as referring to the variable and not to the definedscript
Scilab also needs to know where to find the script to executedIf Scilab cannot find the script it will report that you have anundefined function or variable
The exact procedure for creating and saving Sci-files depends onthe Scilab version the operating system (MS Windows Macin-tosh or UnixLinux ) and the text editor you use The currentversion of Scilab includes a built-in editor which can be used tocreate and edit Sci-files and text files
To create a new Sci-file under the current version of Scilab simplytype edit newfilenamesce This will open up the Scilab editoredit
with a blank document called newfilename
B71 Script input and output
A script is a named file (with the extension sce) containing asequence of Scilab instructions which when its name is executedin a session performs those instructions in order as though theywere entered at the console
There are two common uses for scripts Pragmatically we maywant to save a particular computational sequence for later use orreference If the tasks involved are to be repeated often thenthe nature of the tasks should be analysed formulated into usefulfunctions (which we will discuss later) and the whole procedurepresented in a single function But if the tasks are only likely toget occasional use then the script may be adequately commentedand stored as a script file It is a quick and dirty way of saving
B7 Script files 251
the instructions for doing a particular job when we do not wantto go to the trouble of writing the functions for the job It is nota highly recommended modus operandi but useful all the same
The second use is for the script to act like a storyboard presentinginformation to a user in a structured and controlled way permit-ting the user to see the steps of a computational procedure andtheir outputs up on the screen At a more sophisticated level thescript takes on the proportions of an application controlling a largeand complex interaction with advanced interface features We willnot take our study of Scilab programming to this level but workmostly at the storyboard level using the simplest of Scilabrsquos in-terface tools
All the variables created in a script become global variables definedin the workspace and available for perusal As an application itis assumed that a script controls the entire workspace and thevariables it creates are lsquothe datarsquo of the application When scriptsare used as storyboards on a small scale then it cannot be assumedthat the variables they create deserve global recognition and it iscourteous to erase these variables when the script is finished unlessthey have some possible further use An unfortunate side-effect isthat if variables with the same name already exist in the workspacebefore a script is run then these variables will be over-written andlost
Used as storyboards scripts allow the user to cut paste and re-trythe various steps of a procedure and to modify the script to suithis own needs Storyboards may be used for demonstrations oftoolbox utilities or extended documentationexplanation of func-tions However if there is any danger that variables defined bythe script may over-write workspace variables then the demo ordocumentation should be re-fashioned into a function that leavesthe workspace variables untouched (more on this later)
Scripts will often read data from pre-prepared files and we will dis-cuss the functions provided by Scilab to perform this task below
Gathering data entered by the user really falls into the realm ofinterface design but we only briefly consider the simplest interfacefunction provided by Scilab Similarly we briefly consider thefunctions for providing the simplest forms of output presentation
Keyboard input
Scilab has a built-in function input that can be used to promptinput
the user to enter the value of a variable from the keyboard Forexample
Tc = input(rsquoPlease enter the temperature in degrees Celsiusrsquo)
will print the message
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
234 Appendix B Programming with Scilab
However because it is a matrix operation
gt a + b
ans =
8 10 40
The standard arithmetic operators described in Table B2 are alsoused with matrices The normal rules of matrix addition subtrac-tion and multiplication apply
The left and right division of matrices are also generalisations ofscalar division Right division BA is roughly the same as Binv(A)Left division AB is roughly the same as inv(A)B The differenceis in how the division is performed Different algorithms are usedwhen A or B are square or not
Scilab also defines a transpose operator for matrices The trans-pose of the matrix A is Arsquo For example if
A =
[25 63
12 09
] then A prime =
[25 12
63 09
]
Accessing matrix elements
Individual elements of a matrix can accessed and used in state-ments by specifying the matrix index of an element The index ofan element is its position in the matrix For example consider thefollowing matrix
A =
6 minus1 4
3 5 7
minus3 6 1
Define this in Scilab with the command
A=[ 6 -1 4 3 5 7 -3 6 7]
to display the contents of matrix A in Scilab just type the variablename A To display or use one element from the matrix specify theelement For example A(13) specifies the element in the first rowand third column that is 4 for the above matrix
To change a specific entry just assign a value to it For example
A(31)=-9
changes the value of the third row first column element from -3
to -9
To display or use a whole row or column use the operator Forexample
gt A(2)
ans =
3 5 7
gt A(2)
B4 Arithmetic expressions 235
ans =
-1
5
6
It is also possible to pick out a sub-matrix by the specifying the rowand column entries required For example To pick out the sub-matrix of A containing the elementsA11 A13 A21 A23 the commandA(12123) could be used as shown below
gt A(12123)
ans =
6 4
3 7
Note 123 means from 1 to 3 in steps of 2 This technique alongwith the assignment operator can be used for values of whole rowscolumns or sub-matrices in one command For example
gt A(12123)=[10111213]
A =
10 -1 11
12 5 13
-3 6 7
Note that the elements A11 A13 A21 A23 now contain the values10 11 12 and 13 respectively instead of their previous values
Index notation is logically an unfortunate phenomenon since itclashes with standard function notation On the face of it A(3)
should be the value of a function called A with input 3 but whenit is interpreted as index notation A is a variable However indexnotation is so familiar and useful to those familiar with classicalmathematical notation that many developers of mathematical soft-ware are not prepared to risk a ldquoconsumer backlashrdquo to providinga functional way of extracting elements from an array such as forexample 3 pick A to pick out the third element of A Similarlywe as teachers feel we would be doing students a disservice if wefailed to introduce them to the index notation that abounds inmathematical texts usually in the form of subscripting as in A3Logic does not always win an argument In the commercial worldof computing an established user base carries more weight thanmathematical logic just as in mathematics tradition may well alsofly in the face of logical consistency
B43 Array arithmetic
Scilab has two forms of arithmetic operations that can be usedwith matrices matrix arithmetic or array arithmetic Matrix arith-metic follows the normal rules of matrix algebra Array arithmeticwhich is also called element-by-element arithmetic is where oper-ations are performed on matrices element by element
236 Appendix B Programming with Scilab
Table B4 Scilab array operatorsOperator Meaning Scilab
^ Exponentiation x^y
Multiply xy
Right Division xy
Left Division yx
+ Addition x+y
- Subtraction x-y
Table B4 lists the Scilab array operators
The array addition and subtraction are identical to matrix additionor subtraction Both array and matrix addition and subtractionoperate element by element Array addition and subtraction re-quire that both arrays have the same size The only exception tothis rule in Scilab occurs when we add or subtract a scalar to orfrom an array In this case the scalar is added or subtracted fromeach element in the array
For example
gt [6 5 6] + 3
ans =
[9 8 9]
gt [6 4 5 8] + [3 1 2 4]
ans =
[9 5 7 12]
gt [4 3] - 7
ans =
[-3 -4]
Array multiplication and division is very different to matrix mul-tiplication and division Array multiplication or division is an ele-ment by element multiplication or division that produces an arraywith the same dimensions as the starting arrays As with addi-tion and subtraction array multiplication and division require thatboth arrays have the same dimensions
Example B2 Consider the two row vectors x=[2 4 7 1] andy=[6 1 3 2] Scilab does not recognise the possibility ofmatrix multiplication of these two vectors They are not com-patible the number of columns of the pre-multiplier must bethe same as the number of rows of the post-multiplier Anattempt to execute x y will bring an error message
Obtain the usual dot product of the two row vectors by trans-posing the post-multiplier as in xyrsquo or yxrsquo The matrixproduct xrsquoy produces a matrix of 4 rows and 4 columns
B5 Builtndashin functions 237
Array multiplication produces the result
gt xy
ans =
[12 4 21 2]
by multiplying corresponding elements of each vector Theresult is a row vector with the same dimensions as x and ythat is a 1times 4 vector
Example B3 Consider the row vectors x=[2 4 7 1] and col-umn vector y=[6 1 3 2]rsquo Array multiplication of thesetwo matrices is not possible because their dimensions do notagree
Matrix multiplication xy produces the result 39 that is theinner product or dot product of the vectors Note vectorsas such do not exist in Scilab there are only single rowedor single columned arrays
Scilab not only allows arrays to be raised to powers but alsoallows scalars and arrays to be raised by array powers To performexponentiation on an element-by-element basis use the ^ symbolFor example if x=[3 5 8] then x^3 produces the matrix [33 53
83] = [27 125 512]
A scalar can be raised to an array power For example if p=[2 4 5]then typing 3^p produces the array [32 34 35] = [9 81 243]
With element-by-element array operations it is important to re-member that the dot () is a part of a two-character symbol for aparticular array operator such as () () () (^) Sometimesthis can cause some confusion the dot in 3^p is not a decimalpoint associated with the number 3 The following operations withthe value of p=[2 4 5] are equivalent
3^p
30^p
3^p
(3)^p
3^[2 4 5]
B5 Builtndashin functions
B51 Arithmetic functions
Arithmetic expressions often require computations other than ad-dition subtraction multiplication division and exponentiationFor example many expressions require the use of logarithms ex-ponentials and trigonometric functions Scilab has predefined alarge number of functions that perform these types of computations(see Tables B5 and B6)
238 Appendix B Programming with Scilab
An example of a predefined function is sin which returns the sineof a given angle
m = sin(angle)
where the variable angle contains the angle in radians and theresult is stored in the variable m The variable angle is called theargument or parameter of the function If the angle is stored indegrees it must be converted to radians for example
m = sin(anglepi180)
or
radians =anglepi180
m = sin(radians)
In Scilab the arguments or parameters of the function are con-tained in parentheses following the name of the function This isreferred to as pre-fix notation Many common functions in math-ematics have a syntax using in-fix notation where two argumentsappear to the right and left of the function name as for examplein x+y atimes b nmod r etc Scilab does not support in-fix nota-tion for any but the arithmetic functions Classical mathematicalnotation also has examples of post-fix notationmdash n x squared mdashand other hybrids like |x| while Scilab has ahb to confound thepurist
A function may contain no arguments one argument or many ar-guments depending on its definition If a function contains morethan one argument it is vital to give the arguments in the cor-rect order For example consider the problem of finding θ in thefollowing triangle using the atan command
5
8θ
Now by hand calculation the answer is 05586 radians If we useatan as below we get the answers shown
--gt atan(58)
ans =
05586
--gt atan(85)
ans =
10122
The first answer is the correct one The second answer is incorrect
B5 Builtndashin functions 239
Table B5 Some common mathematical functions defined inScilabFunction Description
abs(x) Computes the absolute value of xsqrt(x) Square root of x The parameter can be nega-
tiveround(x) Round x to the nearest integerfix(x) Round x to the nearest integer toward zerofloor(x) Round x to the nearest integer toward minusinfinceil(x) Round x to the nearest integer toward +infinsign(x) Returns -1 if x is less than zero zero if x equals
zero and 1 otherwisemodulo(xy) Returns the remainder of xy For example
modulo(254) is 1exp(x) Returns ex
log(x) Returns the natural logarithm of x That isthe logarithm to the base e Parameter can benegative but cannot be zero
log10(x) Returns the common logarithm of x That isthe logarithm to the base 10 Parameter can benegative but cannot be zero
since we passed the values to the function in the incorrect order
Some functions also require that the arguments be in specific unitsFor example the trigonometric functions assume that the argu-ments are in radians
In Scilab some functions use the number of arguments to deter-mine the output of the function For example the zeros functioncan have one or two arguments which then determine the output(check this with the command help zeros)
A function reference cannot be made on the left hand side of anassignment since a function call always returns a value(s) whereasan assignment is always to a variable Functions can of courseappear on the right side of an assignment since they return valuesto be assigned to some variable Functions may be applied tothe output of other functions and may be called in the algorithmdefining a function for example
logx = log(abs(x))
When one function is used to compute the argument of anotherfunction be sure to enclose the argument of each function in itsown set of parentheses
Scilab allows recursive functions that is functions that call them-selves in their defining algorithm
Exercise B4 Use examples to show the differences between the
240 Appendix B Programming with Scilab
Table B6 Some common trigonometric functions defined inScilab
Function Description
sin(x) Sine of x where x is in radianscos(x) Cosine of x where x is in radianstan(x) Tangent of x where x is in radiansasin(x) Arcsine or inverse sine of x where x must be
between -1 and 1 The return angle is in radiansbetween minusπ2 and π2
acos(x) Arccosine or inverse cosine of x where x mustbe between minus1 and 1 The return angle is inradians between 0 and π
atan(x) Arctangent or inverse tangent of x The returnangle is in radians between minusπ2 and π2
atan(xy) Arctangent or inverse tangent of xy The re-turn angle is in radians between minusπ and π de-pending on the sign of x and y
sec(x) Secant (1(cos x)) of x where x is in radianscsc(x) Cosecant (1(sin x)) of x where x is in radianscotg(x) Cotangent (1(tan x)) of x where x is in radians
Table B7 Some of Scilabrsquos utility functionsFunction Description
clear Clear variables from memoryhelp Display the first set of comment lines from
scriptswho List the variables in memorywhos List the variables in memory with their sizeerror Terminates the execution of the Sci-file and out-
puts the supplied error message
functions round(x) fix(x) floor(x) and ceil(x)
Exercise B5 The functions listed in Tables B5 B6 are only asmall sample of all of Scilabrsquos mathematical functions
Using Scilabrsquos help command list all of the hyperbolic func-tions available Also list any restrictions on the values of thefunction arguments
B52 Utility functions
In addition to the arithmetic functions Scilab has several utilityfunctions (see Table B7) These functions allow users to trackvariables (whoswho) feedback an error message if an error occursin the code (error) clear the variables form the memory (clear)etc Essential these functions allow users to debug and performbookkeeping operations inside their code
B6 Visualising data in Scilab 241
6
5
4- 04- 8
3
- 02
- 6
00
2- 4 Y- 2
02
Z
10
04
X 02
06
4 - 1
08
6- 28
10
Figure B2 Plot of z = sin(radicx2 + y2)
radicx2 + y2 illustrating the
3d capabilities of Scilab
B6 Visualising data in Scilab
One of the big advantages in using Scilab is the ability to visualisedata and results of complex calculations (see Figure B2 as exampleof the 3d capabilities) While Scilab is capability of complex 3dgraphics (see help pages for mesh and surf) in this section we willmesh amp surf
focus on the 2d plotting command (plot) Hence in this sectionplot
we will focus on
1 plotting mathematical functions from within Scilab
2 changing stylistic elements of the plot
3 adding titles labels and legends to the plot
4 including mathematical notation in titles labels and legendsand
5 plotting data stored in an external ASCII file
B61 Plotting mathematical functions from within Scilab
To plot a mathematical function in Scilab three steps are requiredThese are
1 Define the domain of the function (eg x=linspace(02pi))
2 Calculate the values of function at the selected points in thedomain (eg y=sin(x))
242 Appendix B Programming with Scilab
3 Final step is to plot the function using the plot commandplot
(eg plot(xy))
Using these steps it is possible to produce a basic plot of any math-ematical function (see Figure B3) Note that the plot shown inFigure B3 does not have a title axes labels or a legend all ofwhich are crucial for a good plot The Scilab commands to addthese elements to the plot are given in the following sections
- 10
- 08
- 06
- 04
- 02
00
02
04
06
08
10
0 1 2 3 4 5 6 7
Figure B3 The default plot of y = sin(x) in Scilab Notethat by default axes are not labelled and there is no title orlegend These need to added using the additional commandsxlabelylabeltitle and legend
Changing stylistic elements of the plot
An optional argument can be given to the plot command to cus-tomise each new line aspect This argument is a concatenatedstring containing information about colour line style or markersbased on the specifiers shown in Table B8 To specify the plotbe drawn with a red longdash-dot line with diamond marker thestring that would be passed to plot command would be rsquor-drsquo Thestring which is a concatenation (in any order) of the three typesof properties shown in Table B8 must be unambiguous Further-more the string specification is not case sensitive An exampleof specifying various line styles is shown in Example B6 and Fig-ure B4
Example B6 The following commands illustrate how line stylescan be used to distinguish between the plots of sin(2πt) sin(2πt+π6) and sin(2πt+ π2)
t=linspace(0130)
B6 Visualising data in Scilab 243
Table B8 Plot specifiers required to change line styles coloursand markers in plotsSpecifier Line style Specifier Colour Specifier Marker stylendash Solid line (default) r Red + Plus signndash ndash Dashed line g Green o Circle Dotted line b Blue Asteriskndash Dash-dotted line c Cyan x Cross
m Magenta s Squarey Yellow rsquononersquo No marker(default)k Blackw White
Figure B4 Resulting plotting after issuing the commands givenin Example B6
244 Appendix B Programming with Scilab
Example B6 illustrate a number of key commands relating to thecreation of multiple plots in Scilab The mtlb hold4 commandmtlb hold
holds the current plot and axis properties so that all subsequentplots are added to the existing plot
To distinguish each plot a different line specification is passed eachplot command In the example y = sin(2πt) is plotted using a redsolid line with data points indicated using a red circle as indicatedby the string rsquor-orsquo y = sin(2πt + π6) is plotted using greendashed-dotted line with data points marked by a square (rsquog-srsquo)while y = sin(2πt+ π2) is plotted using a cyan dashed line usingdiamonds to mark the data points (rsquoc--drsquo) Finally to completethe plot axes labels and a legend has been added to plot the com-mands for doing this are discussed in sectB62
B62 Adding titleslabels and legends to the plot
All good plots should have the following properties
1 The style of figure is appropriate to the information beingconveyed
2 The contents are well organised so it is easy for readers tointerpret the information
3 Legends labels etc are clear and concise (eg contain unitsif required do not obstruct parts of the plot etc) and
4 A caption or title that summarises the information presented
Like most mathematical applications Scilab has a wide variety ofdifferent plotting styles that can be used (eg look up the help forbar polarplot etc) This allows you to select the most appropri-barpolarplot
ate style to display and organise your data
The Scilab commands xlabel ylabel title and legend al-xlabel ylabel title and
legend lows you to add axes labels titles and legends to plot As shownin Example B6 the commands xlabel ylabel and title takea string input containing the title or label text For the legend
command a string is required for each separate plot Note theorder of the strings in the legend command reflects the order inwhich the each of plots were made
Including mathematical notation in titles labels and legends
It is possible to include mathematical characters such as π inScilab plots This is done by inserting special commands intoeither the title legend or label string These commands are basedon the abbreviations for mathematical notation used in programssuch as LATEX MathType etc A brief summary of the commands
4 mtlb hold on is equivalent to the Scilab commandset(gca()auto clearoff) while mtlb hold off is equivalentto the Scilab command set(gca()auto clearon)
B6 Visualising data in Scilab 245
Table B9 Basic mathematical symbols available in Scilab forincluding mathematical notation in titles labels and legendscommand symbol command symbol command symbolalpha α upsilon υ sim sim
Im = cup cup wp weierpotimes otimes subseteq sube oslash cap cap in isin supseteq supesupset sup lceil d subset subint
intcdot middot o oslash
rfloor c neg 6= nabla nablalfloor b times times ldots perp perp surd
radicprime prime
wedge and varpi $ O Oslashrceil e rangle 〉 mid |
vee or copyright ccopy langle 〈
available in Scilab for typesetting mathematical notation is givenin Table B9 To insert these commands into a text string simplystart and end the string with $ signs as shown in Example B7
Example B7 Scilab commands to produce the labelled plot ofy = sin
There are a few key concepts to note about the Example B7 Thefirst can be seen in the string being passed to the title commandtitle
In this string sin is prefixed with a backslash () This is doneso that the letters for sin are not typeset in italic which shouldonly be used for variables and not common mathematical func-tions Table B10 shows the command sequences to use for themore common mathematical functions5 To get the fraction in thetitle the command fracab has been used to produce a
b Ifsuperscripts and subscripts are required in a title or label these aresuperscripts amp subscripts
obtained by adding the subscript character ( ) and the superscriptcharacter (^) to the string These characters modify the characteror substring defined in braces immediately following For examplex 2 gives x2 while e^2x gives e2x
Note in Example B7 the commands xlabel and ylabel has beenxlabelylabel
used to set the axes labels in Figure B5 Instead of using thesethree commands the title and the axes labels could have been setusing the single xtitle command (see help xtitle for details)xtitle
In the example we have also customised the font size to make thetext more readable by a) passing an option to the title xlabelylabel commands and b) by using the command xset to set thexset
default font size before executing the xstring command Finallythe xstring command adds the text string to plot at the locationxstring
specified by variables b and c In the example the coordinates wewant to add the text are obtained by executing the xclick functionxclick
and clicking on the point required in the plot window
5 Additional details can be found in books covering Mathematics in LATEX
B6 Visualising data in Scilab 247
Table B10 Command sequences for the common mathematicalfunctions in Scilabarccos arcsin arctan cos cot
csc exp lim ln log
max min sec sin tan
B63 Plotting numeric data stored in an ASCII file
As data can be stored in a wide variety of ways in external filesScilab has a wide variety of functions which can be used to importthe data stored in these files (see help for the commands read
fscanf and xlsread) Since data can be stored in such a wideread fscanf and xlsread
range of formats we will confine our discussions here to the situa-tion where numeric data is stored in a ASCII (text) file where eachrow of the file contains the same number of entries
To illustrate the procedure for plotting data stored in ASCII file wewill consider the data obtained measuring the temperature of a cupof coffee every minute for a total of 100 mins The data recorded isstored in the a file called coffeedatatxt6 which simply containsthe two columns of numbers shown in Table B11 As an examplethe first 10 lines of the text file contains the following numbers7
00000 1000000
10000 950000
20000 903125
30000 859180
40000 817981
50000 779357
60000 743147
70000 709201
80000 677376
90000 647540
Note the current version of Scilab has a builtin text editor whichcan be used to view such text files To access this editor simplytype the command edit at the command prompt Once open youedit
can use the graphical interface to open the required file
The first issue is to get the data into Scilab so it is available forplot command In Scilab the command read can used to importread
the data and assign it to a variable In the current example wewould use the following command to import the data and assignit to a variable called cdata assuming the data is in the currentdirectory8
6 The file coffeedatatxt has been attached and should be available for savingprovided you are using Adobe Reader To save the file right click on thepurple Windows version or the red MacOSLinux version in the margin andselect save file from the options Ensure you save the file as coffeedatatxt
7 normally you will have received the data file in this format8 You can check what directory Scilab is in by typing the command pwd
Figure B6 Temperature of coffee based on the data recorded inthe file coffeedatatxt
cdata=read(rsquocoffeedatatxtrsquo-12)
The -12 following the name of the file tells Scilab that you donrsquotknow how many rows you have but you have two columns of dataSince we know that we have 101 rows of data we could have alsoused the following command to load the data
cdata=read(rsquocoffeedatatxtrsquo1012)
We now have a variable called cdata which has two columns ofdata The first column contains the time while the second containsthe Temperature records Hence we can plot it using the followingcommand
plot(cdata(1)cdata(2))
This gives the plot shown in Figure B6 after appropriate labelshave been added as described above
B7 Script files
Executing commands entered at the keyboard or keypad is themost primitive use of a calculating device It is only when se-quences of operations (to perform complex tasks) are stored awayand re-called for repeated application with different inputs that wehave a programmed operation Complex programs must necessarilyhave an interface with the user The basic interface tool in Scilabcapable of executing sequences of commands that are stored in a fileis the script The file names must have an extension of sce forexample program1sce assignment1sce These files are calledScript Sci-files or simply scripts
A script file is a file of standard keyboard charactersthat define commands to drive a program A script filecan be generated using any text editor or word proces-sor
To invoke a script in Scilab type the commandexec
exec filenamesce
where filename denotes the actual name of the file containing thescript When the script runs it is exactly the same as if you typedthe commands into the command window from the keyboard Thismeans that the Sci-file can use any variables you have defined priorto invoking the script Any variables created in the script are alsoavailable when the script is finished and any variable whose namewas assigned a value in the script will be over-written
If you give a script the same name as any variable that it computesin the running of the script then Scilab will give precedence tothe workspace variable over a script As stated in sectB3 the variablename will hide the script name after the script has run Scilab willtake the name as referring to the variable and not to the definedscript
Scilab also needs to know where to find the script to executedIf Scilab cannot find the script it will report that you have anundefined function or variable
The exact procedure for creating and saving Sci-files depends onthe Scilab version the operating system (MS Windows Macin-tosh or UnixLinux ) and the text editor you use The currentversion of Scilab includes a built-in editor which can be used tocreate and edit Sci-files and text files
To create a new Sci-file under the current version of Scilab simplytype edit newfilenamesce This will open up the Scilab editoredit
with a blank document called newfilename
B71 Script input and output
A script is a named file (with the extension sce) containing asequence of Scilab instructions which when its name is executedin a session performs those instructions in order as though theywere entered at the console
There are two common uses for scripts Pragmatically we maywant to save a particular computational sequence for later use orreference If the tasks involved are to be repeated often thenthe nature of the tasks should be analysed formulated into usefulfunctions (which we will discuss later) and the whole procedurepresented in a single function But if the tasks are only likely toget occasional use then the script may be adequately commentedand stored as a script file It is a quick and dirty way of saving
B7 Script files 251
the instructions for doing a particular job when we do not wantto go to the trouble of writing the functions for the job It is nota highly recommended modus operandi but useful all the same
The second use is for the script to act like a storyboard presentinginformation to a user in a structured and controlled way permit-ting the user to see the steps of a computational procedure andtheir outputs up on the screen At a more sophisticated level thescript takes on the proportions of an application controlling a largeand complex interaction with advanced interface features We willnot take our study of Scilab programming to this level but workmostly at the storyboard level using the simplest of Scilabrsquos in-terface tools
All the variables created in a script become global variables definedin the workspace and available for perusal As an application itis assumed that a script controls the entire workspace and thevariables it creates are lsquothe datarsquo of the application When scriptsare used as storyboards on a small scale then it cannot be assumedthat the variables they create deserve global recognition and it iscourteous to erase these variables when the script is finished unlessthey have some possible further use An unfortunate side-effect isthat if variables with the same name already exist in the workspacebefore a script is run then these variables will be over-written andlost
Used as storyboards scripts allow the user to cut paste and re-trythe various steps of a procedure and to modify the script to suithis own needs Storyboards may be used for demonstrations oftoolbox utilities or extended documentationexplanation of func-tions However if there is any danger that variables defined bythe script may over-write workspace variables then the demo ordocumentation should be re-fashioned into a function that leavesthe workspace variables untouched (more on this later)
Scripts will often read data from pre-prepared files and we will dis-cuss the functions provided by Scilab to perform this task below
Gathering data entered by the user really falls into the realm ofinterface design but we only briefly consider the simplest interfacefunction provided by Scilab Similarly we briefly consider thefunctions for providing the simplest forms of output presentation
Keyboard input
Scilab has a built-in function input that can be used to promptinput
the user to enter the value of a variable from the keyboard Forexample
Tc = input(rsquoPlease enter the temperature in degrees Celsiusrsquo)
will print the message
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
B4 Arithmetic expressions 235
ans =
-1
5
6
It is also possible to pick out a sub-matrix by the specifying the rowand column entries required For example To pick out the sub-matrix of A containing the elementsA11 A13 A21 A23 the commandA(12123) could be used as shown below
gt A(12123)
ans =
6 4
3 7
Note 123 means from 1 to 3 in steps of 2 This technique alongwith the assignment operator can be used for values of whole rowscolumns or sub-matrices in one command For example
gt A(12123)=[10111213]
A =
10 -1 11
12 5 13
-3 6 7
Note that the elements A11 A13 A21 A23 now contain the values10 11 12 and 13 respectively instead of their previous values
Index notation is logically an unfortunate phenomenon since itclashes with standard function notation On the face of it A(3)
should be the value of a function called A with input 3 but whenit is interpreted as index notation A is a variable However indexnotation is so familiar and useful to those familiar with classicalmathematical notation that many developers of mathematical soft-ware are not prepared to risk a ldquoconsumer backlashrdquo to providinga functional way of extracting elements from an array such as forexample 3 pick A to pick out the third element of A Similarlywe as teachers feel we would be doing students a disservice if wefailed to introduce them to the index notation that abounds inmathematical texts usually in the form of subscripting as in A3Logic does not always win an argument In the commercial worldof computing an established user base carries more weight thanmathematical logic just as in mathematics tradition may well alsofly in the face of logical consistency
B43 Array arithmetic
Scilab has two forms of arithmetic operations that can be usedwith matrices matrix arithmetic or array arithmetic Matrix arith-metic follows the normal rules of matrix algebra Array arithmeticwhich is also called element-by-element arithmetic is where oper-ations are performed on matrices element by element
236 Appendix B Programming with Scilab
Table B4 Scilab array operatorsOperator Meaning Scilab
^ Exponentiation x^y
Multiply xy
Right Division xy
Left Division yx
+ Addition x+y
- Subtraction x-y
Table B4 lists the Scilab array operators
The array addition and subtraction are identical to matrix additionor subtraction Both array and matrix addition and subtractionoperate element by element Array addition and subtraction re-quire that both arrays have the same size The only exception tothis rule in Scilab occurs when we add or subtract a scalar to orfrom an array In this case the scalar is added or subtracted fromeach element in the array
For example
gt [6 5 6] + 3
ans =
[9 8 9]
gt [6 4 5 8] + [3 1 2 4]
ans =
[9 5 7 12]
gt [4 3] - 7
ans =
[-3 -4]
Array multiplication and division is very different to matrix mul-tiplication and division Array multiplication or division is an ele-ment by element multiplication or division that produces an arraywith the same dimensions as the starting arrays As with addi-tion and subtraction array multiplication and division require thatboth arrays have the same dimensions
Example B2 Consider the two row vectors x=[2 4 7 1] andy=[6 1 3 2] Scilab does not recognise the possibility ofmatrix multiplication of these two vectors They are not com-patible the number of columns of the pre-multiplier must bethe same as the number of rows of the post-multiplier Anattempt to execute x y will bring an error message
Obtain the usual dot product of the two row vectors by trans-posing the post-multiplier as in xyrsquo or yxrsquo The matrixproduct xrsquoy produces a matrix of 4 rows and 4 columns
B5 Builtndashin functions 237
Array multiplication produces the result
gt xy
ans =
[12 4 21 2]
by multiplying corresponding elements of each vector Theresult is a row vector with the same dimensions as x and ythat is a 1times 4 vector
Example B3 Consider the row vectors x=[2 4 7 1] and col-umn vector y=[6 1 3 2]rsquo Array multiplication of thesetwo matrices is not possible because their dimensions do notagree
Matrix multiplication xy produces the result 39 that is theinner product or dot product of the vectors Note vectorsas such do not exist in Scilab there are only single rowedor single columned arrays
Scilab not only allows arrays to be raised to powers but alsoallows scalars and arrays to be raised by array powers To performexponentiation on an element-by-element basis use the ^ symbolFor example if x=[3 5 8] then x^3 produces the matrix [33 53
83] = [27 125 512]
A scalar can be raised to an array power For example if p=[2 4 5]then typing 3^p produces the array [32 34 35] = [9 81 243]
With element-by-element array operations it is important to re-member that the dot () is a part of a two-character symbol for aparticular array operator such as () () () (^) Sometimesthis can cause some confusion the dot in 3^p is not a decimalpoint associated with the number 3 The following operations withthe value of p=[2 4 5] are equivalent
3^p
30^p
3^p
(3)^p
3^[2 4 5]
B5 Builtndashin functions
B51 Arithmetic functions
Arithmetic expressions often require computations other than ad-dition subtraction multiplication division and exponentiationFor example many expressions require the use of logarithms ex-ponentials and trigonometric functions Scilab has predefined alarge number of functions that perform these types of computations(see Tables B5 and B6)
238 Appendix B Programming with Scilab
An example of a predefined function is sin which returns the sineof a given angle
m = sin(angle)
where the variable angle contains the angle in radians and theresult is stored in the variable m The variable angle is called theargument or parameter of the function If the angle is stored indegrees it must be converted to radians for example
m = sin(anglepi180)
or
radians =anglepi180
m = sin(radians)
In Scilab the arguments or parameters of the function are con-tained in parentheses following the name of the function This isreferred to as pre-fix notation Many common functions in math-ematics have a syntax using in-fix notation where two argumentsappear to the right and left of the function name as for examplein x+y atimes b nmod r etc Scilab does not support in-fix nota-tion for any but the arithmetic functions Classical mathematicalnotation also has examples of post-fix notationmdash n x squared mdashand other hybrids like |x| while Scilab has ahb to confound thepurist
A function may contain no arguments one argument or many ar-guments depending on its definition If a function contains morethan one argument it is vital to give the arguments in the cor-rect order For example consider the problem of finding θ in thefollowing triangle using the atan command
5
8θ
Now by hand calculation the answer is 05586 radians If we useatan as below we get the answers shown
--gt atan(58)
ans =
05586
--gt atan(85)
ans =
10122
The first answer is the correct one The second answer is incorrect
B5 Builtndashin functions 239
Table B5 Some common mathematical functions defined inScilabFunction Description
abs(x) Computes the absolute value of xsqrt(x) Square root of x The parameter can be nega-
tiveround(x) Round x to the nearest integerfix(x) Round x to the nearest integer toward zerofloor(x) Round x to the nearest integer toward minusinfinceil(x) Round x to the nearest integer toward +infinsign(x) Returns -1 if x is less than zero zero if x equals
zero and 1 otherwisemodulo(xy) Returns the remainder of xy For example
modulo(254) is 1exp(x) Returns ex
log(x) Returns the natural logarithm of x That isthe logarithm to the base e Parameter can benegative but cannot be zero
log10(x) Returns the common logarithm of x That isthe logarithm to the base 10 Parameter can benegative but cannot be zero
since we passed the values to the function in the incorrect order
Some functions also require that the arguments be in specific unitsFor example the trigonometric functions assume that the argu-ments are in radians
In Scilab some functions use the number of arguments to deter-mine the output of the function For example the zeros functioncan have one or two arguments which then determine the output(check this with the command help zeros)
A function reference cannot be made on the left hand side of anassignment since a function call always returns a value(s) whereasan assignment is always to a variable Functions can of courseappear on the right side of an assignment since they return valuesto be assigned to some variable Functions may be applied tothe output of other functions and may be called in the algorithmdefining a function for example
logx = log(abs(x))
When one function is used to compute the argument of anotherfunction be sure to enclose the argument of each function in itsown set of parentheses
Scilab allows recursive functions that is functions that call them-selves in their defining algorithm
Exercise B4 Use examples to show the differences between the
240 Appendix B Programming with Scilab
Table B6 Some common trigonometric functions defined inScilab
Function Description
sin(x) Sine of x where x is in radianscos(x) Cosine of x where x is in radianstan(x) Tangent of x where x is in radiansasin(x) Arcsine or inverse sine of x where x must be
between -1 and 1 The return angle is in radiansbetween minusπ2 and π2
acos(x) Arccosine or inverse cosine of x where x mustbe between minus1 and 1 The return angle is inradians between 0 and π
atan(x) Arctangent or inverse tangent of x The returnangle is in radians between minusπ2 and π2
atan(xy) Arctangent or inverse tangent of xy The re-turn angle is in radians between minusπ and π de-pending on the sign of x and y
sec(x) Secant (1(cos x)) of x where x is in radianscsc(x) Cosecant (1(sin x)) of x where x is in radianscotg(x) Cotangent (1(tan x)) of x where x is in radians
Table B7 Some of Scilabrsquos utility functionsFunction Description
clear Clear variables from memoryhelp Display the first set of comment lines from
scriptswho List the variables in memorywhos List the variables in memory with their sizeerror Terminates the execution of the Sci-file and out-
puts the supplied error message
functions round(x) fix(x) floor(x) and ceil(x)
Exercise B5 The functions listed in Tables B5 B6 are only asmall sample of all of Scilabrsquos mathematical functions
Using Scilabrsquos help command list all of the hyperbolic func-tions available Also list any restrictions on the values of thefunction arguments
B52 Utility functions
In addition to the arithmetic functions Scilab has several utilityfunctions (see Table B7) These functions allow users to trackvariables (whoswho) feedback an error message if an error occursin the code (error) clear the variables form the memory (clear)etc Essential these functions allow users to debug and performbookkeeping operations inside their code
B6 Visualising data in Scilab 241
6
5
4- 04- 8
3
- 02
- 6
00
2- 4 Y- 2
02
Z
10
04
X 02
06
4 - 1
08
6- 28
10
Figure B2 Plot of z = sin(radicx2 + y2)
radicx2 + y2 illustrating the
3d capabilities of Scilab
B6 Visualising data in Scilab
One of the big advantages in using Scilab is the ability to visualisedata and results of complex calculations (see Figure B2 as exampleof the 3d capabilities) While Scilab is capability of complex 3dgraphics (see help pages for mesh and surf) in this section we willmesh amp surf
focus on the 2d plotting command (plot) Hence in this sectionplot
we will focus on
1 plotting mathematical functions from within Scilab
2 changing stylistic elements of the plot
3 adding titles labels and legends to the plot
4 including mathematical notation in titles labels and legendsand
5 plotting data stored in an external ASCII file
B61 Plotting mathematical functions from within Scilab
To plot a mathematical function in Scilab three steps are requiredThese are
1 Define the domain of the function (eg x=linspace(02pi))
2 Calculate the values of function at the selected points in thedomain (eg y=sin(x))
242 Appendix B Programming with Scilab
3 Final step is to plot the function using the plot commandplot
(eg plot(xy))
Using these steps it is possible to produce a basic plot of any math-ematical function (see Figure B3) Note that the plot shown inFigure B3 does not have a title axes labels or a legend all ofwhich are crucial for a good plot The Scilab commands to addthese elements to the plot are given in the following sections
- 10
- 08
- 06
- 04
- 02
00
02
04
06
08
10
0 1 2 3 4 5 6 7
Figure B3 The default plot of y = sin(x) in Scilab Notethat by default axes are not labelled and there is no title orlegend These need to added using the additional commandsxlabelylabeltitle and legend
Changing stylistic elements of the plot
An optional argument can be given to the plot command to cus-tomise each new line aspect This argument is a concatenatedstring containing information about colour line style or markersbased on the specifiers shown in Table B8 To specify the plotbe drawn with a red longdash-dot line with diamond marker thestring that would be passed to plot command would be rsquor-drsquo Thestring which is a concatenation (in any order) of the three typesof properties shown in Table B8 must be unambiguous Further-more the string specification is not case sensitive An exampleof specifying various line styles is shown in Example B6 and Fig-ure B4
Example B6 The following commands illustrate how line stylescan be used to distinguish between the plots of sin(2πt) sin(2πt+π6) and sin(2πt+ π2)
t=linspace(0130)
B6 Visualising data in Scilab 243
Table B8 Plot specifiers required to change line styles coloursand markers in plotsSpecifier Line style Specifier Colour Specifier Marker stylendash Solid line (default) r Red + Plus signndash ndash Dashed line g Green o Circle Dotted line b Blue Asteriskndash Dash-dotted line c Cyan x Cross
m Magenta s Squarey Yellow rsquononersquo No marker(default)k Blackw White
Figure B4 Resulting plotting after issuing the commands givenin Example B6
244 Appendix B Programming with Scilab
Example B6 illustrate a number of key commands relating to thecreation of multiple plots in Scilab The mtlb hold4 commandmtlb hold
holds the current plot and axis properties so that all subsequentplots are added to the existing plot
To distinguish each plot a different line specification is passed eachplot command In the example y = sin(2πt) is plotted using a redsolid line with data points indicated using a red circle as indicatedby the string rsquor-orsquo y = sin(2πt + π6) is plotted using greendashed-dotted line with data points marked by a square (rsquog-srsquo)while y = sin(2πt+ π2) is plotted using a cyan dashed line usingdiamonds to mark the data points (rsquoc--drsquo) Finally to completethe plot axes labels and a legend has been added to plot the com-mands for doing this are discussed in sectB62
B62 Adding titleslabels and legends to the plot
All good plots should have the following properties
1 The style of figure is appropriate to the information beingconveyed
2 The contents are well organised so it is easy for readers tointerpret the information
3 Legends labels etc are clear and concise (eg contain unitsif required do not obstruct parts of the plot etc) and
4 A caption or title that summarises the information presented
Like most mathematical applications Scilab has a wide variety ofdifferent plotting styles that can be used (eg look up the help forbar polarplot etc) This allows you to select the most appropri-barpolarplot
ate style to display and organise your data
The Scilab commands xlabel ylabel title and legend al-xlabel ylabel title and
legend lows you to add axes labels titles and legends to plot As shownin Example B6 the commands xlabel ylabel and title takea string input containing the title or label text For the legend
command a string is required for each separate plot Note theorder of the strings in the legend command reflects the order inwhich the each of plots were made
Including mathematical notation in titles labels and legends
It is possible to include mathematical characters such as π inScilab plots This is done by inserting special commands intoeither the title legend or label string These commands are basedon the abbreviations for mathematical notation used in programssuch as LATEX MathType etc A brief summary of the commands
4 mtlb hold on is equivalent to the Scilab commandset(gca()auto clearoff) while mtlb hold off is equivalentto the Scilab command set(gca()auto clearon)
B6 Visualising data in Scilab 245
Table B9 Basic mathematical symbols available in Scilab forincluding mathematical notation in titles labels and legendscommand symbol command symbol command symbolalpha α upsilon υ sim sim
Im = cup cup wp weierpotimes otimes subseteq sube oslash cap cap in isin supseteq supesupset sup lceil d subset subint
intcdot middot o oslash
rfloor c neg 6= nabla nablalfloor b times times ldots perp perp surd
radicprime prime
wedge and varpi $ O Oslashrceil e rangle 〉 mid |
vee or copyright ccopy langle 〈
available in Scilab for typesetting mathematical notation is givenin Table B9 To insert these commands into a text string simplystart and end the string with $ signs as shown in Example B7
Example B7 Scilab commands to produce the labelled plot ofy = sin
There are a few key concepts to note about the Example B7 Thefirst can be seen in the string being passed to the title commandtitle
In this string sin is prefixed with a backslash () This is doneso that the letters for sin are not typeset in italic which shouldonly be used for variables and not common mathematical func-tions Table B10 shows the command sequences to use for themore common mathematical functions5 To get the fraction in thetitle the command fracab has been used to produce a
b Ifsuperscripts and subscripts are required in a title or label these aresuperscripts amp subscripts
obtained by adding the subscript character ( ) and the superscriptcharacter (^) to the string These characters modify the characteror substring defined in braces immediately following For examplex 2 gives x2 while e^2x gives e2x
Note in Example B7 the commands xlabel and ylabel has beenxlabelylabel
used to set the axes labels in Figure B5 Instead of using thesethree commands the title and the axes labels could have been setusing the single xtitle command (see help xtitle for details)xtitle
In the example we have also customised the font size to make thetext more readable by a) passing an option to the title xlabelylabel commands and b) by using the command xset to set thexset
default font size before executing the xstring command Finallythe xstring command adds the text string to plot at the locationxstring
specified by variables b and c In the example the coordinates wewant to add the text are obtained by executing the xclick functionxclick
and clicking on the point required in the plot window
5 Additional details can be found in books covering Mathematics in LATEX
B6 Visualising data in Scilab 247
Table B10 Command sequences for the common mathematicalfunctions in Scilabarccos arcsin arctan cos cot
csc exp lim ln log
max min sec sin tan
B63 Plotting numeric data stored in an ASCII file
As data can be stored in a wide variety of ways in external filesScilab has a wide variety of functions which can be used to importthe data stored in these files (see help for the commands read
fscanf and xlsread) Since data can be stored in such a wideread fscanf and xlsread
range of formats we will confine our discussions here to the situa-tion where numeric data is stored in a ASCII (text) file where eachrow of the file contains the same number of entries
To illustrate the procedure for plotting data stored in ASCII file wewill consider the data obtained measuring the temperature of a cupof coffee every minute for a total of 100 mins The data recorded isstored in the a file called coffeedatatxt6 which simply containsthe two columns of numbers shown in Table B11 As an examplethe first 10 lines of the text file contains the following numbers7
00000 1000000
10000 950000
20000 903125
30000 859180
40000 817981
50000 779357
60000 743147
70000 709201
80000 677376
90000 647540
Note the current version of Scilab has a builtin text editor whichcan be used to view such text files To access this editor simplytype the command edit at the command prompt Once open youedit
can use the graphical interface to open the required file
The first issue is to get the data into Scilab so it is available forplot command In Scilab the command read can used to importread
the data and assign it to a variable In the current example wewould use the following command to import the data and assignit to a variable called cdata assuming the data is in the currentdirectory8
6 The file coffeedatatxt has been attached and should be available for savingprovided you are using Adobe Reader To save the file right click on thepurple Windows version or the red MacOSLinux version in the margin andselect save file from the options Ensure you save the file as coffeedatatxt
7 normally you will have received the data file in this format8 You can check what directory Scilab is in by typing the command pwd
Figure B6 Temperature of coffee based on the data recorded inthe file coffeedatatxt
cdata=read(rsquocoffeedatatxtrsquo-12)
The -12 following the name of the file tells Scilab that you donrsquotknow how many rows you have but you have two columns of dataSince we know that we have 101 rows of data we could have alsoused the following command to load the data
cdata=read(rsquocoffeedatatxtrsquo1012)
We now have a variable called cdata which has two columns ofdata The first column contains the time while the second containsthe Temperature records Hence we can plot it using the followingcommand
plot(cdata(1)cdata(2))
This gives the plot shown in Figure B6 after appropriate labelshave been added as described above
B7 Script files
Executing commands entered at the keyboard or keypad is themost primitive use of a calculating device It is only when se-quences of operations (to perform complex tasks) are stored awayand re-called for repeated application with different inputs that wehave a programmed operation Complex programs must necessarilyhave an interface with the user The basic interface tool in Scilabcapable of executing sequences of commands that are stored in a fileis the script The file names must have an extension of sce forexample program1sce assignment1sce These files are calledScript Sci-files or simply scripts
A script file is a file of standard keyboard charactersthat define commands to drive a program A script filecan be generated using any text editor or word proces-sor
To invoke a script in Scilab type the commandexec
exec filenamesce
where filename denotes the actual name of the file containing thescript When the script runs it is exactly the same as if you typedthe commands into the command window from the keyboard Thismeans that the Sci-file can use any variables you have defined priorto invoking the script Any variables created in the script are alsoavailable when the script is finished and any variable whose namewas assigned a value in the script will be over-written
If you give a script the same name as any variable that it computesin the running of the script then Scilab will give precedence tothe workspace variable over a script As stated in sectB3 the variablename will hide the script name after the script has run Scilab willtake the name as referring to the variable and not to the definedscript
Scilab also needs to know where to find the script to executedIf Scilab cannot find the script it will report that you have anundefined function or variable
The exact procedure for creating and saving Sci-files depends onthe Scilab version the operating system (MS Windows Macin-tosh or UnixLinux ) and the text editor you use The currentversion of Scilab includes a built-in editor which can be used tocreate and edit Sci-files and text files
To create a new Sci-file under the current version of Scilab simplytype edit newfilenamesce This will open up the Scilab editoredit
with a blank document called newfilename
B71 Script input and output
A script is a named file (with the extension sce) containing asequence of Scilab instructions which when its name is executedin a session performs those instructions in order as though theywere entered at the console
There are two common uses for scripts Pragmatically we maywant to save a particular computational sequence for later use orreference If the tasks involved are to be repeated often thenthe nature of the tasks should be analysed formulated into usefulfunctions (which we will discuss later) and the whole procedurepresented in a single function But if the tasks are only likely toget occasional use then the script may be adequately commentedand stored as a script file It is a quick and dirty way of saving
B7 Script files 251
the instructions for doing a particular job when we do not wantto go to the trouble of writing the functions for the job It is nota highly recommended modus operandi but useful all the same
The second use is for the script to act like a storyboard presentinginformation to a user in a structured and controlled way permit-ting the user to see the steps of a computational procedure andtheir outputs up on the screen At a more sophisticated level thescript takes on the proportions of an application controlling a largeand complex interaction with advanced interface features We willnot take our study of Scilab programming to this level but workmostly at the storyboard level using the simplest of Scilabrsquos in-terface tools
All the variables created in a script become global variables definedin the workspace and available for perusal As an application itis assumed that a script controls the entire workspace and thevariables it creates are lsquothe datarsquo of the application When scriptsare used as storyboards on a small scale then it cannot be assumedthat the variables they create deserve global recognition and it iscourteous to erase these variables when the script is finished unlessthey have some possible further use An unfortunate side-effect isthat if variables with the same name already exist in the workspacebefore a script is run then these variables will be over-written andlost
Used as storyboards scripts allow the user to cut paste and re-trythe various steps of a procedure and to modify the script to suithis own needs Storyboards may be used for demonstrations oftoolbox utilities or extended documentationexplanation of func-tions However if there is any danger that variables defined bythe script may over-write workspace variables then the demo ordocumentation should be re-fashioned into a function that leavesthe workspace variables untouched (more on this later)
Scripts will often read data from pre-prepared files and we will dis-cuss the functions provided by Scilab to perform this task below
Gathering data entered by the user really falls into the realm ofinterface design but we only briefly consider the simplest interfacefunction provided by Scilab Similarly we briefly consider thefunctions for providing the simplest forms of output presentation
Keyboard input
Scilab has a built-in function input that can be used to promptinput
the user to enter the value of a variable from the keyboard Forexample
Tc = input(rsquoPlease enter the temperature in degrees Celsiusrsquo)
will print the message
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
236 Appendix B Programming with Scilab
Table B4 Scilab array operatorsOperator Meaning Scilab
^ Exponentiation x^y
Multiply xy
Right Division xy
Left Division yx
+ Addition x+y
- Subtraction x-y
Table B4 lists the Scilab array operators
The array addition and subtraction are identical to matrix additionor subtraction Both array and matrix addition and subtractionoperate element by element Array addition and subtraction re-quire that both arrays have the same size The only exception tothis rule in Scilab occurs when we add or subtract a scalar to orfrom an array In this case the scalar is added or subtracted fromeach element in the array
For example
gt [6 5 6] + 3
ans =
[9 8 9]
gt [6 4 5 8] + [3 1 2 4]
ans =
[9 5 7 12]
gt [4 3] - 7
ans =
[-3 -4]
Array multiplication and division is very different to matrix mul-tiplication and division Array multiplication or division is an ele-ment by element multiplication or division that produces an arraywith the same dimensions as the starting arrays As with addi-tion and subtraction array multiplication and division require thatboth arrays have the same dimensions
Example B2 Consider the two row vectors x=[2 4 7 1] andy=[6 1 3 2] Scilab does not recognise the possibility ofmatrix multiplication of these two vectors They are not com-patible the number of columns of the pre-multiplier must bethe same as the number of rows of the post-multiplier Anattempt to execute x y will bring an error message
Obtain the usual dot product of the two row vectors by trans-posing the post-multiplier as in xyrsquo or yxrsquo The matrixproduct xrsquoy produces a matrix of 4 rows and 4 columns
B5 Builtndashin functions 237
Array multiplication produces the result
gt xy
ans =
[12 4 21 2]
by multiplying corresponding elements of each vector Theresult is a row vector with the same dimensions as x and ythat is a 1times 4 vector
Example B3 Consider the row vectors x=[2 4 7 1] and col-umn vector y=[6 1 3 2]rsquo Array multiplication of thesetwo matrices is not possible because their dimensions do notagree
Matrix multiplication xy produces the result 39 that is theinner product or dot product of the vectors Note vectorsas such do not exist in Scilab there are only single rowedor single columned arrays
Scilab not only allows arrays to be raised to powers but alsoallows scalars and arrays to be raised by array powers To performexponentiation on an element-by-element basis use the ^ symbolFor example if x=[3 5 8] then x^3 produces the matrix [33 53
83] = [27 125 512]
A scalar can be raised to an array power For example if p=[2 4 5]then typing 3^p produces the array [32 34 35] = [9 81 243]
With element-by-element array operations it is important to re-member that the dot () is a part of a two-character symbol for aparticular array operator such as () () () (^) Sometimesthis can cause some confusion the dot in 3^p is not a decimalpoint associated with the number 3 The following operations withthe value of p=[2 4 5] are equivalent
3^p
30^p
3^p
(3)^p
3^[2 4 5]
B5 Builtndashin functions
B51 Arithmetic functions
Arithmetic expressions often require computations other than ad-dition subtraction multiplication division and exponentiationFor example many expressions require the use of logarithms ex-ponentials and trigonometric functions Scilab has predefined alarge number of functions that perform these types of computations(see Tables B5 and B6)
238 Appendix B Programming with Scilab
An example of a predefined function is sin which returns the sineof a given angle
m = sin(angle)
where the variable angle contains the angle in radians and theresult is stored in the variable m The variable angle is called theargument or parameter of the function If the angle is stored indegrees it must be converted to radians for example
m = sin(anglepi180)
or
radians =anglepi180
m = sin(radians)
In Scilab the arguments or parameters of the function are con-tained in parentheses following the name of the function This isreferred to as pre-fix notation Many common functions in math-ematics have a syntax using in-fix notation where two argumentsappear to the right and left of the function name as for examplein x+y atimes b nmod r etc Scilab does not support in-fix nota-tion for any but the arithmetic functions Classical mathematicalnotation also has examples of post-fix notationmdash n x squared mdashand other hybrids like |x| while Scilab has ahb to confound thepurist
A function may contain no arguments one argument or many ar-guments depending on its definition If a function contains morethan one argument it is vital to give the arguments in the cor-rect order For example consider the problem of finding θ in thefollowing triangle using the atan command
5
8θ
Now by hand calculation the answer is 05586 radians If we useatan as below we get the answers shown
--gt atan(58)
ans =
05586
--gt atan(85)
ans =
10122
The first answer is the correct one The second answer is incorrect
B5 Builtndashin functions 239
Table B5 Some common mathematical functions defined inScilabFunction Description
abs(x) Computes the absolute value of xsqrt(x) Square root of x The parameter can be nega-
tiveround(x) Round x to the nearest integerfix(x) Round x to the nearest integer toward zerofloor(x) Round x to the nearest integer toward minusinfinceil(x) Round x to the nearest integer toward +infinsign(x) Returns -1 if x is less than zero zero if x equals
zero and 1 otherwisemodulo(xy) Returns the remainder of xy For example
modulo(254) is 1exp(x) Returns ex
log(x) Returns the natural logarithm of x That isthe logarithm to the base e Parameter can benegative but cannot be zero
log10(x) Returns the common logarithm of x That isthe logarithm to the base 10 Parameter can benegative but cannot be zero
since we passed the values to the function in the incorrect order
Some functions also require that the arguments be in specific unitsFor example the trigonometric functions assume that the argu-ments are in radians
In Scilab some functions use the number of arguments to deter-mine the output of the function For example the zeros functioncan have one or two arguments which then determine the output(check this with the command help zeros)
A function reference cannot be made on the left hand side of anassignment since a function call always returns a value(s) whereasan assignment is always to a variable Functions can of courseappear on the right side of an assignment since they return valuesto be assigned to some variable Functions may be applied tothe output of other functions and may be called in the algorithmdefining a function for example
logx = log(abs(x))
When one function is used to compute the argument of anotherfunction be sure to enclose the argument of each function in itsown set of parentheses
Scilab allows recursive functions that is functions that call them-selves in their defining algorithm
Exercise B4 Use examples to show the differences between the
240 Appendix B Programming with Scilab
Table B6 Some common trigonometric functions defined inScilab
Function Description
sin(x) Sine of x where x is in radianscos(x) Cosine of x where x is in radianstan(x) Tangent of x where x is in radiansasin(x) Arcsine or inverse sine of x where x must be
between -1 and 1 The return angle is in radiansbetween minusπ2 and π2
acos(x) Arccosine or inverse cosine of x where x mustbe between minus1 and 1 The return angle is inradians between 0 and π
atan(x) Arctangent or inverse tangent of x The returnangle is in radians between minusπ2 and π2
atan(xy) Arctangent or inverse tangent of xy The re-turn angle is in radians between minusπ and π de-pending on the sign of x and y
sec(x) Secant (1(cos x)) of x where x is in radianscsc(x) Cosecant (1(sin x)) of x where x is in radianscotg(x) Cotangent (1(tan x)) of x where x is in radians
Table B7 Some of Scilabrsquos utility functionsFunction Description
clear Clear variables from memoryhelp Display the first set of comment lines from
scriptswho List the variables in memorywhos List the variables in memory with their sizeerror Terminates the execution of the Sci-file and out-
puts the supplied error message
functions round(x) fix(x) floor(x) and ceil(x)
Exercise B5 The functions listed in Tables B5 B6 are only asmall sample of all of Scilabrsquos mathematical functions
Using Scilabrsquos help command list all of the hyperbolic func-tions available Also list any restrictions on the values of thefunction arguments
B52 Utility functions
In addition to the arithmetic functions Scilab has several utilityfunctions (see Table B7) These functions allow users to trackvariables (whoswho) feedback an error message if an error occursin the code (error) clear the variables form the memory (clear)etc Essential these functions allow users to debug and performbookkeeping operations inside their code
B6 Visualising data in Scilab 241
6
5
4- 04- 8
3
- 02
- 6
00
2- 4 Y- 2
02
Z
10
04
X 02
06
4 - 1
08
6- 28
10
Figure B2 Plot of z = sin(radicx2 + y2)
radicx2 + y2 illustrating the
3d capabilities of Scilab
B6 Visualising data in Scilab
One of the big advantages in using Scilab is the ability to visualisedata and results of complex calculations (see Figure B2 as exampleof the 3d capabilities) While Scilab is capability of complex 3dgraphics (see help pages for mesh and surf) in this section we willmesh amp surf
focus on the 2d plotting command (plot) Hence in this sectionplot
we will focus on
1 plotting mathematical functions from within Scilab
2 changing stylistic elements of the plot
3 adding titles labels and legends to the plot
4 including mathematical notation in titles labels and legendsand
5 plotting data stored in an external ASCII file
B61 Plotting mathematical functions from within Scilab
To plot a mathematical function in Scilab three steps are requiredThese are
1 Define the domain of the function (eg x=linspace(02pi))
2 Calculate the values of function at the selected points in thedomain (eg y=sin(x))
242 Appendix B Programming with Scilab
3 Final step is to plot the function using the plot commandplot
(eg plot(xy))
Using these steps it is possible to produce a basic plot of any math-ematical function (see Figure B3) Note that the plot shown inFigure B3 does not have a title axes labels or a legend all ofwhich are crucial for a good plot The Scilab commands to addthese elements to the plot are given in the following sections
- 10
- 08
- 06
- 04
- 02
00
02
04
06
08
10
0 1 2 3 4 5 6 7
Figure B3 The default plot of y = sin(x) in Scilab Notethat by default axes are not labelled and there is no title orlegend These need to added using the additional commandsxlabelylabeltitle and legend
Changing stylistic elements of the plot
An optional argument can be given to the plot command to cus-tomise each new line aspect This argument is a concatenatedstring containing information about colour line style or markersbased on the specifiers shown in Table B8 To specify the plotbe drawn with a red longdash-dot line with diamond marker thestring that would be passed to plot command would be rsquor-drsquo Thestring which is a concatenation (in any order) of the three typesof properties shown in Table B8 must be unambiguous Further-more the string specification is not case sensitive An exampleof specifying various line styles is shown in Example B6 and Fig-ure B4
Example B6 The following commands illustrate how line stylescan be used to distinguish between the plots of sin(2πt) sin(2πt+π6) and sin(2πt+ π2)
t=linspace(0130)
B6 Visualising data in Scilab 243
Table B8 Plot specifiers required to change line styles coloursand markers in plotsSpecifier Line style Specifier Colour Specifier Marker stylendash Solid line (default) r Red + Plus signndash ndash Dashed line g Green o Circle Dotted line b Blue Asteriskndash Dash-dotted line c Cyan x Cross
m Magenta s Squarey Yellow rsquononersquo No marker(default)k Blackw White
Figure B4 Resulting plotting after issuing the commands givenin Example B6
244 Appendix B Programming with Scilab
Example B6 illustrate a number of key commands relating to thecreation of multiple plots in Scilab The mtlb hold4 commandmtlb hold
holds the current plot and axis properties so that all subsequentplots are added to the existing plot
To distinguish each plot a different line specification is passed eachplot command In the example y = sin(2πt) is plotted using a redsolid line with data points indicated using a red circle as indicatedby the string rsquor-orsquo y = sin(2πt + π6) is plotted using greendashed-dotted line with data points marked by a square (rsquog-srsquo)while y = sin(2πt+ π2) is plotted using a cyan dashed line usingdiamonds to mark the data points (rsquoc--drsquo) Finally to completethe plot axes labels and a legend has been added to plot the com-mands for doing this are discussed in sectB62
B62 Adding titleslabels and legends to the plot
All good plots should have the following properties
1 The style of figure is appropriate to the information beingconveyed
2 The contents are well organised so it is easy for readers tointerpret the information
3 Legends labels etc are clear and concise (eg contain unitsif required do not obstruct parts of the plot etc) and
4 A caption or title that summarises the information presented
Like most mathematical applications Scilab has a wide variety ofdifferent plotting styles that can be used (eg look up the help forbar polarplot etc) This allows you to select the most appropri-barpolarplot
ate style to display and organise your data
The Scilab commands xlabel ylabel title and legend al-xlabel ylabel title and
legend lows you to add axes labels titles and legends to plot As shownin Example B6 the commands xlabel ylabel and title takea string input containing the title or label text For the legend
command a string is required for each separate plot Note theorder of the strings in the legend command reflects the order inwhich the each of plots were made
Including mathematical notation in titles labels and legends
It is possible to include mathematical characters such as π inScilab plots This is done by inserting special commands intoeither the title legend or label string These commands are basedon the abbreviations for mathematical notation used in programssuch as LATEX MathType etc A brief summary of the commands
4 mtlb hold on is equivalent to the Scilab commandset(gca()auto clearoff) while mtlb hold off is equivalentto the Scilab command set(gca()auto clearon)
B6 Visualising data in Scilab 245
Table B9 Basic mathematical symbols available in Scilab forincluding mathematical notation in titles labels and legendscommand symbol command symbol command symbolalpha α upsilon υ sim sim
Im = cup cup wp weierpotimes otimes subseteq sube oslash cap cap in isin supseteq supesupset sup lceil d subset subint
intcdot middot o oslash
rfloor c neg 6= nabla nablalfloor b times times ldots perp perp surd
radicprime prime
wedge and varpi $ O Oslashrceil e rangle 〉 mid |
vee or copyright ccopy langle 〈
available in Scilab for typesetting mathematical notation is givenin Table B9 To insert these commands into a text string simplystart and end the string with $ signs as shown in Example B7
Example B7 Scilab commands to produce the labelled plot ofy = sin
There are a few key concepts to note about the Example B7 Thefirst can be seen in the string being passed to the title commandtitle
In this string sin is prefixed with a backslash () This is doneso that the letters for sin are not typeset in italic which shouldonly be used for variables and not common mathematical func-tions Table B10 shows the command sequences to use for themore common mathematical functions5 To get the fraction in thetitle the command fracab has been used to produce a
b Ifsuperscripts and subscripts are required in a title or label these aresuperscripts amp subscripts
obtained by adding the subscript character ( ) and the superscriptcharacter (^) to the string These characters modify the characteror substring defined in braces immediately following For examplex 2 gives x2 while e^2x gives e2x
Note in Example B7 the commands xlabel and ylabel has beenxlabelylabel
used to set the axes labels in Figure B5 Instead of using thesethree commands the title and the axes labels could have been setusing the single xtitle command (see help xtitle for details)xtitle
In the example we have also customised the font size to make thetext more readable by a) passing an option to the title xlabelylabel commands and b) by using the command xset to set thexset
default font size before executing the xstring command Finallythe xstring command adds the text string to plot at the locationxstring
specified by variables b and c In the example the coordinates wewant to add the text are obtained by executing the xclick functionxclick
and clicking on the point required in the plot window
5 Additional details can be found in books covering Mathematics in LATEX
B6 Visualising data in Scilab 247
Table B10 Command sequences for the common mathematicalfunctions in Scilabarccos arcsin arctan cos cot
csc exp lim ln log
max min sec sin tan
B63 Plotting numeric data stored in an ASCII file
As data can be stored in a wide variety of ways in external filesScilab has a wide variety of functions which can be used to importthe data stored in these files (see help for the commands read
fscanf and xlsread) Since data can be stored in such a wideread fscanf and xlsread
range of formats we will confine our discussions here to the situa-tion where numeric data is stored in a ASCII (text) file where eachrow of the file contains the same number of entries
To illustrate the procedure for plotting data stored in ASCII file wewill consider the data obtained measuring the temperature of a cupof coffee every minute for a total of 100 mins The data recorded isstored in the a file called coffeedatatxt6 which simply containsthe two columns of numbers shown in Table B11 As an examplethe first 10 lines of the text file contains the following numbers7
00000 1000000
10000 950000
20000 903125
30000 859180
40000 817981
50000 779357
60000 743147
70000 709201
80000 677376
90000 647540
Note the current version of Scilab has a builtin text editor whichcan be used to view such text files To access this editor simplytype the command edit at the command prompt Once open youedit
can use the graphical interface to open the required file
The first issue is to get the data into Scilab so it is available forplot command In Scilab the command read can used to importread
the data and assign it to a variable In the current example wewould use the following command to import the data and assignit to a variable called cdata assuming the data is in the currentdirectory8
6 The file coffeedatatxt has been attached and should be available for savingprovided you are using Adobe Reader To save the file right click on thepurple Windows version or the red MacOSLinux version in the margin andselect save file from the options Ensure you save the file as coffeedatatxt
7 normally you will have received the data file in this format8 You can check what directory Scilab is in by typing the command pwd
Figure B6 Temperature of coffee based on the data recorded inthe file coffeedatatxt
cdata=read(rsquocoffeedatatxtrsquo-12)
The -12 following the name of the file tells Scilab that you donrsquotknow how many rows you have but you have two columns of dataSince we know that we have 101 rows of data we could have alsoused the following command to load the data
cdata=read(rsquocoffeedatatxtrsquo1012)
We now have a variable called cdata which has two columns ofdata The first column contains the time while the second containsthe Temperature records Hence we can plot it using the followingcommand
plot(cdata(1)cdata(2))
This gives the plot shown in Figure B6 after appropriate labelshave been added as described above
B7 Script files
Executing commands entered at the keyboard or keypad is themost primitive use of a calculating device It is only when se-quences of operations (to perform complex tasks) are stored awayand re-called for repeated application with different inputs that wehave a programmed operation Complex programs must necessarilyhave an interface with the user The basic interface tool in Scilabcapable of executing sequences of commands that are stored in a fileis the script The file names must have an extension of sce forexample program1sce assignment1sce These files are calledScript Sci-files or simply scripts
A script file is a file of standard keyboard charactersthat define commands to drive a program A script filecan be generated using any text editor or word proces-sor
To invoke a script in Scilab type the commandexec
exec filenamesce
where filename denotes the actual name of the file containing thescript When the script runs it is exactly the same as if you typedthe commands into the command window from the keyboard Thismeans that the Sci-file can use any variables you have defined priorto invoking the script Any variables created in the script are alsoavailable when the script is finished and any variable whose namewas assigned a value in the script will be over-written
If you give a script the same name as any variable that it computesin the running of the script then Scilab will give precedence tothe workspace variable over a script As stated in sectB3 the variablename will hide the script name after the script has run Scilab willtake the name as referring to the variable and not to the definedscript
Scilab also needs to know where to find the script to executedIf Scilab cannot find the script it will report that you have anundefined function or variable
The exact procedure for creating and saving Sci-files depends onthe Scilab version the operating system (MS Windows Macin-tosh or UnixLinux ) and the text editor you use The currentversion of Scilab includes a built-in editor which can be used tocreate and edit Sci-files and text files
To create a new Sci-file under the current version of Scilab simplytype edit newfilenamesce This will open up the Scilab editoredit
with a blank document called newfilename
B71 Script input and output
A script is a named file (with the extension sce) containing asequence of Scilab instructions which when its name is executedin a session performs those instructions in order as though theywere entered at the console
There are two common uses for scripts Pragmatically we maywant to save a particular computational sequence for later use orreference If the tasks involved are to be repeated often thenthe nature of the tasks should be analysed formulated into usefulfunctions (which we will discuss later) and the whole procedurepresented in a single function But if the tasks are only likely toget occasional use then the script may be adequately commentedand stored as a script file It is a quick and dirty way of saving
B7 Script files 251
the instructions for doing a particular job when we do not wantto go to the trouble of writing the functions for the job It is nota highly recommended modus operandi but useful all the same
The second use is for the script to act like a storyboard presentinginformation to a user in a structured and controlled way permit-ting the user to see the steps of a computational procedure andtheir outputs up on the screen At a more sophisticated level thescript takes on the proportions of an application controlling a largeand complex interaction with advanced interface features We willnot take our study of Scilab programming to this level but workmostly at the storyboard level using the simplest of Scilabrsquos in-terface tools
All the variables created in a script become global variables definedin the workspace and available for perusal As an application itis assumed that a script controls the entire workspace and thevariables it creates are lsquothe datarsquo of the application When scriptsare used as storyboards on a small scale then it cannot be assumedthat the variables they create deserve global recognition and it iscourteous to erase these variables when the script is finished unlessthey have some possible further use An unfortunate side-effect isthat if variables with the same name already exist in the workspacebefore a script is run then these variables will be over-written andlost
Used as storyboards scripts allow the user to cut paste and re-trythe various steps of a procedure and to modify the script to suithis own needs Storyboards may be used for demonstrations oftoolbox utilities or extended documentationexplanation of func-tions However if there is any danger that variables defined bythe script may over-write workspace variables then the demo ordocumentation should be re-fashioned into a function that leavesthe workspace variables untouched (more on this later)
Scripts will often read data from pre-prepared files and we will dis-cuss the functions provided by Scilab to perform this task below
Gathering data entered by the user really falls into the realm ofinterface design but we only briefly consider the simplest interfacefunction provided by Scilab Similarly we briefly consider thefunctions for providing the simplest forms of output presentation
Keyboard input
Scilab has a built-in function input that can be used to promptinput
the user to enter the value of a variable from the keyboard Forexample
Tc = input(rsquoPlease enter the temperature in degrees Celsiusrsquo)
will print the message
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
B5 Builtndashin functions 237
Array multiplication produces the result
gt xy
ans =
[12 4 21 2]
by multiplying corresponding elements of each vector Theresult is a row vector with the same dimensions as x and ythat is a 1times 4 vector
Example B3 Consider the row vectors x=[2 4 7 1] and col-umn vector y=[6 1 3 2]rsquo Array multiplication of thesetwo matrices is not possible because their dimensions do notagree
Matrix multiplication xy produces the result 39 that is theinner product or dot product of the vectors Note vectorsas such do not exist in Scilab there are only single rowedor single columned arrays
Scilab not only allows arrays to be raised to powers but alsoallows scalars and arrays to be raised by array powers To performexponentiation on an element-by-element basis use the ^ symbolFor example if x=[3 5 8] then x^3 produces the matrix [33 53
83] = [27 125 512]
A scalar can be raised to an array power For example if p=[2 4 5]then typing 3^p produces the array [32 34 35] = [9 81 243]
With element-by-element array operations it is important to re-member that the dot () is a part of a two-character symbol for aparticular array operator such as () () () (^) Sometimesthis can cause some confusion the dot in 3^p is not a decimalpoint associated with the number 3 The following operations withthe value of p=[2 4 5] are equivalent
3^p
30^p
3^p
(3)^p
3^[2 4 5]
B5 Builtndashin functions
B51 Arithmetic functions
Arithmetic expressions often require computations other than ad-dition subtraction multiplication division and exponentiationFor example many expressions require the use of logarithms ex-ponentials and trigonometric functions Scilab has predefined alarge number of functions that perform these types of computations(see Tables B5 and B6)
238 Appendix B Programming with Scilab
An example of a predefined function is sin which returns the sineof a given angle
m = sin(angle)
where the variable angle contains the angle in radians and theresult is stored in the variable m The variable angle is called theargument or parameter of the function If the angle is stored indegrees it must be converted to radians for example
m = sin(anglepi180)
or
radians =anglepi180
m = sin(radians)
In Scilab the arguments or parameters of the function are con-tained in parentheses following the name of the function This isreferred to as pre-fix notation Many common functions in math-ematics have a syntax using in-fix notation where two argumentsappear to the right and left of the function name as for examplein x+y atimes b nmod r etc Scilab does not support in-fix nota-tion for any but the arithmetic functions Classical mathematicalnotation also has examples of post-fix notationmdash n x squared mdashand other hybrids like |x| while Scilab has ahb to confound thepurist
A function may contain no arguments one argument or many ar-guments depending on its definition If a function contains morethan one argument it is vital to give the arguments in the cor-rect order For example consider the problem of finding θ in thefollowing triangle using the atan command
5
8θ
Now by hand calculation the answer is 05586 radians If we useatan as below we get the answers shown
--gt atan(58)
ans =
05586
--gt atan(85)
ans =
10122
The first answer is the correct one The second answer is incorrect
B5 Builtndashin functions 239
Table B5 Some common mathematical functions defined inScilabFunction Description
abs(x) Computes the absolute value of xsqrt(x) Square root of x The parameter can be nega-
tiveround(x) Round x to the nearest integerfix(x) Round x to the nearest integer toward zerofloor(x) Round x to the nearest integer toward minusinfinceil(x) Round x to the nearest integer toward +infinsign(x) Returns -1 if x is less than zero zero if x equals
zero and 1 otherwisemodulo(xy) Returns the remainder of xy For example
modulo(254) is 1exp(x) Returns ex
log(x) Returns the natural logarithm of x That isthe logarithm to the base e Parameter can benegative but cannot be zero
log10(x) Returns the common logarithm of x That isthe logarithm to the base 10 Parameter can benegative but cannot be zero
since we passed the values to the function in the incorrect order
Some functions also require that the arguments be in specific unitsFor example the trigonometric functions assume that the argu-ments are in radians
In Scilab some functions use the number of arguments to deter-mine the output of the function For example the zeros functioncan have one or two arguments which then determine the output(check this with the command help zeros)
A function reference cannot be made on the left hand side of anassignment since a function call always returns a value(s) whereasan assignment is always to a variable Functions can of courseappear on the right side of an assignment since they return valuesto be assigned to some variable Functions may be applied tothe output of other functions and may be called in the algorithmdefining a function for example
logx = log(abs(x))
When one function is used to compute the argument of anotherfunction be sure to enclose the argument of each function in itsown set of parentheses
Scilab allows recursive functions that is functions that call them-selves in their defining algorithm
Exercise B4 Use examples to show the differences between the
240 Appendix B Programming with Scilab
Table B6 Some common trigonometric functions defined inScilab
Function Description
sin(x) Sine of x where x is in radianscos(x) Cosine of x where x is in radianstan(x) Tangent of x where x is in radiansasin(x) Arcsine or inverse sine of x where x must be
between -1 and 1 The return angle is in radiansbetween minusπ2 and π2
acos(x) Arccosine or inverse cosine of x where x mustbe between minus1 and 1 The return angle is inradians between 0 and π
atan(x) Arctangent or inverse tangent of x The returnangle is in radians between minusπ2 and π2
atan(xy) Arctangent or inverse tangent of xy The re-turn angle is in radians between minusπ and π de-pending on the sign of x and y
sec(x) Secant (1(cos x)) of x where x is in radianscsc(x) Cosecant (1(sin x)) of x where x is in radianscotg(x) Cotangent (1(tan x)) of x where x is in radians
Table B7 Some of Scilabrsquos utility functionsFunction Description
clear Clear variables from memoryhelp Display the first set of comment lines from
scriptswho List the variables in memorywhos List the variables in memory with their sizeerror Terminates the execution of the Sci-file and out-
puts the supplied error message
functions round(x) fix(x) floor(x) and ceil(x)
Exercise B5 The functions listed in Tables B5 B6 are only asmall sample of all of Scilabrsquos mathematical functions
Using Scilabrsquos help command list all of the hyperbolic func-tions available Also list any restrictions on the values of thefunction arguments
B52 Utility functions
In addition to the arithmetic functions Scilab has several utilityfunctions (see Table B7) These functions allow users to trackvariables (whoswho) feedback an error message if an error occursin the code (error) clear the variables form the memory (clear)etc Essential these functions allow users to debug and performbookkeeping operations inside their code
B6 Visualising data in Scilab 241
6
5
4- 04- 8
3
- 02
- 6
00
2- 4 Y- 2
02
Z
10
04
X 02
06
4 - 1
08
6- 28
10
Figure B2 Plot of z = sin(radicx2 + y2)
radicx2 + y2 illustrating the
3d capabilities of Scilab
B6 Visualising data in Scilab
One of the big advantages in using Scilab is the ability to visualisedata and results of complex calculations (see Figure B2 as exampleof the 3d capabilities) While Scilab is capability of complex 3dgraphics (see help pages for mesh and surf) in this section we willmesh amp surf
focus on the 2d plotting command (plot) Hence in this sectionplot
we will focus on
1 plotting mathematical functions from within Scilab
2 changing stylistic elements of the plot
3 adding titles labels and legends to the plot
4 including mathematical notation in titles labels and legendsand
5 plotting data stored in an external ASCII file
B61 Plotting mathematical functions from within Scilab
To plot a mathematical function in Scilab three steps are requiredThese are
1 Define the domain of the function (eg x=linspace(02pi))
2 Calculate the values of function at the selected points in thedomain (eg y=sin(x))
242 Appendix B Programming with Scilab
3 Final step is to plot the function using the plot commandplot
(eg plot(xy))
Using these steps it is possible to produce a basic plot of any math-ematical function (see Figure B3) Note that the plot shown inFigure B3 does not have a title axes labels or a legend all ofwhich are crucial for a good plot The Scilab commands to addthese elements to the plot are given in the following sections
- 10
- 08
- 06
- 04
- 02
00
02
04
06
08
10
0 1 2 3 4 5 6 7
Figure B3 The default plot of y = sin(x) in Scilab Notethat by default axes are not labelled and there is no title orlegend These need to added using the additional commandsxlabelylabeltitle and legend
Changing stylistic elements of the plot
An optional argument can be given to the plot command to cus-tomise each new line aspect This argument is a concatenatedstring containing information about colour line style or markersbased on the specifiers shown in Table B8 To specify the plotbe drawn with a red longdash-dot line with diamond marker thestring that would be passed to plot command would be rsquor-drsquo Thestring which is a concatenation (in any order) of the three typesof properties shown in Table B8 must be unambiguous Further-more the string specification is not case sensitive An exampleof specifying various line styles is shown in Example B6 and Fig-ure B4
Example B6 The following commands illustrate how line stylescan be used to distinguish between the plots of sin(2πt) sin(2πt+π6) and sin(2πt+ π2)
t=linspace(0130)
B6 Visualising data in Scilab 243
Table B8 Plot specifiers required to change line styles coloursand markers in plotsSpecifier Line style Specifier Colour Specifier Marker stylendash Solid line (default) r Red + Plus signndash ndash Dashed line g Green o Circle Dotted line b Blue Asteriskndash Dash-dotted line c Cyan x Cross
m Magenta s Squarey Yellow rsquononersquo No marker(default)k Blackw White
Figure B4 Resulting plotting after issuing the commands givenin Example B6
244 Appendix B Programming with Scilab
Example B6 illustrate a number of key commands relating to thecreation of multiple plots in Scilab The mtlb hold4 commandmtlb hold
holds the current plot and axis properties so that all subsequentplots are added to the existing plot
To distinguish each plot a different line specification is passed eachplot command In the example y = sin(2πt) is plotted using a redsolid line with data points indicated using a red circle as indicatedby the string rsquor-orsquo y = sin(2πt + π6) is plotted using greendashed-dotted line with data points marked by a square (rsquog-srsquo)while y = sin(2πt+ π2) is plotted using a cyan dashed line usingdiamonds to mark the data points (rsquoc--drsquo) Finally to completethe plot axes labels and a legend has been added to plot the com-mands for doing this are discussed in sectB62
B62 Adding titleslabels and legends to the plot
All good plots should have the following properties
1 The style of figure is appropriate to the information beingconveyed
2 The contents are well organised so it is easy for readers tointerpret the information
3 Legends labels etc are clear and concise (eg contain unitsif required do not obstruct parts of the plot etc) and
4 A caption or title that summarises the information presented
Like most mathematical applications Scilab has a wide variety ofdifferent plotting styles that can be used (eg look up the help forbar polarplot etc) This allows you to select the most appropri-barpolarplot
ate style to display and organise your data
The Scilab commands xlabel ylabel title and legend al-xlabel ylabel title and
legend lows you to add axes labels titles and legends to plot As shownin Example B6 the commands xlabel ylabel and title takea string input containing the title or label text For the legend
command a string is required for each separate plot Note theorder of the strings in the legend command reflects the order inwhich the each of plots were made
Including mathematical notation in titles labels and legends
It is possible to include mathematical characters such as π inScilab plots This is done by inserting special commands intoeither the title legend or label string These commands are basedon the abbreviations for mathematical notation used in programssuch as LATEX MathType etc A brief summary of the commands
4 mtlb hold on is equivalent to the Scilab commandset(gca()auto clearoff) while mtlb hold off is equivalentto the Scilab command set(gca()auto clearon)
B6 Visualising data in Scilab 245
Table B9 Basic mathematical symbols available in Scilab forincluding mathematical notation in titles labels and legendscommand symbol command symbol command symbolalpha α upsilon υ sim sim
Im = cup cup wp weierpotimes otimes subseteq sube oslash cap cap in isin supseteq supesupset sup lceil d subset subint
intcdot middot o oslash
rfloor c neg 6= nabla nablalfloor b times times ldots perp perp surd
radicprime prime
wedge and varpi $ O Oslashrceil e rangle 〉 mid |
vee or copyright ccopy langle 〈
available in Scilab for typesetting mathematical notation is givenin Table B9 To insert these commands into a text string simplystart and end the string with $ signs as shown in Example B7
Example B7 Scilab commands to produce the labelled plot ofy = sin
There are a few key concepts to note about the Example B7 Thefirst can be seen in the string being passed to the title commandtitle
In this string sin is prefixed with a backslash () This is doneso that the letters for sin are not typeset in italic which shouldonly be used for variables and not common mathematical func-tions Table B10 shows the command sequences to use for themore common mathematical functions5 To get the fraction in thetitle the command fracab has been used to produce a
b Ifsuperscripts and subscripts are required in a title or label these aresuperscripts amp subscripts
obtained by adding the subscript character ( ) and the superscriptcharacter (^) to the string These characters modify the characteror substring defined in braces immediately following For examplex 2 gives x2 while e^2x gives e2x
Note in Example B7 the commands xlabel and ylabel has beenxlabelylabel
used to set the axes labels in Figure B5 Instead of using thesethree commands the title and the axes labels could have been setusing the single xtitle command (see help xtitle for details)xtitle
In the example we have also customised the font size to make thetext more readable by a) passing an option to the title xlabelylabel commands and b) by using the command xset to set thexset
default font size before executing the xstring command Finallythe xstring command adds the text string to plot at the locationxstring
specified by variables b and c In the example the coordinates wewant to add the text are obtained by executing the xclick functionxclick
and clicking on the point required in the plot window
5 Additional details can be found in books covering Mathematics in LATEX
B6 Visualising data in Scilab 247
Table B10 Command sequences for the common mathematicalfunctions in Scilabarccos arcsin arctan cos cot
csc exp lim ln log
max min sec sin tan
B63 Plotting numeric data stored in an ASCII file
As data can be stored in a wide variety of ways in external filesScilab has a wide variety of functions which can be used to importthe data stored in these files (see help for the commands read
fscanf and xlsread) Since data can be stored in such a wideread fscanf and xlsread
range of formats we will confine our discussions here to the situa-tion where numeric data is stored in a ASCII (text) file where eachrow of the file contains the same number of entries
To illustrate the procedure for plotting data stored in ASCII file wewill consider the data obtained measuring the temperature of a cupof coffee every minute for a total of 100 mins The data recorded isstored in the a file called coffeedatatxt6 which simply containsthe two columns of numbers shown in Table B11 As an examplethe first 10 lines of the text file contains the following numbers7
00000 1000000
10000 950000
20000 903125
30000 859180
40000 817981
50000 779357
60000 743147
70000 709201
80000 677376
90000 647540
Note the current version of Scilab has a builtin text editor whichcan be used to view such text files To access this editor simplytype the command edit at the command prompt Once open youedit
can use the graphical interface to open the required file
The first issue is to get the data into Scilab so it is available forplot command In Scilab the command read can used to importread
the data and assign it to a variable In the current example wewould use the following command to import the data and assignit to a variable called cdata assuming the data is in the currentdirectory8
6 The file coffeedatatxt has been attached and should be available for savingprovided you are using Adobe Reader To save the file right click on thepurple Windows version or the red MacOSLinux version in the margin andselect save file from the options Ensure you save the file as coffeedatatxt
7 normally you will have received the data file in this format8 You can check what directory Scilab is in by typing the command pwd
Figure B6 Temperature of coffee based on the data recorded inthe file coffeedatatxt
cdata=read(rsquocoffeedatatxtrsquo-12)
The -12 following the name of the file tells Scilab that you donrsquotknow how many rows you have but you have two columns of dataSince we know that we have 101 rows of data we could have alsoused the following command to load the data
cdata=read(rsquocoffeedatatxtrsquo1012)
We now have a variable called cdata which has two columns ofdata The first column contains the time while the second containsthe Temperature records Hence we can plot it using the followingcommand
plot(cdata(1)cdata(2))
This gives the plot shown in Figure B6 after appropriate labelshave been added as described above
B7 Script files
Executing commands entered at the keyboard or keypad is themost primitive use of a calculating device It is only when se-quences of operations (to perform complex tasks) are stored awayand re-called for repeated application with different inputs that wehave a programmed operation Complex programs must necessarilyhave an interface with the user The basic interface tool in Scilabcapable of executing sequences of commands that are stored in a fileis the script The file names must have an extension of sce forexample program1sce assignment1sce These files are calledScript Sci-files or simply scripts
A script file is a file of standard keyboard charactersthat define commands to drive a program A script filecan be generated using any text editor or word proces-sor
To invoke a script in Scilab type the commandexec
exec filenamesce
where filename denotes the actual name of the file containing thescript When the script runs it is exactly the same as if you typedthe commands into the command window from the keyboard Thismeans that the Sci-file can use any variables you have defined priorto invoking the script Any variables created in the script are alsoavailable when the script is finished and any variable whose namewas assigned a value in the script will be over-written
If you give a script the same name as any variable that it computesin the running of the script then Scilab will give precedence tothe workspace variable over a script As stated in sectB3 the variablename will hide the script name after the script has run Scilab willtake the name as referring to the variable and not to the definedscript
Scilab also needs to know where to find the script to executedIf Scilab cannot find the script it will report that you have anundefined function or variable
The exact procedure for creating and saving Sci-files depends onthe Scilab version the operating system (MS Windows Macin-tosh or UnixLinux ) and the text editor you use The currentversion of Scilab includes a built-in editor which can be used tocreate and edit Sci-files and text files
To create a new Sci-file under the current version of Scilab simplytype edit newfilenamesce This will open up the Scilab editoredit
with a blank document called newfilename
B71 Script input and output
A script is a named file (with the extension sce) containing asequence of Scilab instructions which when its name is executedin a session performs those instructions in order as though theywere entered at the console
There are two common uses for scripts Pragmatically we maywant to save a particular computational sequence for later use orreference If the tasks involved are to be repeated often thenthe nature of the tasks should be analysed formulated into usefulfunctions (which we will discuss later) and the whole procedurepresented in a single function But if the tasks are only likely toget occasional use then the script may be adequately commentedand stored as a script file It is a quick and dirty way of saving
B7 Script files 251
the instructions for doing a particular job when we do not wantto go to the trouble of writing the functions for the job It is nota highly recommended modus operandi but useful all the same
The second use is for the script to act like a storyboard presentinginformation to a user in a structured and controlled way permit-ting the user to see the steps of a computational procedure andtheir outputs up on the screen At a more sophisticated level thescript takes on the proportions of an application controlling a largeand complex interaction with advanced interface features We willnot take our study of Scilab programming to this level but workmostly at the storyboard level using the simplest of Scilabrsquos in-terface tools
All the variables created in a script become global variables definedin the workspace and available for perusal As an application itis assumed that a script controls the entire workspace and thevariables it creates are lsquothe datarsquo of the application When scriptsare used as storyboards on a small scale then it cannot be assumedthat the variables they create deserve global recognition and it iscourteous to erase these variables when the script is finished unlessthey have some possible further use An unfortunate side-effect isthat if variables with the same name already exist in the workspacebefore a script is run then these variables will be over-written andlost
Used as storyboards scripts allow the user to cut paste and re-trythe various steps of a procedure and to modify the script to suithis own needs Storyboards may be used for demonstrations oftoolbox utilities or extended documentationexplanation of func-tions However if there is any danger that variables defined bythe script may over-write workspace variables then the demo ordocumentation should be re-fashioned into a function that leavesthe workspace variables untouched (more on this later)
Scripts will often read data from pre-prepared files and we will dis-cuss the functions provided by Scilab to perform this task below
Gathering data entered by the user really falls into the realm ofinterface design but we only briefly consider the simplest interfacefunction provided by Scilab Similarly we briefly consider thefunctions for providing the simplest forms of output presentation
Keyboard input
Scilab has a built-in function input that can be used to promptinput
the user to enter the value of a variable from the keyboard Forexample
Tc = input(rsquoPlease enter the temperature in degrees Celsiusrsquo)
will print the message
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
238 Appendix B Programming with Scilab
An example of a predefined function is sin which returns the sineof a given angle
m = sin(angle)
where the variable angle contains the angle in radians and theresult is stored in the variable m The variable angle is called theargument or parameter of the function If the angle is stored indegrees it must be converted to radians for example
m = sin(anglepi180)
or
radians =anglepi180
m = sin(radians)
In Scilab the arguments or parameters of the function are con-tained in parentheses following the name of the function This isreferred to as pre-fix notation Many common functions in math-ematics have a syntax using in-fix notation where two argumentsappear to the right and left of the function name as for examplein x+y atimes b nmod r etc Scilab does not support in-fix nota-tion for any but the arithmetic functions Classical mathematicalnotation also has examples of post-fix notationmdash n x squared mdashand other hybrids like |x| while Scilab has ahb to confound thepurist
A function may contain no arguments one argument or many ar-guments depending on its definition If a function contains morethan one argument it is vital to give the arguments in the cor-rect order For example consider the problem of finding θ in thefollowing triangle using the atan command
5
8θ
Now by hand calculation the answer is 05586 radians If we useatan as below we get the answers shown
--gt atan(58)
ans =
05586
--gt atan(85)
ans =
10122
The first answer is the correct one The second answer is incorrect
B5 Builtndashin functions 239
Table B5 Some common mathematical functions defined inScilabFunction Description
abs(x) Computes the absolute value of xsqrt(x) Square root of x The parameter can be nega-
tiveround(x) Round x to the nearest integerfix(x) Round x to the nearest integer toward zerofloor(x) Round x to the nearest integer toward minusinfinceil(x) Round x to the nearest integer toward +infinsign(x) Returns -1 if x is less than zero zero if x equals
zero and 1 otherwisemodulo(xy) Returns the remainder of xy For example
modulo(254) is 1exp(x) Returns ex
log(x) Returns the natural logarithm of x That isthe logarithm to the base e Parameter can benegative but cannot be zero
log10(x) Returns the common logarithm of x That isthe logarithm to the base 10 Parameter can benegative but cannot be zero
since we passed the values to the function in the incorrect order
Some functions also require that the arguments be in specific unitsFor example the trigonometric functions assume that the argu-ments are in radians
In Scilab some functions use the number of arguments to deter-mine the output of the function For example the zeros functioncan have one or two arguments which then determine the output(check this with the command help zeros)
A function reference cannot be made on the left hand side of anassignment since a function call always returns a value(s) whereasan assignment is always to a variable Functions can of courseappear on the right side of an assignment since they return valuesto be assigned to some variable Functions may be applied tothe output of other functions and may be called in the algorithmdefining a function for example
logx = log(abs(x))
When one function is used to compute the argument of anotherfunction be sure to enclose the argument of each function in itsown set of parentheses
Scilab allows recursive functions that is functions that call them-selves in their defining algorithm
Exercise B4 Use examples to show the differences between the
240 Appendix B Programming with Scilab
Table B6 Some common trigonometric functions defined inScilab
Function Description
sin(x) Sine of x where x is in radianscos(x) Cosine of x where x is in radianstan(x) Tangent of x where x is in radiansasin(x) Arcsine or inverse sine of x where x must be
between -1 and 1 The return angle is in radiansbetween minusπ2 and π2
acos(x) Arccosine or inverse cosine of x where x mustbe between minus1 and 1 The return angle is inradians between 0 and π
atan(x) Arctangent or inverse tangent of x The returnangle is in radians between minusπ2 and π2
atan(xy) Arctangent or inverse tangent of xy The re-turn angle is in radians between minusπ and π de-pending on the sign of x and y
sec(x) Secant (1(cos x)) of x where x is in radianscsc(x) Cosecant (1(sin x)) of x where x is in radianscotg(x) Cotangent (1(tan x)) of x where x is in radians
Table B7 Some of Scilabrsquos utility functionsFunction Description
clear Clear variables from memoryhelp Display the first set of comment lines from
scriptswho List the variables in memorywhos List the variables in memory with their sizeerror Terminates the execution of the Sci-file and out-
puts the supplied error message
functions round(x) fix(x) floor(x) and ceil(x)
Exercise B5 The functions listed in Tables B5 B6 are only asmall sample of all of Scilabrsquos mathematical functions
Using Scilabrsquos help command list all of the hyperbolic func-tions available Also list any restrictions on the values of thefunction arguments
B52 Utility functions
In addition to the arithmetic functions Scilab has several utilityfunctions (see Table B7) These functions allow users to trackvariables (whoswho) feedback an error message if an error occursin the code (error) clear the variables form the memory (clear)etc Essential these functions allow users to debug and performbookkeeping operations inside their code
B6 Visualising data in Scilab 241
6
5
4- 04- 8
3
- 02
- 6
00
2- 4 Y- 2
02
Z
10
04
X 02
06
4 - 1
08
6- 28
10
Figure B2 Plot of z = sin(radicx2 + y2)
radicx2 + y2 illustrating the
3d capabilities of Scilab
B6 Visualising data in Scilab
One of the big advantages in using Scilab is the ability to visualisedata and results of complex calculations (see Figure B2 as exampleof the 3d capabilities) While Scilab is capability of complex 3dgraphics (see help pages for mesh and surf) in this section we willmesh amp surf
focus on the 2d plotting command (plot) Hence in this sectionplot
we will focus on
1 plotting mathematical functions from within Scilab
2 changing stylistic elements of the plot
3 adding titles labels and legends to the plot
4 including mathematical notation in titles labels and legendsand
5 plotting data stored in an external ASCII file
B61 Plotting mathematical functions from within Scilab
To plot a mathematical function in Scilab three steps are requiredThese are
1 Define the domain of the function (eg x=linspace(02pi))
2 Calculate the values of function at the selected points in thedomain (eg y=sin(x))
242 Appendix B Programming with Scilab
3 Final step is to plot the function using the plot commandplot
(eg plot(xy))
Using these steps it is possible to produce a basic plot of any math-ematical function (see Figure B3) Note that the plot shown inFigure B3 does not have a title axes labels or a legend all ofwhich are crucial for a good plot The Scilab commands to addthese elements to the plot are given in the following sections
- 10
- 08
- 06
- 04
- 02
00
02
04
06
08
10
0 1 2 3 4 5 6 7
Figure B3 The default plot of y = sin(x) in Scilab Notethat by default axes are not labelled and there is no title orlegend These need to added using the additional commandsxlabelylabeltitle and legend
Changing stylistic elements of the plot
An optional argument can be given to the plot command to cus-tomise each new line aspect This argument is a concatenatedstring containing information about colour line style or markersbased on the specifiers shown in Table B8 To specify the plotbe drawn with a red longdash-dot line with diamond marker thestring that would be passed to plot command would be rsquor-drsquo Thestring which is a concatenation (in any order) of the three typesof properties shown in Table B8 must be unambiguous Further-more the string specification is not case sensitive An exampleof specifying various line styles is shown in Example B6 and Fig-ure B4
Example B6 The following commands illustrate how line stylescan be used to distinguish between the plots of sin(2πt) sin(2πt+π6) and sin(2πt+ π2)
t=linspace(0130)
B6 Visualising data in Scilab 243
Table B8 Plot specifiers required to change line styles coloursand markers in plotsSpecifier Line style Specifier Colour Specifier Marker stylendash Solid line (default) r Red + Plus signndash ndash Dashed line g Green o Circle Dotted line b Blue Asteriskndash Dash-dotted line c Cyan x Cross
m Magenta s Squarey Yellow rsquononersquo No marker(default)k Blackw White
Figure B4 Resulting plotting after issuing the commands givenin Example B6
244 Appendix B Programming with Scilab
Example B6 illustrate a number of key commands relating to thecreation of multiple plots in Scilab The mtlb hold4 commandmtlb hold
holds the current plot and axis properties so that all subsequentplots are added to the existing plot
To distinguish each plot a different line specification is passed eachplot command In the example y = sin(2πt) is plotted using a redsolid line with data points indicated using a red circle as indicatedby the string rsquor-orsquo y = sin(2πt + π6) is plotted using greendashed-dotted line with data points marked by a square (rsquog-srsquo)while y = sin(2πt+ π2) is plotted using a cyan dashed line usingdiamonds to mark the data points (rsquoc--drsquo) Finally to completethe plot axes labels and a legend has been added to plot the com-mands for doing this are discussed in sectB62
B62 Adding titleslabels and legends to the plot
All good plots should have the following properties
1 The style of figure is appropriate to the information beingconveyed
2 The contents are well organised so it is easy for readers tointerpret the information
3 Legends labels etc are clear and concise (eg contain unitsif required do not obstruct parts of the plot etc) and
4 A caption or title that summarises the information presented
Like most mathematical applications Scilab has a wide variety ofdifferent plotting styles that can be used (eg look up the help forbar polarplot etc) This allows you to select the most appropri-barpolarplot
ate style to display and organise your data
The Scilab commands xlabel ylabel title and legend al-xlabel ylabel title and
legend lows you to add axes labels titles and legends to plot As shownin Example B6 the commands xlabel ylabel and title takea string input containing the title or label text For the legend
command a string is required for each separate plot Note theorder of the strings in the legend command reflects the order inwhich the each of plots were made
Including mathematical notation in titles labels and legends
It is possible to include mathematical characters such as π inScilab plots This is done by inserting special commands intoeither the title legend or label string These commands are basedon the abbreviations for mathematical notation used in programssuch as LATEX MathType etc A brief summary of the commands
4 mtlb hold on is equivalent to the Scilab commandset(gca()auto clearoff) while mtlb hold off is equivalentto the Scilab command set(gca()auto clearon)
B6 Visualising data in Scilab 245
Table B9 Basic mathematical symbols available in Scilab forincluding mathematical notation in titles labels and legendscommand symbol command symbol command symbolalpha α upsilon υ sim sim
Im = cup cup wp weierpotimes otimes subseteq sube oslash cap cap in isin supseteq supesupset sup lceil d subset subint
intcdot middot o oslash
rfloor c neg 6= nabla nablalfloor b times times ldots perp perp surd
radicprime prime
wedge and varpi $ O Oslashrceil e rangle 〉 mid |
vee or copyright ccopy langle 〈
available in Scilab for typesetting mathematical notation is givenin Table B9 To insert these commands into a text string simplystart and end the string with $ signs as shown in Example B7
Example B7 Scilab commands to produce the labelled plot ofy = sin
There are a few key concepts to note about the Example B7 Thefirst can be seen in the string being passed to the title commandtitle
In this string sin is prefixed with a backslash () This is doneso that the letters for sin are not typeset in italic which shouldonly be used for variables and not common mathematical func-tions Table B10 shows the command sequences to use for themore common mathematical functions5 To get the fraction in thetitle the command fracab has been used to produce a
b Ifsuperscripts and subscripts are required in a title or label these aresuperscripts amp subscripts
obtained by adding the subscript character ( ) and the superscriptcharacter (^) to the string These characters modify the characteror substring defined in braces immediately following For examplex 2 gives x2 while e^2x gives e2x
Note in Example B7 the commands xlabel and ylabel has beenxlabelylabel
used to set the axes labels in Figure B5 Instead of using thesethree commands the title and the axes labels could have been setusing the single xtitle command (see help xtitle for details)xtitle
In the example we have also customised the font size to make thetext more readable by a) passing an option to the title xlabelylabel commands and b) by using the command xset to set thexset
default font size before executing the xstring command Finallythe xstring command adds the text string to plot at the locationxstring
specified by variables b and c In the example the coordinates wewant to add the text are obtained by executing the xclick functionxclick
and clicking on the point required in the plot window
5 Additional details can be found in books covering Mathematics in LATEX
B6 Visualising data in Scilab 247
Table B10 Command sequences for the common mathematicalfunctions in Scilabarccos arcsin arctan cos cot
csc exp lim ln log
max min sec sin tan
B63 Plotting numeric data stored in an ASCII file
As data can be stored in a wide variety of ways in external filesScilab has a wide variety of functions which can be used to importthe data stored in these files (see help for the commands read
fscanf and xlsread) Since data can be stored in such a wideread fscanf and xlsread
range of formats we will confine our discussions here to the situa-tion where numeric data is stored in a ASCII (text) file where eachrow of the file contains the same number of entries
To illustrate the procedure for plotting data stored in ASCII file wewill consider the data obtained measuring the temperature of a cupof coffee every minute for a total of 100 mins The data recorded isstored in the a file called coffeedatatxt6 which simply containsthe two columns of numbers shown in Table B11 As an examplethe first 10 lines of the text file contains the following numbers7
00000 1000000
10000 950000
20000 903125
30000 859180
40000 817981
50000 779357
60000 743147
70000 709201
80000 677376
90000 647540
Note the current version of Scilab has a builtin text editor whichcan be used to view such text files To access this editor simplytype the command edit at the command prompt Once open youedit
can use the graphical interface to open the required file
The first issue is to get the data into Scilab so it is available forplot command In Scilab the command read can used to importread
the data and assign it to a variable In the current example wewould use the following command to import the data and assignit to a variable called cdata assuming the data is in the currentdirectory8
6 The file coffeedatatxt has been attached and should be available for savingprovided you are using Adobe Reader To save the file right click on thepurple Windows version or the red MacOSLinux version in the margin andselect save file from the options Ensure you save the file as coffeedatatxt
7 normally you will have received the data file in this format8 You can check what directory Scilab is in by typing the command pwd
Figure B6 Temperature of coffee based on the data recorded inthe file coffeedatatxt
cdata=read(rsquocoffeedatatxtrsquo-12)
The -12 following the name of the file tells Scilab that you donrsquotknow how many rows you have but you have two columns of dataSince we know that we have 101 rows of data we could have alsoused the following command to load the data
cdata=read(rsquocoffeedatatxtrsquo1012)
We now have a variable called cdata which has two columns ofdata The first column contains the time while the second containsthe Temperature records Hence we can plot it using the followingcommand
plot(cdata(1)cdata(2))
This gives the plot shown in Figure B6 after appropriate labelshave been added as described above
B7 Script files
Executing commands entered at the keyboard or keypad is themost primitive use of a calculating device It is only when se-quences of operations (to perform complex tasks) are stored awayand re-called for repeated application with different inputs that wehave a programmed operation Complex programs must necessarilyhave an interface with the user The basic interface tool in Scilabcapable of executing sequences of commands that are stored in a fileis the script The file names must have an extension of sce forexample program1sce assignment1sce These files are calledScript Sci-files or simply scripts
A script file is a file of standard keyboard charactersthat define commands to drive a program A script filecan be generated using any text editor or word proces-sor
To invoke a script in Scilab type the commandexec
exec filenamesce
where filename denotes the actual name of the file containing thescript When the script runs it is exactly the same as if you typedthe commands into the command window from the keyboard Thismeans that the Sci-file can use any variables you have defined priorto invoking the script Any variables created in the script are alsoavailable when the script is finished and any variable whose namewas assigned a value in the script will be over-written
If you give a script the same name as any variable that it computesin the running of the script then Scilab will give precedence tothe workspace variable over a script As stated in sectB3 the variablename will hide the script name after the script has run Scilab willtake the name as referring to the variable and not to the definedscript
Scilab also needs to know where to find the script to executedIf Scilab cannot find the script it will report that you have anundefined function or variable
The exact procedure for creating and saving Sci-files depends onthe Scilab version the operating system (MS Windows Macin-tosh or UnixLinux ) and the text editor you use The currentversion of Scilab includes a built-in editor which can be used tocreate and edit Sci-files and text files
To create a new Sci-file under the current version of Scilab simplytype edit newfilenamesce This will open up the Scilab editoredit
with a blank document called newfilename
B71 Script input and output
A script is a named file (with the extension sce) containing asequence of Scilab instructions which when its name is executedin a session performs those instructions in order as though theywere entered at the console
There are two common uses for scripts Pragmatically we maywant to save a particular computational sequence for later use orreference If the tasks involved are to be repeated often thenthe nature of the tasks should be analysed formulated into usefulfunctions (which we will discuss later) and the whole procedurepresented in a single function But if the tasks are only likely toget occasional use then the script may be adequately commentedand stored as a script file It is a quick and dirty way of saving
B7 Script files 251
the instructions for doing a particular job when we do not wantto go to the trouble of writing the functions for the job It is nota highly recommended modus operandi but useful all the same
The second use is for the script to act like a storyboard presentinginformation to a user in a structured and controlled way permit-ting the user to see the steps of a computational procedure andtheir outputs up on the screen At a more sophisticated level thescript takes on the proportions of an application controlling a largeand complex interaction with advanced interface features We willnot take our study of Scilab programming to this level but workmostly at the storyboard level using the simplest of Scilabrsquos in-terface tools
All the variables created in a script become global variables definedin the workspace and available for perusal As an application itis assumed that a script controls the entire workspace and thevariables it creates are lsquothe datarsquo of the application When scriptsare used as storyboards on a small scale then it cannot be assumedthat the variables they create deserve global recognition and it iscourteous to erase these variables when the script is finished unlessthey have some possible further use An unfortunate side-effect isthat if variables with the same name already exist in the workspacebefore a script is run then these variables will be over-written andlost
Used as storyboards scripts allow the user to cut paste and re-trythe various steps of a procedure and to modify the script to suithis own needs Storyboards may be used for demonstrations oftoolbox utilities or extended documentationexplanation of func-tions However if there is any danger that variables defined bythe script may over-write workspace variables then the demo ordocumentation should be re-fashioned into a function that leavesthe workspace variables untouched (more on this later)
Scripts will often read data from pre-prepared files and we will dis-cuss the functions provided by Scilab to perform this task below
Gathering data entered by the user really falls into the realm ofinterface design but we only briefly consider the simplest interfacefunction provided by Scilab Similarly we briefly consider thefunctions for providing the simplest forms of output presentation
Keyboard input
Scilab has a built-in function input that can be used to promptinput
the user to enter the value of a variable from the keyboard Forexample
Tc = input(rsquoPlease enter the temperature in degrees Celsiusrsquo)
will print the message
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
B5 Builtndashin functions 239
Table B5 Some common mathematical functions defined inScilabFunction Description
abs(x) Computes the absolute value of xsqrt(x) Square root of x The parameter can be nega-
tiveround(x) Round x to the nearest integerfix(x) Round x to the nearest integer toward zerofloor(x) Round x to the nearest integer toward minusinfinceil(x) Round x to the nearest integer toward +infinsign(x) Returns -1 if x is less than zero zero if x equals
zero and 1 otherwisemodulo(xy) Returns the remainder of xy For example
modulo(254) is 1exp(x) Returns ex
log(x) Returns the natural logarithm of x That isthe logarithm to the base e Parameter can benegative but cannot be zero
log10(x) Returns the common logarithm of x That isthe logarithm to the base 10 Parameter can benegative but cannot be zero
since we passed the values to the function in the incorrect order
Some functions also require that the arguments be in specific unitsFor example the trigonometric functions assume that the argu-ments are in radians
In Scilab some functions use the number of arguments to deter-mine the output of the function For example the zeros functioncan have one or two arguments which then determine the output(check this with the command help zeros)
A function reference cannot be made on the left hand side of anassignment since a function call always returns a value(s) whereasan assignment is always to a variable Functions can of courseappear on the right side of an assignment since they return valuesto be assigned to some variable Functions may be applied tothe output of other functions and may be called in the algorithmdefining a function for example
logx = log(abs(x))
When one function is used to compute the argument of anotherfunction be sure to enclose the argument of each function in itsown set of parentheses
Scilab allows recursive functions that is functions that call them-selves in their defining algorithm
Exercise B4 Use examples to show the differences between the
240 Appendix B Programming with Scilab
Table B6 Some common trigonometric functions defined inScilab
Function Description
sin(x) Sine of x where x is in radianscos(x) Cosine of x where x is in radianstan(x) Tangent of x where x is in radiansasin(x) Arcsine or inverse sine of x where x must be
between -1 and 1 The return angle is in radiansbetween minusπ2 and π2
acos(x) Arccosine or inverse cosine of x where x mustbe between minus1 and 1 The return angle is inradians between 0 and π
atan(x) Arctangent or inverse tangent of x The returnangle is in radians between minusπ2 and π2
atan(xy) Arctangent or inverse tangent of xy The re-turn angle is in radians between minusπ and π de-pending on the sign of x and y
sec(x) Secant (1(cos x)) of x where x is in radianscsc(x) Cosecant (1(sin x)) of x where x is in radianscotg(x) Cotangent (1(tan x)) of x where x is in radians
Table B7 Some of Scilabrsquos utility functionsFunction Description
clear Clear variables from memoryhelp Display the first set of comment lines from
scriptswho List the variables in memorywhos List the variables in memory with their sizeerror Terminates the execution of the Sci-file and out-
puts the supplied error message
functions round(x) fix(x) floor(x) and ceil(x)
Exercise B5 The functions listed in Tables B5 B6 are only asmall sample of all of Scilabrsquos mathematical functions
Using Scilabrsquos help command list all of the hyperbolic func-tions available Also list any restrictions on the values of thefunction arguments
B52 Utility functions
In addition to the arithmetic functions Scilab has several utilityfunctions (see Table B7) These functions allow users to trackvariables (whoswho) feedback an error message if an error occursin the code (error) clear the variables form the memory (clear)etc Essential these functions allow users to debug and performbookkeeping operations inside their code
B6 Visualising data in Scilab 241
6
5
4- 04- 8
3
- 02
- 6
00
2- 4 Y- 2
02
Z
10
04
X 02
06
4 - 1
08
6- 28
10
Figure B2 Plot of z = sin(radicx2 + y2)
radicx2 + y2 illustrating the
3d capabilities of Scilab
B6 Visualising data in Scilab
One of the big advantages in using Scilab is the ability to visualisedata and results of complex calculations (see Figure B2 as exampleof the 3d capabilities) While Scilab is capability of complex 3dgraphics (see help pages for mesh and surf) in this section we willmesh amp surf
focus on the 2d plotting command (plot) Hence in this sectionplot
we will focus on
1 plotting mathematical functions from within Scilab
2 changing stylistic elements of the plot
3 adding titles labels and legends to the plot
4 including mathematical notation in titles labels and legendsand
5 plotting data stored in an external ASCII file
B61 Plotting mathematical functions from within Scilab
To plot a mathematical function in Scilab three steps are requiredThese are
1 Define the domain of the function (eg x=linspace(02pi))
2 Calculate the values of function at the selected points in thedomain (eg y=sin(x))
242 Appendix B Programming with Scilab
3 Final step is to plot the function using the plot commandplot
(eg plot(xy))
Using these steps it is possible to produce a basic plot of any math-ematical function (see Figure B3) Note that the plot shown inFigure B3 does not have a title axes labels or a legend all ofwhich are crucial for a good plot The Scilab commands to addthese elements to the plot are given in the following sections
- 10
- 08
- 06
- 04
- 02
00
02
04
06
08
10
0 1 2 3 4 5 6 7
Figure B3 The default plot of y = sin(x) in Scilab Notethat by default axes are not labelled and there is no title orlegend These need to added using the additional commandsxlabelylabeltitle and legend
Changing stylistic elements of the plot
An optional argument can be given to the plot command to cus-tomise each new line aspect This argument is a concatenatedstring containing information about colour line style or markersbased on the specifiers shown in Table B8 To specify the plotbe drawn with a red longdash-dot line with diamond marker thestring that would be passed to plot command would be rsquor-drsquo Thestring which is a concatenation (in any order) of the three typesof properties shown in Table B8 must be unambiguous Further-more the string specification is not case sensitive An exampleof specifying various line styles is shown in Example B6 and Fig-ure B4
Example B6 The following commands illustrate how line stylescan be used to distinguish between the plots of sin(2πt) sin(2πt+π6) and sin(2πt+ π2)
t=linspace(0130)
B6 Visualising data in Scilab 243
Table B8 Plot specifiers required to change line styles coloursand markers in plotsSpecifier Line style Specifier Colour Specifier Marker stylendash Solid line (default) r Red + Plus signndash ndash Dashed line g Green o Circle Dotted line b Blue Asteriskndash Dash-dotted line c Cyan x Cross
m Magenta s Squarey Yellow rsquononersquo No marker(default)k Blackw White
Figure B4 Resulting plotting after issuing the commands givenin Example B6
244 Appendix B Programming with Scilab
Example B6 illustrate a number of key commands relating to thecreation of multiple plots in Scilab The mtlb hold4 commandmtlb hold
holds the current plot and axis properties so that all subsequentplots are added to the existing plot
To distinguish each plot a different line specification is passed eachplot command In the example y = sin(2πt) is plotted using a redsolid line with data points indicated using a red circle as indicatedby the string rsquor-orsquo y = sin(2πt + π6) is plotted using greendashed-dotted line with data points marked by a square (rsquog-srsquo)while y = sin(2πt+ π2) is plotted using a cyan dashed line usingdiamonds to mark the data points (rsquoc--drsquo) Finally to completethe plot axes labels and a legend has been added to plot the com-mands for doing this are discussed in sectB62
B62 Adding titleslabels and legends to the plot
All good plots should have the following properties
1 The style of figure is appropriate to the information beingconveyed
2 The contents are well organised so it is easy for readers tointerpret the information
3 Legends labels etc are clear and concise (eg contain unitsif required do not obstruct parts of the plot etc) and
4 A caption or title that summarises the information presented
Like most mathematical applications Scilab has a wide variety ofdifferent plotting styles that can be used (eg look up the help forbar polarplot etc) This allows you to select the most appropri-barpolarplot
ate style to display and organise your data
The Scilab commands xlabel ylabel title and legend al-xlabel ylabel title and
legend lows you to add axes labels titles and legends to plot As shownin Example B6 the commands xlabel ylabel and title takea string input containing the title or label text For the legend
command a string is required for each separate plot Note theorder of the strings in the legend command reflects the order inwhich the each of plots were made
Including mathematical notation in titles labels and legends
It is possible to include mathematical characters such as π inScilab plots This is done by inserting special commands intoeither the title legend or label string These commands are basedon the abbreviations for mathematical notation used in programssuch as LATEX MathType etc A brief summary of the commands
4 mtlb hold on is equivalent to the Scilab commandset(gca()auto clearoff) while mtlb hold off is equivalentto the Scilab command set(gca()auto clearon)
B6 Visualising data in Scilab 245
Table B9 Basic mathematical symbols available in Scilab forincluding mathematical notation in titles labels and legendscommand symbol command symbol command symbolalpha α upsilon υ sim sim
Im = cup cup wp weierpotimes otimes subseteq sube oslash cap cap in isin supseteq supesupset sup lceil d subset subint
intcdot middot o oslash
rfloor c neg 6= nabla nablalfloor b times times ldots perp perp surd
radicprime prime
wedge and varpi $ O Oslashrceil e rangle 〉 mid |
vee or copyright ccopy langle 〈
available in Scilab for typesetting mathematical notation is givenin Table B9 To insert these commands into a text string simplystart and end the string with $ signs as shown in Example B7
Example B7 Scilab commands to produce the labelled plot ofy = sin
There are a few key concepts to note about the Example B7 Thefirst can be seen in the string being passed to the title commandtitle
In this string sin is prefixed with a backslash () This is doneso that the letters for sin are not typeset in italic which shouldonly be used for variables and not common mathematical func-tions Table B10 shows the command sequences to use for themore common mathematical functions5 To get the fraction in thetitle the command fracab has been used to produce a
b Ifsuperscripts and subscripts are required in a title or label these aresuperscripts amp subscripts
obtained by adding the subscript character ( ) and the superscriptcharacter (^) to the string These characters modify the characteror substring defined in braces immediately following For examplex 2 gives x2 while e^2x gives e2x
Note in Example B7 the commands xlabel and ylabel has beenxlabelylabel
used to set the axes labels in Figure B5 Instead of using thesethree commands the title and the axes labels could have been setusing the single xtitle command (see help xtitle for details)xtitle
In the example we have also customised the font size to make thetext more readable by a) passing an option to the title xlabelylabel commands and b) by using the command xset to set thexset
default font size before executing the xstring command Finallythe xstring command adds the text string to plot at the locationxstring
specified by variables b and c In the example the coordinates wewant to add the text are obtained by executing the xclick functionxclick
and clicking on the point required in the plot window
5 Additional details can be found in books covering Mathematics in LATEX
B6 Visualising data in Scilab 247
Table B10 Command sequences for the common mathematicalfunctions in Scilabarccos arcsin arctan cos cot
csc exp lim ln log
max min sec sin tan
B63 Plotting numeric data stored in an ASCII file
As data can be stored in a wide variety of ways in external filesScilab has a wide variety of functions which can be used to importthe data stored in these files (see help for the commands read
fscanf and xlsread) Since data can be stored in such a wideread fscanf and xlsread
range of formats we will confine our discussions here to the situa-tion where numeric data is stored in a ASCII (text) file where eachrow of the file contains the same number of entries
To illustrate the procedure for plotting data stored in ASCII file wewill consider the data obtained measuring the temperature of a cupof coffee every minute for a total of 100 mins The data recorded isstored in the a file called coffeedatatxt6 which simply containsthe two columns of numbers shown in Table B11 As an examplethe first 10 lines of the text file contains the following numbers7
00000 1000000
10000 950000
20000 903125
30000 859180
40000 817981
50000 779357
60000 743147
70000 709201
80000 677376
90000 647540
Note the current version of Scilab has a builtin text editor whichcan be used to view such text files To access this editor simplytype the command edit at the command prompt Once open youedit
can use the graphical interface to open the required file
The first issue is to get the data into Scilab so it is available forplot command In Scilab the command read can used to importread
the data and assign it to a variable In the current example wewould use the following command to import the data and assignit to a variable called cdata assuming the data is in the currentdirectory8
6 The file coffeedatatxt has been attached and should be available for savingprovided you are using Adobe Reader To save the file right click on thepurple Windows version or the red MacOSLinux version in the margin andselect save file from the options Ensure you save the file as coffeedatatxt
7 normally you will have received the data file in this format8 You can check what directory Scilab is in by typing the command pwd
Figure B6 Temperature of coffee based on the data recorded inthe file coffeedatatxt
cdata=read(rsquocoffeedatatxtrsquo-12)
The -12 following the name of the file tells Scilab that you donrsquotknow how many rows you have but you have two columns of dataSince we know that we have 101 rows of data we could have alsoused the following command to load the data
cdata=read(rsquocoffeedatatxtrsquo1012)
We now have a variable called cdata which has two columns ofdata The first column contains the time while the second containsthe Temperature records Hence we can plot it using the followingcommand
plot(cdata(1)cdata(2))
This gives the plot shown in Figure B6 after appropriate labelshave been added as described above
B7 Script files
Executing commands entered at the keyboard or keypad is themost primitive use of a calculating device It is only when se-quences of operations (to perform complex tasks) are stored awayand re-called for repeated application with different inputs that wehave a programmed operation Complex programs must necessarilyhave an interface with the user The basic interface tool in Scilabcapable of executing sequences of commands that are stored in a fileis the script The file names must have an extension of sce forexample program1sce assignment1sce These files are calledScript Sci-files or simply scripts
A script file is a file of standard keyboard charactersthat define commands to drive a program A script filecan be generated using any text editor or word proces-sor
To invoke a script in Scilab type the commandexec
exec filenamesce
where filename denotes the actual name of the file containing thescript When the script runs it is exactly the same as if you typedthe commands into the command window from the keyboard Thismeans that the Sci-file can use any variables you have defined priorto invoking the script Any variables created in the script are alsoavailable when the script is finished and any variable whose namewas assigned a value in the script will be over-written
If you give a script the same name as any variable that it computesin the running of the script then Scilab will give precedence tothe workspace variable over a script As stated in sectB3 the variablename will hide the script name after the script has run Scilab willtake the name as referring to the variable and not to the definedscript
Scilab also needs to know where to find the script to executedIf Scilab cannot find the script it will report that you have anundefined function or variable
The exact procedure for creating and saving Sci-files depends onthe Scilab version the operating system (MS Windows Macin-tosh or UnixLinux ) and the text editor you use The currentversion of Scilab includes a built-in editor which can be used tocreate and edit Sci-files and text files
To create a new Sci-file under the current version of Scilab simplytype edit newfilenamesce This will open up the Scilab editoredit
with a blank document called newfilename
B71 Script input and output
A script is a named file (with the extension sce) containing asequence of Scilab instructions which when its name is executedin a session performs those instructions in order as though theywere entered at the console
There are two common uses for scripts Pragmatically we maywant to save a particular computational sequence for later use orreference If the tasks involved are to be repeated often thenthe nature of the tasks should be analysed formulated into usefulfunctions (which we will discuss later) and the whole procedurepresented in a single function But if the tasks are only likely toget occasional use then the script may be adequately commentedand stored as a script file It is a quick and dirty way of saving
B7 Script files 251
the instructions for doing a particular job when we do not wantto go to the trouble of writing the functions for the job It is nota highly recommended modus operandi but useful all the same
The second use is for the script to act like a storyboard presentinginformation to a user in a structured and controlled way permit-ting the user to see the steps of a computational procedure andtheir outputs up on the screen At a more sophisticated level thescript takes on the proportions of an application controlling a largeand complex interaction with advanced interface features We willnot take our study of Scilab programming to this level but workmostly at the storyboard level using the simplest of Scilabrsquos in-terface tools
All the variables created in a script become global variables definedin the workspace and available for perusal As an application itis assumed that a script controls the entire workspace and thevariables it creates are lsquothe datarsquo of the application When scriptsare used as storyboards on a small scale then it cannot be assumedthat the variables they create deserve global recognition and it iscourteous to erase these variables when the script is finished unlessthey have some possible further use An unfortunate side-effect isthat if variables with the same name already exist in the workspacebefore a script is run then these variables will be over-written andlost
Used as storyboards scripts allow the user to cut paste and re-trythe various steps of a procedure and to modify the script to suithis own needs Storyboards may be used for demonstrations oftoolbox utilities or extended documentationexplanation of func-tions However if there is any danger that variables defined bythe script may over-write workspace variables then the demo ordocumentation should be re-fashioned into a function that leavesthe workspace variables untouched (more on this later)
Scripts will often read data from pre-prepared files and we will dis-cuss the functions provided by Scilab to perform this task below
Gathering data entered by the user really falls into the realm ofinterface design but we only briefly consider the simplest interfacefunction provided by Scilab Similarly we briefly consider thefunctions for providing the simplest forms of output presentation
Keyboard input
Scilab has a built-in function input that can be used to promptinput
the user to enter the value of a variable from the keyboard Forexample
Tc = input(rsquoPlease enter the temperature in degrees Celsiusrsquo)
will print the message
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
240 Appendix B Programming with Scilab
Table B6 Some common trigonometric functions defined inScilab
Function Description
sin(x) Sine of x where x is in radianscos(x) Cosine of x where x is in radianstan(x) Tangent of x where x is in radiansasin(x) Arcsine or inverse sine of x where x must be
between -1 and 1 The return angle is in radiansbetween minusπ2 and π2
acos(x) Arccosine or inverse cosine of x where x mustbe between minus1 and 1 The return angle is inradians between 0 and π
atan(x) Arctangent or inverse tangent of x The returnangle is in radians between minusπ2 and π2
atan(xy) Arctangent or inverse tangent of xy The re-turn angle is in radians between minusπ and π de-pending on the sign of x and y
sec(x) Secant (1(cos x)) of x where x is in radianscsc(x) Cosecant (1(sin x)) of x where x is in radianscotg(x) Cotangent (1(tan x)) of x where x is in radians
Table B7 Some of Scilabrsquos utility functionsFunction Description
clear Clear variables from memoryhelp Display the first set of comment lines from
scriptswho List the variables in memorywhos List the variables in memory with their sizeerror Terminates the execution of the Sci-file and out-
puts the supplied error message
functions round(x) fix(x) floor(x) and ceil(x)
Exercise B5 The functions listed in Tables B5 B6 are only asmall sample of all of Scilabrsquos mathematical functions
Using Scilabrsquos help command list all of the hyperbolic func-tions available Also list any restrictions on the values of thefunction arguments
B52 Utility functions
In addition to the arithmetic functions Scilab has several utilityfunctions (see Table B7) These functions allow users to trackvariables (whoswho) feedback an error message if an error occursin the code (error) clear the variables form the memory (clear)etc Essential these functions allow users to debug and performbookkeeping operations inside their code
B6 Visualising data in Scilab 241
6
5
4- 04- 8
3
- 02
- 6
00
2- 4 Y- 2
02
Z
10
04
X 02
06
4 - 1
08
6- 28
10
Figure B2 Plot of z = sin(radicx2 + y2)
radicx2 + y2 illustrating the
3d capabilities of Scilab
B6 Visualising data in Scilab
One of the big advantages in using Scilab is the ability to visualisedata and results of complex calculations (see Figure B2 as exampleof the 3d capabilities) While Scilab is capability of complex 3dgraphics (see help pages for mesh and surf) in this section we willmesh amp surf
focus on the 2d plotting command (plot) Hence in this sectionplot
we will focus on
1 plotting mathematical functions from within Scilab
2 changing stylistic elements of the plot
3 adding titles labels and legends to the plot
4 including mathematical notation in titles labels and legendsand
5 plotting data stored in an external ASCII file
B61 Plotting mathematical functions from within Scilab
To plot a mathematical function in Scilab three steps are requiredThese are
1 Define the domain of the function (eg x=linspace(02pi))
2 Calculate the values of function at the selected points in thedomain (eg y=sin(x))
242 Appendix B Programming with Scilab
3 Final step is to plot the function using the plot commandplot
(eg plot(xy))
Using these steps it is possible to produce a basic plot of any math-ematical function (see Figure B3) Note that the plot shown inFigure B3 does not have a title axes labels or a legend all ofwhich are crucial for a good plot The Scilab commands to addthese elements to the plot are given in the following sections
- 10
- 08
- 06
- 04
- 02
00
02
04
06
08
10
0 1 2 3 4 5 6 7
Figure B3 The default plot of y = sin(x) in Scilab Notethat by default axes are not labelled and there is no title orlegend These need to added using the additional commandsxlabelylabeltitle and legend
Changing stylistic elements of the plot
An optional argument can be given to the plot command to cus-tomise each new line aspect This argument is a concatenatedstring containing information about colour line style or markersbased on the specifiers shown in Table B8 To specify the plotbe drawn with a red longdash-dot line with diamond marker thestring that would be passed to plot command would be rsquor-drsquo Thestring which is a concatenation (in any order) of the three typesof properties shown in Table B8 must be unambiguous Further-more the string specification is not case sensitive An exampleof specifying various line styles is shown in Example B6 and Fig-ure B4
Example B6 The following commands illustrate how line stylescan be used to distinguish between the plots of sin(2πt) sin(2πt+π6) and sin(2πt+ π2)
t=linspace(0130)
B6 Visualising data in Scilab 243
Table B8 Plot specifiers required to change line styles coloursand markers in plotsSpecifier Line style Specifier Colour Specifier Marker stylendash Solid line (default) r Red + Plus signndash ndash Dashed line g Green o Circle Dotted line b Blue Asteriskndash Dash-dotted line c Cyan x Cross
m Magenta s Squarey Yellow rsquononersquo No marker(default)k Blackw White
Figure B4 Resulting plotting after issuing the commands givenin Example B6
244 Appendix B Programming with Scilab
Example B6 illustrate a number of key commands relating to thecreation of multiple plots in Scilab The mtlb hold4 commandmtlb hold
holds the current plot and axis properties so that all subsequentplots are added to the existing plot
To distinguish each plot a different line specification is passed eachplot command In the example y = sin(2πt) is plotted using a redsolid line with data points indicated using a red circle as indicatedby the string rsquor-orsquo y = sin(2πt + π6) is plotted using greendashed-dotted line with data points marked by a square (rsquog-srsquo)while y = sin(2πt+ π2) is plotted using a cyan dashed line usingdiamonds to mark the data points (rsquoc--drsquo) Finally to completethe plot axes labels and a legend has been added to plot the com-mands for doing this are discussed in sectB62
B62 Adding titleslabels and legends to the plot
All good plots should have the following properties
1 The style of figure is appropriate to the information beingconveyed
2 The contents are well organised so it is easy for readers tointerpret the information
3 Legends labels etc are clear and concise (eg contain unitsif required do not obstruct parts of the plot etc) and
4 A caption or title that summarises the information presented
Like most mathematical applications Scilab has a wide variety ofdifferent plotting styles that can be used (eg look up the help forbar polarplot etc) This allows you to select the most appropri-barpolarplot
ate style to display and organise your data
The Scilab commands xlabel ylabel title and legend al-xlabel ylabel title and
legend lows you to add axes labels titles and legends to plot As shownin Example B6 the commands xlabel ylabel and title takea string input containing the title or label text For the legend
command a string is required for each separate plot Note theorder of the strings in the legend command reflects the order inwhich the each of plots were made
Including mathematical notation in titles labels and legends
It is possible to include mathematical characters such as π inScilab plots This is done by inserting special commands intoeither the title legend or label string These commands are basedon the abbreviations for mathematical notation used in programssuch as LATEX MathType etc A brief summary of the commands
4 mtlb hold on is equivalent to the Scilab commandset(gca()auto clearoff) while mtlb hold off is equivalentto the Scilab command set(gca()auto clearon)
B6 Visualising data in Scilab 245
Table B9 Basic mathematical symbols available in Scilab forincluding mathematical notation in titles labels and legendscommand symbol command symbol command symbolalpha α upsilon υ sim sim
Im = cup cup wp weierpotimes otimes subseteq sube oslash cap cap in isin supseteq supesupset sup lceil d subset subint
intcdot middot o oslash
rfloor c neg 6= nabla nablalfloor b times times ldots perp perp surd
radicprime prime
wedge and varpi $ O Oslashrceil e rangle 〉 mid |
vee or copyright ccopy langle 〈
available in Scilab for typesetting mathematical notation is givenin Table B9 To insert these commands into a text string simplystart and end the string with $ signs as shown in Example B7
Example B7 Scilab commands to produce the labelled plot ofy = sin
There are a few key concepts to note about the Example B7 Thefirst can be seen in the string being passed to the title commandtitle
In this string sin is prefixed with a backslash () This is doneso that the letters for sin are not typeset in italic which shouldonly be used for variables and not common mathematical func-tions Table B10 shows the command sequences to use for themore common mathematical functions5 To get the fraction in thetitle the command fracab has been used to produce a
b Ifsuperscripts and subscripts are required in a title or label these aresuperscripts amp subscripts
obtained by adding the subscript character ( ) and the superscriptcharacter (^) to the string These characters modify the characteror substring defined in braces immediately following For examplex 2 gives x2 while e^2x gives e2x
Note in Example B7 the commands xlabel and ylabel has beenxlabelylabel
used to set the axes labels in Figure B5 Instead of using thesethree commands the title and the axes labels could have been setusing the single xtitle command (see help xtitle for details)xtitle
In the example we have also customised the font size to make thetext more readable by a) passing an option to the title xlabelylabel commands and b) by using the command xset to set thexset
default font size before executing the xstring command Finallythe xstring command adds the text string to plot at the locationxstring
specified by variables b and c In the example the coordinates wewant to add the text are obtained by executing the xclick functionxclick
and clicking on the point required in the plot window
5 Additional details can be found in books covering Mathematics in LATEX
B6 Visualising data in Scilab 247
Table B10 Command sequences for the common mathematicalfunctions in Scilabarccos arcsin arctan cos cot
csc exp lim ln log
max min sec sin tan
B63 Plotting numeric data stored in an ASCII file
As data can be stored in a wide variety of ways in external filesScilab has a wide variety of functions which can be used to importthe data stored in these files (see help for the commands read
fscanf and xlsread) Since data can be stored in such a wideread fscanf and xlsread
range of formats we will confine our discussions here to the situa-tion where numeric data is stored in a ASCII (text) file where eachrow of the file contains the same number of entries
To illustrate the procedure for plotting data stored in ASCII file wewill consider the data obtained measuring the temperature of a cupof coffee every minute for a total of 100 mins The data recorded isstored in the a file called coffeedatatxt6 which simply containsthe two columns of numbers shown in Table B11 As an examplethe first 10 lines of the text file contains the following numbers7
00000 1000000
10000 950000
20000 903125
30000 859180
40000 817981
50000 779357
60000 743147
70000 709201
80000 677376
90000 647540
Note the current version of Scilab has a builtin text editor whichcan be used to view such text files To access this editor simplytype the command edit at the command prompt Once open youedit
can use the graphical interface to open the required file
The first issue is to get the data into Scilab so it is available forplot command In Scilab the command read can used to importread
the data and assign it to a variable In the current example wewould use the following command to import the data and assignit to a variable called cdata assuming the data is in the currentdirectory8
6 The file coffeedatatxt has been attached and should be available for savingprovided you are using Adobe Reader To save the file right click on thepurple Windows version or the red MacOSLinux version in the margin andselect save file from the options Ensure you save the file as coffeedatatxt
7 normally you will have received the data file in this format8 You can check what directory Scilab is in by typing the command pwd
Figure B6 Temperature of coffee based on the data recorded inthe file coffeedatatxt
cdata=read(rsquocoffeedatatxtrsquo-12)
The -12 following the name of the file tells Scilab that you donrsquotknow how many rows you have but you have two columns of dataSince we know that we have 101 rows of data we could have alsoused the following command to load the data
cdata=read(rsquocoffeedatatxtrsquo1012)
We now have a variable called cdata which has two columns ofdata The first column contains the time while the second containsthe Temperature records Hence we can plot it using the followingcommand
plot(cdata(1)cdata(2))
This gives the plot shown in Figure B6 after appropriate labelshave been added as described above
B7 Script files
Executing commands entered at the keyboard or keypad is themost primitive use of a calculating device It is only when se-quences of operations (to perform complex tasks) are stored awayand re-called for repeated application with different inputs that wehave a programmed operation Complex programs must necessarilyhave an interface with the user The basic interface tool in Scilabcapable of executing sequences of commands that are stored in a fileis the script The file names must have an extension of sce forexample program1sce assignment1sce These files are calledScript Sci-files or simply scripts
A script file is a file of standard keyboard charactersthat define commands to drive a program A script filecan be generated using any text editor or word proces-sor
To invoke a script in Scilab type the commandexec
exec filenamesce
where filename denotes the actual name of the file containing thescript When the script runs it is exactly the same as if you typedthe commands into the command window from the keyboard Thismeans that the Sci-file can use any variables you have defined priorto invoking the script Any variables created in the script are alsoavailable when the script is finished and any variable whose namewas assigned a value in the script will be over-written
If you give a script the same name as any variable that it computesin the running of the script then Scilab will give precedence tothe workspace variable over a script As stated in sectB3 the variablename will hide the script name after the script has run Scilab willtake the name as referring to the variable and not to the definedscript
Scilab also needs to know where to find the script to executedIf Scilab cannot find the script it will report that you have anundefined function or variable
The exact procedure for creating and saving Sci-files depends onthe Scilab version the operating system (MS Windows Macin-tosh or UnixLinux ) and the text editor you use The currentversion of Scilab includes a built-in editor which can be used tocreate and edit Sci-files and text files
To create a new Sci-file under the current version of Scilab simplytype edit newfilenamesce This will open up the Scilab editoredit
with a blank document called newfilename
B71 Script input and output
A script is a named file (with the extension sce) containing asequence of Scilab instructions which when its name is executedin a session performs those instructions in order as though theywere entered at the console
There are two common uses for scripts Pragmatically we maywant to save a particular computational sequence for later use orreference If the tasks involved are to be repeated often thenthe nature of the tasks should be analysed formulated into usefulfunctions (which we will discuss later) and the whole procedurepresented in a single function But if the tasks are only likely toget occasional use then the script may be adequately commentedand stored as a script file It is a quick and dirty way of saving
B7 Script files 251
the instructions for doing a particular job when we do not wantto go to the trouble of writing the functions for the job It is nota highly recommended modus operandi but useful all the same
The second use is for the script to act like a storyboard presentinginformation to a user in a structured and controlled way permit-ting the user to see the steps of a computational procedure andtheir outputs up on the screen At a more sophisticated level thescript takes on the proportions of an application controlling a largeand complex interaction with advanced interface features We willnot take our study of Scilab programming to this level but workmostly at the storyboard level using the simplest of Scilabrsquos in-terface tools
All the variables created in a script become global variables definedin the workspace and available for perusal As an application itis assumed that a script controls the entire workspace and thevariables it creates are lsquothe datarsquo of the application When scriptsare used as storyboards on a small scale then it cannot be assumedthat the variables they create deserve global recognition and it iscourteous to erase these variables when the script is finished unlessthey have some possible further use An unfortunate side-effect isthat if variables with the same name already exist in the workspacebefore a script is run then these variables will be over-written andlost
Used as storyboards scripts allow the user to cut paste and re-trythe various steps of a procedure and to modify the script to suithis own needs Storyboards may be used for demonstrations oftoolbox utilities or extended documentationexplanation of func-tions However if there is any danger that variables defined bythe script may over-write workspace variables then the demo ordocumentation should be re-fashioned into a function that leavesthe workspace variables untouched (more on this later)
Scripts will often read data from pre-prepared files and we will dis-cuss the functions provided by Scilab to perform this task below
Gathering data entered by the user really falls into the realm ofinterface design but we only briefly consider the simplest interfacefunction provided by Scilab Similarly we briefly consider thefunctions for providing the simplest forms of output presentation
Keyboard input
Scilab has a built-in function input that can be used to promptinput
the user to enter the value of a variable from the keyboard Forexample
Tc = input(rsquoPlease enter the temperature in degrees Celsiusrsquo)
will print the message
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
B6 Visualising data in Scilab 241
6
5
4- 04- 8
3
- 02
- 6
00
2- 4 Y- 2
02
Z
10
04
X 02
06
4 - 1
08
6- 28
10
Figure B2 Plot of z = sin(radicx2 + y2)
radicx2 + y2 illustrating the
3d capabilities of Scilab
B6 Visualising data in Scilab
One of the big advantages in using Scilab is the ability to visualisedata and results of complex calculations (see Figure B2 as exampleof the 3d capabilities) While Scilab is capability of complex 3dgraphics (see help pages for mesh and surf) in this section we willmesh amp surf
focus on the 2d plotting command (plot) Hence in this sectionplot
we will focus on
1 plotting mathematical functions from within Scilab
2 changing stylistic elements of the plot
3 adding titles labels and legends to the plot
4 including mathematical notation in titles labels and legendsand
5 plotting data stored in an external ASCII file
B61 Plotting mathematical functions from within Scilab
To plot a mathematical function in Scilab three steps are requiredThese are
1 Define the domain of the function (eg x=linspace(02pi))
2 Calculate the values of function at the selected points in thedomain (eg y=sin(x))
242 Appendix B Programming with Scilab
3 Final step is to plot the function using the plot commandplot
(eg plot(xy))
Using these steps it is possible to produce a basic plot of any math-ematical function (see Figure B3) Note that the plot shown inFigure B3 does not have a title axes labels or a legend all ofwhich are crucial for a good plot The Scilab commands to addthese elements to the plot are given in the following sections
- 10
- 08
- 06
- 04
- 02
00
02
04
06
08
10
0 1 2 3 4 5 6 7
Figure B3 The default plot of y = sin(x) in Scilab Notethat by default axes are not labelled and there is no title orlegend These need to added using the additional commandsxlabelylabeltitle and legend
Changing stylistic elements of the plot
An optional argument can be given to the plot command to cus-tomise each new line aspect This argument is a concatenatedstring containing information about colour line style or markersbased on the specifiers shown in Table B8 To specify the plotbe drawn with a red longdash-dot line with diamond marker thestring that would be passed to plot command would be rsquor-drsquo Thestring which is a concatenation (in any order) of the three typesof properties shown in Table B8 must be unambiguous Further-more the string specification is not case sensitive An exampleof specifying various line styles is shown in Example B6 and Fig-ure B4
Example B6 The following commands illustrate how line stylescan be used to distinguish between the plots of sin(2πt) sin(2πt+π6) and sin(2πt+ π2)
t=linspace(0130)
B6 Visualising data in Scilab 243
Table B8 Plot specifiers required to change line styles coloursand markers in plotsSpecifier Line style Specifier Colour Specifier Marker stylendash Solid line (default) r Red + Plus signndash ndash Dashed line g Green o Circle Dotted line b Blue Asteriskndash Dash-dotted line c Cyan x Cross
m Magenta s Squarey Yellow rsquononersquo No marker(default)k Blackw White
Figure B4 Resulting plotting after issuing the commands givenin Example B6
244 Appendix B Programming with Scilab
Example B6 illustrate a number of key commands relating to thecreation of multiple plots in Scilab The mtlb hold4 commandmtlb hold
holds the current plot and axis properties so that all subsequentplots are added to the existing plot
To distinguish each plot a different line specification is passed eachplot command In the example y = sin(2πt) is plotted using a redsolid line with data points indicated using a red circle as indicatedby the string rsquor-orsquo y = sin(2πt + π6) is plotted using greendashed-dotted line with data points marked by a square (rsquog-srsquo)while y = sin(2πt+ π2) is plotted using a cyan dashed line usingdiamonds to mark the data points (rsquoc--drsquo) Finally to completethe plot axes labels and a legend has been added to plot the com-mands for doing this are discussed in sectB62
B62 Adding titleslabels and legends to the plot
All good plots should have the following properties
1 The style of figure is appropriate to the information beingconveyed
2 The contents are well organised so it is easy for readers tointerpret the information
3 Legends labels etc are clear and concise (eg contain unitsif required do not obstruct parts of the plot etc) and
4 A caption or title that summarises the information presented
Like most mathematical applications Scilab has a wide variety ofdifferent plotting styles that can be used (eg look up the help forbar polarplot etc) This allows you to select the most appropri-barpolarplot
ate style to display and organise your data
The Scilab commands xlabel ylabel title and legend al-xlabel ylabel title and
legend lows you to add axes labels titles and legends to plot As shownin Example B6 the commands xlabel ylabel and title takea string input containing the title or label text For the legend
command a string is required for each separate plot Note theorder of the strings in the legend command reflects the order inwhich the each of plots were made
Including mathematical notation in titles labels and legends
It is possible to include mathematical characters such as π inScilab plots This is done by inserting special commands intoeither the title legend or label string These commands are basedon the abbreviations for mathematical notation used in programssuch as LATEX MathType etc A brief summary of the commands
4 mtlb hold on is equivalent to the Scilab commandset(gca()auto clearoff) while mtlb hold off is equivalentto the Scilab command set(gca()auto clearon)
B6 Visualising data in Scilab 245
Table B9 Basic mathematical symbols available in Scilab forincluding mathematical notation in titles labels and legendscommand symbol command symbol command symbolalpha α upsilon υ sim sim
Im = cup cup wp weierpotimes otimes subseteq sube oslash cap cap in isin supseteq supesupset sup lceil d subset subint
intcdot middot o oslash
rfloor c neg 6= nabla nablalfloor b times times ldots perp perp surd
radicprime prime
wedge and varpi $ O Oslashrceil e rangle 〉 mid |
vee or copyright ccopy langle 〈
available in Scilab for typesetting mathematical notation is givenin Table B9 To insert these commands into a text string simplystart and end the string with $ signs as shown in Example B7
Example B7 Scilab commands to produce the labelled plot ofy = sin
There are a few key concepts to note about the Example B7 Thefirst can be seen in the string being passed to the title commandtitle
In this string sin is prefixed with a backslash () This is doneso that the letters for sin are not typeset in italic which shouldonly be used for variables and not common mathematical func-tions Table B10 shows the command sequences to use for themore common mathematical functions5 To get the fraction in thetitle the command fracab has been used to produce a
b Ifsuperscripts and subscripts are required in a title or label these aresuperscripts amp subscripts
obtained by adding the subscript character ( ) and the superscriptcharacter (^) to the string These characters modify the characteror substring defined in braces immediately following For examplex 2 gives x2 while e^2x gives e2x
Note in Example B7 the commands xlabel and ylabel has beenxlabelylabel
used to set the axes labels in Figure B5 Instead of using thesethree commands the title and the axes labels could have been setusing the single xtitle command (see help xtitle for details)xtitle
In the example we have also customised the font size to make thetext more readable by a) passing an option to the title xlabelylabel commands and b) by using the command xset to set thexset
default font size before executing the xstring command Finallythe xstring command adds the text string to plot at the locationxstring
specified by variables b and c In the example the coordinates wewant to add the text are obtained by executing the xclick functionxclick
and clicking on the point required in the plot window
5 Additional details can be found in books covering Mathematics in LATEX
B6 Visualising data in Scilab 247
Table B10 Command sequences for the common mathematicalfunctions in Scilabarccos arcsin arctan cos cot
csc exp lim ln log
max min sec sin tan
B63 Plotting numeric data stored in an ASCII file
As data can be stored in a wide variety of ways in external filesScilab has a wide variety of functions which can be used to importthe data stored in these files (see help for the commands read
fscanf and xlsread) Since data can be stored in such a wideread fscanf and xlsread
range of formats we will confine our discussions here to the situa-tion where numeric data is stored in a ASCII (text) file where eachrow of the file contains the same number of entries
To illustrate the procedure for plotting data stored in ASCII file wewill consider the data obtained measuring the temperature of a cupof coffee every minute for a total of 100 mins The data recorded isstored in the a file called coffeedatatxt6 which simply containsthe two columns of numbers shown in Table B11 As an examplethe first 10 lines of the text file contains the following numbers7
00000 1000000
10000 950000
20000 903125
30000 859180
40000 817981
50000 779357
60000 743147
70000 709201
80000 677376
90000 647540
Note the current version of Scilab has a builtin text editor whichcan be used to view such text files To access this editor simplytype the command edit at the command prompt Once open youedit
can use the graphical interface to open the required file
The first issue is to get the data into Scilab so it is available forplot command In Scilab the command read can used to importread
the data and assign it to a variable In the current example wewould use the following command to import the data and assignit to a variable called cdata assuming the data is in the currentdirectory8
6 The file coffeedatatxt has been attached and should be available for savingprovided you are using Adobe Reader To save the file right click on thepurple Windows version or the red MacOSLinux version in the margin andselect save file from the options Ensure you save the file as coffeedatatxt
7 normally you will have received the data file in this format8 You can check what directory Scilab is in by typing the command pwd
Figure B6 Temperature of coffee based on the data recorded inthe file coffeedatatxt
cdata=read(rsquocoffeedatatxtrsquo-12)
The -12 following the name of the file tells Scilab that you donrsquotknow how many rows you have but you have two columns of dataSince we know that we have 101 rows of data we could have alsoused the following command to load the data
cdata=read(rsquocoffeedatatxtrsquo1012)
We now have a variable called cdata which has two columns ofdata The first column contains the time while the second containsthe Temperature records Hence we can plot it using the followingcommand
plot(cdata(1)cdata(2))
This gives the plot shown in Figure B6 after appropriate labelshave been added as described above
B7 Script files
Executing commands entered at the keyboard or keypad is themost primitive use of a calculating device It is only when se-quences of operations (to perform complex tasks) are stored awayand re-called for repeated application with different inputs that wehave a programmed operation Complex programs must necessarilyhave an interface with the user The basic interface tool in Scilabcapable of executing sequences of commands that are stored in a fileis the script The file names must have an extension of sce forexample program1sce assignment1sce These files are calledScript Sci-files or simply scripts
A script file is a file of standard keyboard charactersthat define commands to drive a program A script filecan be generated using any text editor or word proces-sor
To invoke a script in Scilab type the commandexec
exec filenamesce
where filename denotes the actual name of the file containing thescript When the script runs it is exactly the same as if you typedthe commands into the command window from the keyboard Thismeans that the Sci-file can use any variables you have defined priorto invoking the script Any variables created in the script are alsoavailable when the script is finished and any variable whose namewas assigned a value in the script will be over-written
If you give a script the same name as any variable that it computesin the running of the script then Scilab will give precedence tothe workspace variable over a script As stated in sectB3 the variablename will hide the script name after the script has run Scilab willtake the name as referring to the variable and not to the definedscript
Scilab also needs to know where to find the script to executedIf Scilab cannot find the script it will report that you have anundefined function or variable
The exact procedure for creating and saving Sci-files depends onthe Scilab version the operating system (MS Windows Macin-tosh or UnixLinux ) and the text editor you use The currentversion of Scilab includes a built-in editor which can be used tocreate and edit Sci-files and text files
To create a new Sci-file under the current version of Scilab simplytype edit newfilenamesce This will open up the Scilab editoredit
with a blank document called newfilename
B71 Script input and output
A script is a named file (with the extension sce) containing asequence of Scilab instructions which when its name is executedin a session performs those instructions in order as though theywere entered at the console
There are two common uses for scripts Pragmatically we maywant to save a particular computational sequence for later use orreference If the tasks involved are to be repeated often thenthe nature of the tasks should be analysed formulated into usefulfunctions (which we will discuss later) and the whole procedurepresented in a single function But if the tasks are only likely toget occasional use then the script may be adequately commentedand stored as a script file It is a quick and dirty way of saving
B7 Script files 251
the instructions for doing a particular job when we do not wantto go to the trouble of writing the functions for the job It is nota highly recommended modus operandi but useful all the same
The second use is for the script to act like a storyboard presentinginformation to a user in a structured and controlled way permit-ting the user to see the steps of a computational procedure andtheir outputs up on the screen At a more sophisticated level thescript takes on the proportions of an application controlling a largeand complex interaction with advanced interface features We willnot take our study of Scilab programming to this level but workmostly at the storyboard level using the simplest of Scilabrsquos in-terface tools
All the variables created in a script become global variables definedin the workspace and available for perusal As an application itis assumed that a script controls the entire workspace and thevariables it creates are lsquothe datarsquo of the application When scriptsare used as storyboards on a small scale then it cannot be assumedthat the variables they create deserve global recognition and it iscourteous to erase these variables when the script is finished unlessthey have some possible further use An unfortunate side-effect isthat if variables with the same name already exist in the workspacebefore a script is run then these variables will be over-written andlost
Used as storyboards scripts allow the user to cut paste and re-trythe various steps of a procedure and to modify the script to suithis own needs Storyboards may be used for demonstrations oftoolbox utilities or extended documentationexplanation of func-tions However if there is any danger that variables defined bythe script may over-write workspace variables then the demo ordocumentation should be re-fashioned into a function that leavesthe workspace variables untouched (more on this later)
Scripts will often read data from pre-prepared files and we will dis-cuss the functions provided by Scilab to perform this task below
Gathering data entered by the user really falls into the realm ofinterface design but we only briefly consider the simplest interfacefunction provided by Scilab Similarly we briefly consider thefunctions for providing the simplest forms of output presentation
Keyboard input
Scilab has a built-in function input that can be used to promptinput
the user to enter the value of a variable from the keyboard Forexample
Tc = input(rsquoPlease enter the temperature in degrees Celsiusrsquo)
will print the message
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
242 Appendix B Programming with Scilab
3 Final step is to plot the function using the plot commandplot
(eg plot(xy))
Using these steps it is possible to produce a basic plot of any math-ematical function (see Figure B3) Note that the plot shown inFigure B3 does not have a title axes labels or a legend all ofwhich are crucial for a good plot The Scilab commands to addthese elements to the plot are given in the following sections
- 10
- 08
- 06
- 04
- 02
00
02
04
06
08
10
0 1 2 3 4 5 6 7
Figure B3 The default plot of y = sin(x) in Scilab Notethat by default axes are not labelled and there is no title orlegend These need to added using the additional commandsxlabelylabeltitle and legend
Changing stylistic elements of the plot
An optional argument can be given to the plot command to cus-tomise each new line aspect This argument is a concatenatedstring containing information about colour line style or markersbased on the specifiers shown in Table B8 To specify the plotbe drawn with a red longdash-dot line with diamond marker thestring that would be passed to plot command would be rsquor-drsquo Thestring which is a concatenation (in any order) of the three typesof properties shown in Table B8 must be unambiguous Further-more the string specification is not case sensitive An exampleof specifying various line styles is shown in Example B6 and Fig-ure B4
Example B6 The following commands illustrate how line stylescan be used to distinguish between the plots of sin(2πt) sin(2πt+π6) and sin(2πt+ π2)
t=linspace(0130)
B6 Visualising data in Scilab 243
Table B8 Plot specifiers required to change line styles coloursand markers in plotsSpecifier Line style Specifier Colour Specifier Marker stylendash Solid line (default) r Red + Plus signndash ndash Dashed line g Green o Circle Dotted line b Blue Asteriskndash Dash-dotted line c Cyan x Cross
m Magenta s Squarey Yellow rsquononersquo No marker(default)k Blackw White
Figure B4 Resulting plotting after issuing the commands givenin Example B6
244 Appendix B Programming with Scilab
Example B6 illustrate a number of key commands relating to thecreation of multiple plots in Scilab The mtlb hold4 commandmtlb hold
holds the current plot and axis properties so that all subsequentplots are added to the existing plot
To distinguish each plot a different line specification is passed eachplot command In the example y = sin(2πt) is plotted using a redsolid line with data points indicated using a red circle as indicatedby the string rsquor-orsquo y = sin(2πt + π6) is plotted using greendashed-dotted line with data points marked by a square (rsquog-srsquo)while y = sin(2πt+ π2) is plotted using a cyan dashed line usingdiamonds to mark the data points (rsquoc--drsquo) Finally to completethe plot axes labels and a legend has been added to plot the com-mands for doing this are discussed in sectB62
B62 Adding titleslabels and legends to the plot
All good plots should have the following properties
1 The style of figure is appropriate to the information beingconveyed
2 The contents are well organised so it is easy for readers tointerpret the information
3 Legends labels etc are clear and concise (eg contain unitsif required do not obstruct parts of the plot etc) and
4 A caption or title that summarises the information presented
Like most mathematical applications Scilab has a wide variety ofdifferent plotting styles that can be used (eg look up the help forbar polarplot etc) This allows you to select the most appropri-barpolarplot
ate style to display and organise your data
The Scilab commands xlabel ylabel title and legend al-xlabel ylabel title and
legend lows you to add axes labels titles and legends to plot As shownin Example B6 the commands xlabel ylabel and title takea string input containing the title or label text For the legend
command a string is required for each separate plot Note theorder of the strings in the legend command reflects the order inwhich the each of plots were made
Including mathematical notation in titles labels and legends
It is possible to include mathematical characters such as π inScilab plots This is done by inserting special commands intoeither the title legend or label string These commands are basedon the abbreviations for mathematical notation used in programssuch as LATEX MathType etc A brief summary of the commands
4 mtlb hold on is equivalent to the Scilab commandset(gca()auto clearoff) while mtlb hold off is equivalentto the Scilab command set(gca()auto clearon)
B6 Visualising data in Scilab 245
Table B9 Basic mathematical symbols available in Scilab forincluding mathematical notation in titles labels and legendscommand symbol command symbol command symbolalpha α upsilon υ sim sim
Im = cup cup wp weierpotimes otimes subseteq sube oslash cap cap in isin supseteq supesupset sup lceil d subset subint
intcdot middot o oslash
rfloor c neg 6= nabla nablalfloor b times times ldots perp perp surd
radicprime prime
wedge and varpi $ O Oslashrceil e rangle 〉 mid |
vee or copyright ccopy langle 〈
available in Scilab for typesetting mathematical notation is givenin Table B9 To insert these commands into a text string simplystart and end the string with $ signs as shown in Example B7
Example B7 Scilab commands to produce the labelled plot ofy = sin
There are a few key concepts to note about the Example B7 Thefirst can be seen in the string being passed to the title commandtitle
In this string sin is prefixed with a backslash () This is doneso that the letters for sin are not typeset in italic which shouldonly be used for variables and not common mathematical func-tions Table B10 shows the command sequences to use for themore common mathematical functions5 To get the fraction in thetitle the command fracab has been used to produce a
b Ifsuperscripts and subscripts are required in a title or label these aresuperscripts amp subscripts
obtained by adding the subscript character ( ) and the superscriptcharacter (^) to the string These characters modify the characteror substring defined in braces immediately following For examplex 2 gives x2 while e^2x gives e2x
Note in Example B7 the commands xlabel and ylabel has beenxlabelylabel
used to set the axes labels in Figure B5 Instead of using thesethree commands the title and the axes labels could have been setusing the single xtitle command (see help xtitle for details)xtitle
In the example we have also customised the font size to make thetext more readable by a) passing an option to the title xlabelylabel commands and b) by using the command xset to set thexset
default font size before executing the xstring command Finallythe xstring command adds the text string to plot at the locationxstring
specified by variables b and c In the example the coordinates wewant to add the text are obtained by executing the xclick functionxclick
and clicking on the point required in the plot window
5 Additional details can be found in books covering Mathematics in LATEX
B6 Visualising data in Scilab 247
Table B10 Command sequences for the common mathematicalfunctions in Scilabarccos arcsin arctan cos cot
csc exp lim ln log
max min sec sin tan
B63 Plotting numeric data stored in an ASCII file
As data can be stored in a wide variety of ways in external filesScilab has a wide variety of functions which can be used to importthe data stored in these files (see help for the commands read
fscanf and xlsread) Since data can be stored in such a wideread fscanf and xlsread
range of formats we will confine our discussions here to the situa-tion where numeric data is stored in a ASCII (text) file where eachrow of the file contains the same number of entries
To illustrate the procedure for plotting data stored in ASCII file wewill consider the data obtained measuring the temperature of a cupof coffee every minute for a total of 100 mins The data recorded isstored in the a file called coffeedatatxt6 which simply containsthe two columns of numbers shown in Table B11 As an examplethe first 10 lines of the text file contains the following numbers7
00000 1000000
10000 950000
20000 903125
30000 859180
40000 817981
50000 779357
60000 743147
70000 709201
80000 677376
90000 647540
Note the current version of Scilab has a builtin text editor whichcan be used to view such text files To access this editor simplytype the command edit at the command prompt Once open youedit
can use the graphical interface to open the required file
The first issue is to get the data into Scilab so it is available forplot command In Scilab the command read can used to importread
the data and assign it to a variable In the current example wewould use the following command to import the data and assignit to a variable called cdata assuming the data is in the currentdirectory8
6 The file coffeedatatxt has been attached and should be available for savingprovided you are using Adobe Reader To save the file right click on thepurple Windows version or the red MacOSLinux version in the margin andselect save file from the options Ensure you save the file as coffeedatatxt
7 normally you will have received the data file in this format8 You can check what directory Scilab is in by typing the command pwd
Figure B6 Temperature of coffee based on the data recorded inthe file coffeedatatxt
cdata=read(rsquocoffeedatatxtrsquo-12)
The -12 following the name of the file tells Scilab that you donrsquotknow how many rows you have but you have two columns of dataSince we know that we have 101 rows of data we could have alsoused the following command to load the data
cdata=read(rsquocoffeedatatxtrsquo1012)
We now have a variable called cdata which has two columns ofdata The first column contains the time while the second containsthe Temperature records Hence we can plot it using the followingcommand
plot(cdata(1)cdata(2))
This gives the plot shown in Figure B6 after appropriate labelshave been added as described above
B7 Script files
Executing commands entered at the keyboard or keypad is themost primitive use of a calculating device It is only when se-quences of operations (to perform complex tasks) are stored awayand re-called for repeated application with different inputs that wehave a programmed operation Complex programs must necessarilyhave an interface with the user The basic interface tool in Scilabcapable of executing sequences of commands that are stored in a fileis the script The file names must have an extension of sce forexample program1sce assignment1sce These files are calledScript Sci-files or simply scripts
A script file is a file of standard keyboard charactersthat define commands to drive a program A script filecan be generated using any text editor or word proces-sor
To invoke a script in Scilab type the commandexec
exec filenamesce
where filename denotes the actual name of the file containing thescript When the script runs it is exactly the same as if you typedthe commands into the command window from the keyboard Thismeans that the Sci-file can use any variables you have defined priorto invoking the script Any variables created in the script are alsoavailable when the script is finished and any variable whose namewas assigned a value in the script will be over-written
If you give a script the same name as any variable that it computesin the running of the script then Scilab will give precedence tothe workspace variable over a script As stated in sectB3 the variablename will hide the script name after the script has run Scilab willtake the name as referring to the variable and not to the definedscript
Scilab also needs to know where to find the script to executedIf Scilab cannot find the script it will report that you have anundefined function or variable
The exact procedure for creating and saving Sci-files depends onthe Scilab version the operating system (MS Windows Macin-tosh or UnixLinux ) and the text editor you use The currentversion of Scilab includes a built-in editor which can be used tocreate and edit Sci-files and text files
To create a new Sci-file under the current version of Scilab simplytype edit newfilenamesce This will open up the Scilab editoredit
with a blank document called newfilename
B71 Script input and output
A script is a named file (with the extension sce) containing asequence of Scilab instructions which when its name is executedin a session performs those instructions in order as though theywere entered at the console
There are two common uses for scripts Pragmatically we maywant to save a particular computational sequence for later use orreference If the tasks involved are to be repeated often thenthe nature of the tasks should be analysed formulated into usefulfunctions (which we will discuss later) and the whole procedurepresented in a single function But if the tasks are only likely toget occasional use then the script may be adequately commentedand stored as a script file It is a quick and dirty way of saving
B7 Script files 251
the instructions for doing a particular job when we do not wantto go to the trouble of writing the functions for the job It is nota highly recommended modus operandi but useful all the same
The second use is for the script to act like a storyboard presentinginformation to a user in a structured and controlled way permit-ting the user to see the steps of a computational procedure andtheir outputs up on the screen At a more sophisticated level thescript takes on the proportions of an application controlling a largeand complex interaction with advanced interface features We willnot take our study of Scilab programming to this level but workmostly at the storyboard level using the simplest of Scilabrsquos in-terface tools
All the variables created in a script become global variables definedin the workspace and available for perusal As an application itis assumed that a script controls the entire workspace and thevariables it creates are lsquothe datarsquo of the application When scriptsare used as storyboards on a small scale then it cannot be assumedthat the variables they create deserve global recognition and it iscourteous to erase these variables when the script is finished unlessthey have some possible further use An unfortunate side-effect isthat if variables with the same name already exist in the workspacebefore a script is run then these variables will be over-written andlost
Used as storyboards scripts allow the user to cut paste and re-trythe various steps of a procedure and to modify the script to suithis own needs Storyboards may be used for demonstrations oftoolbox utilities or extended documentationexplanation of func-tions However if there is any danger that variables defined bythe script may over-write workspace variables then the demo ordocumentation should be re-fashioned into a function that leavesthe workspace variables untouched (more on this later)
Scripts will often read data from pre-prepared files and we will dis-cuss the functions provided by Scilab to perform this task below
Gathering data entered by the user really falls into the realm ofinterface design but we only briefly consider the simplest interfacefunction provided by Scilab Similarly we briefly consider thefunctions for providing the simplest forms of output presentation
Keyboard input
Scilab has a built-in function input that can be used to promptinput
the user to enter the value of a variable from the keyboard Forexample
Tc = input(rsquoPlease enter the temperature in degrees Celsiusrsquo)
will print the message
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
B6 Visualising data in Scilab 243
Table B8 Plot specifiers required to change line styles coloursand markers in plotsSpecifier Line style Specifier Colour Specifier Marker stylendash Solid line (default) r Red + Plus signndash ndash Dashed line g Green o Circle Dotted line b Blue Asteriskndash Dash-dotted line c Cyan x Cross
m Magenta s Squarey Yellow rsquononersquo No marker(default)k Blackw White
Figure B4 Resulting plotting after issuing the commands givenin Example B6
244 Appendix B Programming with Scilab
Example B6 illustrate a number of key commands relating to thecreation of multiple plots in Scilab The mtlb hold4 commandmtlb hold
holds the current plot and axis properties so that all subsequentplots are added to the existing plot
To distinguish each plot a different line specification is passed eachplot command In the example y = sin(2πt) is plotted using a redsolid line with data points indicated using a red circle as indicatedby the string rsquor-orsquo y = sin(2πt + π6) is plotted using greendashed-dotted line with data points marked by a square (rsquog-srsquo)while y = sin(2πt+ π2) is plotted using a cyan dashed line usingdiamonds to mark the data points (rsquoc--drsquo) Finally to completethe plot axes labels and a legend has been added to plot the com-mands for doing this are discussed in sectB62
B62 Adding titleslabels and legends to the plot
All good plots should have the following properties
1 The style of figure is appropriate to the information beingconveyed
2 The contents are well organised so it is easy for readers tointerpret the information
3 Legends labels etc are clear and concise (eg contain unitsif required do not obstruct parts of the plot etc) and
4 A caption or title that summarises the information presented
Like most mathematical applications Scilab has a wide variety ofdifferent plotting styles that can be used (eg look up the help forbar polarplot etc) This allows you to select the most appropri-barpolarplot
ate style to display and organise your data
The Scilab commands xlabel ylabel title and legend al-xlabel ylabel title and
legend lows you to add axes labels titles and legends to plot As shownin Example B6 the commands xlabel ylabel and title takea string input containing the title or label text For the legend
command a string is required for each separate plot Note theorder of the strings in the legend command reflects the order inwhich the each of plots were made
Including mathematical notation in titles labels and legends
It is possible to include mathematical characters such as π inScilab plots This is done by inserting special commands intoeither the title legend or label string These commands are basedon the abbreviations for mathematical notation used in programssuch as LATEX MathType etc A brief summary of the commands
4 mtlb hold on is equivalent to the Scilab commandset(gca()auto clearoff) while mtlb hold off is equivalentto the Scilab command set(gca()auto clearon)
B6 Visualising data in Scilab 245
Table B9 Basic mathematical symbols available in Scilab forincluding mathematical notation in titles labels and legendscommand symbol command symbol command symbolalpha α upsilon υ sim sim
Im = cup cup wp weierpotimes otimes subseteq sube oslash cap cap in isin supseteq supesupset sup lceil d subset subint
intcdot middot o oslash
rfloor c neg 6= nabla nablalfloor b times times ldots perp perp surd
radicprime prime
wedge and varpi $ O Oslashrceil e rangle 〉 mid |
vee or copyright ccopy langle 〈
available in Scilab for typesetting mathematical notation is givenin Table B9 To insert these commands into a text string simplystart and end the string with $ signs as shown in Example B7
Example B7 Scilab commands to produce the labelled plot ofy = sin
There are a few key concepts to note about the Example B7 Thefirst can be seen in the string being passed to the title commandtitle
In this string sin is prefixed with a backslash () This is doneso that the letters for sin are not typeset in italic which shouldonly be used for variables and not common mathematical func-tions Table B10 shows the command sequences to use for themore common mathematical functions5 To get the fraction in thetitle the command fracab has been used to produce a
b Ifsuperscripts and subscripts are required in a title or label these aresuperscripts amp subscripts
obtained by adding the subscript character ( ) and the superscriptcharacter (^) to the string These characters modify the characteror substring defined in braces immediately following For examplex 2 gives x2 while e^2x gives e2x
Note in Example B7 the commands xlabel and ylabel has beenxlabelylabel
used to set the axes labels in Figure B5 Instead of using thesethree commands the title and the axes labels could have been setusing the single xtitle command (see help xtitle for details)xtitle
In the example we have also customised the font size to make thetext more readable by a) passing an option to the title xlabelylabel commands and b) by using the command xset to set thexset
default font size before executing the xstring command Finallythe xstring command adds the text string to plot at the locationxstring
specified by variables b and c In the example the coordinates wewant to add the text are obtained by executing the xclick functionxclick
and clicking on the point required in the plot window
5 Additional details can be found in books covering Mathematics in LATEX
B6 Visualising data in Scilab 247
Table B10 Command sequences for the common mathematicalfunctions in Scilabarccos arcsin arctan cos cot
csc exp lim ln log
max min sec sin tan
B63 Plotting numeric data stored in an ASCII file
As data can be stored in a wide variety of ways in external filesScilab has a wide variety of functions which can be used to importthe data stored in these files (see help for the commands read
fscanf and xlsread) Since data can be stored in such a wideread fscanf and xlsread
range of formats we will confine our discussions here to the situa-tion where numeric data is stored in a ASCII (text) file where eachrow of the file contains the same number of entries
To illustrate the procedure for plotting data stored in ASCII file wewill consider the data obtained measuring the temperature of a cupof coffee every minute for a total of 100 mins The data recorded isstored in the a file called coffeedatatxt6 which simply containsthe two columns of numbers shown in Table B11 As an examplethe first 10 lines of the text file contains the following numbers7
00000 1000000
10000 950000
20000 903125
30000 859180
40000 817981
50000 779357
60000 743147
70000 709201
80000 677376
90000 647540
Note the current version of Scilab has a builtin text editor whichcan be used to view such text files To access this editor simplytype the command edit at the command prompt Once open youedit
can use the graphical interface to open the required file
The first issue is to get the data into Scilab so it is available forplot command In Scilab the command read can used to importread
the data and assign it to a variable In the current example wewould use the following command to import the data and assignit to a variable called cdata assuming the data is in the currentdirectory8
6 The file coffeedatatxt has been attached and should be available for savingprovided you are using Adobe Reader To save the file right click on thepurple Windows version or the red MacOSLinux version in the margin andselect save file from the options Ensure you save the file as coffeedatatxt
7 normally you will have received the data file in this format8 You can check what directory Scilab is in by typing the command pwd
Figure B6 Temperature of coffee based on the data recorded inthe file coffeedatatxt
cdata=read(rsquocoffeedatatxtrsquo-12)
The -12 following the name of the file tells Scilab that you donrsquotknow how many rows you have but you have two columns of dataSince we know that we have 101 rows of data we could have alsoused the following command to load the data
cdata=read(rsquocoffeedatatxtrsquo1012)
We now have a variable called cdata which has two columns ofdata The first column contains the time while the second containsthe Temperature records Hence we can plot it using the followingcommand
plot(cdata(1)cdata(2))
This gives the plot shown in Figure B6 after appropriate labelshave been added as described above
B7 Script files
Executing commands entered at the keyboard or keypad is themost primitive use of a calculating device It is only when se-quences of operations (to perform complex tasks) are stored awayand re-called for repeated application with different inputs that wehave a programmed operation Complex programs must necessarilyhave an interface with the user The basic interface tool in Scilabcapable of executing sequences of commands that are stored in a fileis the script The file names must have an extension of sce forexample program1sce assignment1sce These files are calledScript Sci-files or simply scripts
A script file is a file of standard keyboard charactersthat define commands to drive a program A script filecan be generated using any text editor or word proces-sor
To invoke a script in Scilab type the commandexec
exec filenamesce
where filename denotes the actual name of the file containing thescript When the script runs it is exactly the same as if you typedthe commands into the command window from the keyboard Thismeans that the Sci-file can use any variables you have defined priorto invoking the script Any variables created in the script are alsoavailable when the script is finished and any variable whose namewas assigned a value in the script will be over-written
If you give a script the same name as any variable that it computesin the running of the script then Scilab will give precedence tothe workspace variable over a script As stated in sectB3 the variablename will hide the script name after the script has run Scilab willtake the name as referring to the variable and not to the definedscript
Scilab also needs to know where to find the script to executedIf Scilab cannot find the script it will report that you have anundefined function or variable
The exact procedure for creating and saving Sci-files depends onthe Scilab version the operating system (MS Windows Macin-tosh or UnixLinux ) and the text editor you use The currentversion of Scilab includes a built-in editor which can be used tocreate and edit Sci-files and text files
To create a new Sci-file under the current version of Scilab simplytype edit newfilenamesce This will open up the Scilab editoredit
with a blank document called newfilename
B71 Script input and output
A script is a named file (with the extension sce) containing asequence of Scilab instructions which when its name is executedin a session performs those instructions in order as though theywere entered at the console
There are two common uses for scripts Pragmatically we maywant to save a particular computational sequence for later use orreference If the tasks involved are to be repeated often thenthe nature of the tasks should be analysed formulated into usefulfunctions (which we will discuss later) and the whole procedurepresented in a single function But if the tasks are only likely toget occasional use then the script may be adequately commentedand stored as a script file It is a quick and dirty way of saving
B7 Script files 251
the instructions for doing a particular job when we do not wantto go to the trouble of writing the functions for the job It is nota highly recommended modus operandi but useful all the same
The second use is for the script to act like a storyboard presentinginformation to a user in a structured and controlled way permit-ting the user to see the steps of a computational procedure andtheir outputs up on the screen At a more sophisticated level thescript takes on the proportions of an application controlling a largeand complex interaction with advanced interface features We willnot take our study of Scilab programming to this level but workmostly at the storyboard level using the simplest of Scilabrsquos in-terface tools
All the variables created in a script become global variables definedin the workspace and available for perusal As an application itis assumed that a script controls the entire workspace and thevariables it creates are lsquothe datarsquo of the application When scriptsare used as storyboards on a small scale then it cannot be assumedthat the variables they create deserve global recognition and it iscourteous to erase these variables when the script is finished unlessthey have some possible further use An unfortunate side-effect isthat if variables with the same name already exist in the workspacebefore a script is run then these variables will be over-written andlost
Used as storyboards scripts allow the user to cut paste and re-trythe various steps of a procedure and to modify the script to suithis own needs Storyboards may be used for demonstrations oftoolbox utilities or extended documentationexplanation of func-tions However if there is any danger that variables defined bythe script may over-write workspace variables then the demo ordocumentation should be re-fashioned into a function that leavesthe workspace variables untouched (more on this later)
Scripts will often read data from pre-prepared files and we will dis-cuss the functions provided by Scilab to perform this task below
Gathering data entered by the user really falls into the realm ofinterface design but we only briefly consider the simplest interfacefunction provided by Scilab Similarly we briefly consider thefunctions for providing the simplest forms of output presentation
Keyboard input
Scilab has a built-in function input that can be used to promptinput
the user to enter the value of a variable from the keyboard Forexample
Tc = input(rsquoPlease enter the temperature in degrees Celsiusrsquo)
will print the message
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
244 Appendix B Programming with Scilab
Example B6 illustrate a number of key commands relating to thecreation of multiple plots in Scilab The mtlb hold4 commandmtlb hold
holds the current plot and axis properties so that all subsequentplots are added to the existing plot
To distinguish each plot a different line specification is passed eachplot command In the example y = sin(2πt) is plotted using a redsolid line with data points indicated using a red circle as indicatedby the string rsquor-orsquo y = sin(2πt + π6) is plotted using greendashed-dotted line with data points marked by a square (rsquog-srsquo)while y = sin(2πt+ π2) is plotted using a cyan dashed line usingdiamonds to mark the data points (rsquoc--drsquo) Finally to completethe plot axes labels and a legend has been added to plot the com-mands for doing this are discussed in sectB62
B62 Adding titleslabels and legends to the plot
All good plots should have the following properties
1 The style of figure is appropriate to the information beingconveyed
2 The contents are well organised so it is easy for readers tointerpret the information
3 Legends labels etc are clear and concise (eg contain unitsif required do not obstruct parts of the plot etc) and
4 A caption or title that summarises the information presented
Like most mathematical applications Scilab has a wide variety ofdifferent plotting styles that can be used (eg look up the help forbar polarplot etc) This allows you to select the most appropri-barpolarplot
ate style to display and organise your data
The Scilab commands xlabel ylabel title and legend al-xlabel ylabel title and
legend lows you to add axes labels titles and legends to plot As shownin Example B6 the commands xlabel ylabel and title takea string input containing the title or label text For the legend
command a string is required for each separate plot Note theorder of the strings in the legend command reflects the order inwhich the each of plots were made
Including mathematical notation in titles labels and legends
It is possible to include mathematical characters such as π inScilab plots This is done by inserting special commands intoeither the title legend or label string These commands are basedon the abbreviations for mathematical notation used in programssuch as LATEX MathType etc A brief summary of the commands
4 mtlb hold on is equivalent to the Scilab commandset(gca()auto clearoff) while mtlb hold off is equivalentto the Scilab command set(gca()auto clearon)
B6 Visualising data in Scilab 245
Table B9 Basic mathematical symbols available in Scilab forincluding mathematical notation in titles labels and legendscommand symbol command symbol command symbolalpha α upsilon υ sim sim
Im = cup cup wp weierpotimes otimes subseteq sube oslash cap cap in isin supseteq supesupset sup lceil d subset subint
intcdot middot o oslash
rfloor c neg 6= nabla nablalfloor b times times ldots perp perp surd
radicprime prime
wedge and varpi $ O Oslashrceil e rangle 〉 mid |
vee or copyright ccopy langle 〈
available in Scilab for typesetting mathematical notation is givenin Table B9 To insert these commands into a text string simplystart and end the string with $ signs as shown in Example B7
Example B7 Scilab commands to produce the labelled plot ofy = sin
There are a few key concepts to note about the Example B7 Thefirst can be seen in the string being passed to the title commandtitle
In this string sin is prefixed with a backslash () This is doneso that the letters for sin are not typeset in italic which shouldonly be used for variables and not common mathematical func-tions Table B10 shows the command sequences to use for themore common mathematical functions5 To get the fraction in thetitle the command fracab has been used to produce a
b Ifsuperscripts and subscripts are required in a title or label these aresuperscripts amp subscripts
obtained by adding the subscript character ( ) and the superscriptcharacter (^) to the string These characters modify the characteror substring defined in braces immediately following For examplex 2 gives x2 while e^2x gives e2x
Note in Example B7 the commands xlabel and ylabel has beenxlabelylabel
used to set the axes labels in Figure B5 Instead of using thesethree commands the title and the axes labels could have been setusing the single xtitle command (see help xtitle for details)xtitle
In the example we have also customised the font size to make thetext more readable by a) passing an option to the title xlabelylabel commands and b) by using the command xset to set thexset
default font size before executing the xstring command Finallythe xstring command adds the text string to plot at the locationxstring
specified by variables b and c In the example the coordinates wewant to add the text are obtained by executing the xclick functionxclick
and clicking on the point required in the plot window
5 Additional details can be found in books covering Mathematics in LATEX
B6 Visualising data in Scilab 247
Table B10 Command sequences for the common mathematicalfunctions in Scilabarccos arcsin arctan cos cot
csc exp lim ln log
max min sec sin tan
B63 Plotting numeric data stored in an ASCII file
As data can be stored in a wide variety of ways in external filesScilab has a wide variety of functions which can be used to importthe data stored in these files (see help for the commands read
fscanf and xlsread) Since data can be stored in such a wideread fscanf and xlsread
range of formats we will confine our discussions here to the situa-tion where numeric data is stored in a ASCII (text) file where eachrow of the file contains the same number of entries
To illustrate the procedure for plotting data stored in ASCII file wewill consider the data obtained measuring the temperature of a cupof coffee every minute for a total of 100 mins The data recorded isstored in the a file called coffeedatatxt6 which simply containsthe two columns of numbers shown in Table B11 As an examplethe first 10 lines of the text file contains the following numbers7
00000 1000000
10000 950000
20000 903125
30000 859180
40000 817981
50000 779357
60000 743147
70000 709201
80000 677376
90000 647540
Note the current version of Scilab has a builtin text editor whichcan be used to view such text files To access this editor simplytype the command edit at the command prompt Once open youedit
can use the graphical interface to open the required file
The first issue is to get the data into Scilab so it is available forplot command In Scilab the command read can used to importread
the data and assign it to a variable In the current example wewould use the following command to import the data and assignit to a variable called cdata assuming the data is in the currentdirectory8
6 The file coffeedatatxt has been attached and should be available for savingprovided you are using Adobe Reader To save the file right click on thepurple Windows version or the red MacOSLinux version in the margin andselect save file from the options Ensure you save the file as coffeedatatxt
7 normally you will have received the data file in this format8 You can check what directory Scilab is in by typing the command pwd
Figure B6 Temperature of coffee based on the data recorded inthe file coffeedatatxt
cdata=read(rsquocoffeedatatxtrsquo-12)
The -12 following the name of the file tells Scilab that you donrsquotknow how many rows you have but you have two columns of dataSince we know that we have 101 rows of data we could have alsoused the following command to load the data
cdata=read(rsquocoffeedatatxtrsquo1012)
We now have a variable called cdata which has two columns ofdata The first column contains the time while the second containsthe Temperature records Hence we can plot it using the followingcommand
plot(cdata(1)cdata(2))
This gives the plot shown in Figure B6 after appropriate labelshave been added as described above
B7 Script files
Executing commands entered at the keyboard or keypad is themost primitive use of a calculating device It is only when se-quences of operations (to perform complex tasks) are stored awayand re-called for repeated application with different inputs that wehave a programmed operation Complex programs must necessarilyhave an interface with the user The basic interface tool in Scilabcapable of executing sequences of commands that are stored in a fileis the script The file names must have an extension of sce forexample program1sce assignment1sce These files are calledScript Sci-files or simply scripts
A script file is a file of standard keyboard charactersthat define commands to drive a program A script filecan be generated using any text editor or word proces-sor
To invoke a script in Scilab type the commandexec
exec filenamesce
where filename denotes the actual name of the file containing thescript When the script runs it is exactly the same as if you typedthe commands into the command window from the keyboard Thismeans that the Sci-file can use any variables you have defined priorto invoking the script Any variables created in the script are alsoavailable when the script is finished and any variable whose namewas assigned a value in the script will be over-written
If you give a script the same name as any variable that it computesin the running of the script then Scilab will give precedence tothe workspace variable over a script As stated in sectB3 the variablename will hide the script name after the script has run Scilab willtake the name as referring to the variable and not to the definedscript
Scilab also needs to know where to find the script to executedIf Scilab cannot find the script it will report that you have anundefined function or variable
The exact procedure for creating and saving Sci-files depends onthe Scilab version the operating system (MS Windows Macin-tosh or UnixLinux ) and the text editor you use The currentversion of Scilab includes a built-in editor which can be used tocreate and edit Sci-files and text files
To create a new Sci-file under the current version of Scilab simplytype edit newfilenamesce This will open up the Scilab editoredit
with a blank document called newfilename
B71 Script input and output
A script is a named file (with the extension sce) containing asequence of Scilab instructions which when its name is executedin a session performs those instructions in order as though theywere entered at the console
There are two common uses for scripts Pragmatically we maywant to save a particular computational sequence for later use orreference If the tasks involved are to be repeated often thenthe nature of the tasks should be analysed formulated into usefulfunctions (which we will discuss later) and the whole procedurepresented in a single function But if the tasks are only likely toget occasional use then the script may be adequately commentedand stored as a script file It is a quick and dirty way of saving
B7 Script files 251
the instructions for doing a particular job when we do not wantto go to the trouble of writing the functions for the job It is nota highly recommended modus operandi but useful all the same
The second use is for the script to act like a storyboard presentinginformation to a user in a structured and controlled way permit-ting the user to see the steps of a computational procedure andtheir outputs up on the screen At a more sophisticated level thescript takes on the proportions of an application controlling a largeand complex interaction with advanced interface features We willnot take our study of Scilab programming to this level but workmostly at the storyboard level using the simplest of Scilabrsquos in-terface tools
All the variables created in a script become global variables definedin the workspace and available for perusal As an application itis assumed that a script controls the entire workspace and thevariables it creates are lsquothe datarsquo of the application When scriptsare used as storyboards on a small scale then it cannot be assumedthat the variables they create deserve global recognition and it iscourteous to erase these variables when the script is finished unlessthey have some possible further use An unfortunate side-effect isthat if variables with the same name already exist in the workspacebefore a script is run then these variables will be over-written andlost
Used as storyboards scripts allow the user to cut paste and re-trythe various steps of a procedure and to modify the script to suithis own needs Storyboards may be used for demonstrations oftoolbox utilities or extended documentationexplanation of func-tions However if there is any danger that variables defined bythe script may over-write workspace variables then the demo ordocumentation should be re-fashioned into a function that leavesthe workspace variables untouched (more on this later)
Scripts will often read data from pre-prepared files and we will dis-cuss the functions provided by Scilab to perform this task below
Gathering data entered by the user really falls into the realm ofinterface design but we only briefly consider the simplest interfacefunction provided by Scilab Similarly we briefly consider thefunctions for providing the simplest forms of output presentation
Keyboard input
Scilab has a built-in function input that can be used to promptinput
the user to enter the value of a variable from the keyboard Forexample
Tc = input(rsquoPlease enter the temperature in degrees Celsiusrsquo)
will print the message
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
B6 Visualising data in Scilab 245
Table B9 Basic mathematical symbols available in Scilab forincluding mathematical notation in titles labels and legendscommand symbol command symbol command symbolalpha α upsilon υ sim sim
Im = cup cup wp weierpotimes otimes subseteq sube oslash cap cap in isin supseteq supesupset sup lceil d subset subint
intcdot middot o oslash
rfloor c neg 6= nabla nablalfloor b times times ldots perp perp surd
radicprime prime
wedge and varpi $ O Oslashrceil e rangle 〉 mid |
vee or copyright ccopy langle 〈
available in Scilab for typesetting mathematical notation is givenin Table B9 To insert these commands into a text string simplystart and end the string with $ signs as shown in Example B7
Example B7 Scilab commands to produce the labelled plot ofy = sin
There are a few key concepts to note about the Example B7 Thefirst can be seen in the string being passed to the title commandtitle
In this string sin is prefixed with a backslash () This is doneso that the letters for sin are not typeset in italic which shouldonly be used for variables and not common mathematical func-tions Table B10 shows the command sequences to use for themore common mathematical functions5 To get the fraction in thetitle the command fracab has been used to produce a
b Ifsuperscripts and subscripts are required in a title or label these aresuperscripts amp subscripts
obtained by adding the subscript character ( ) and the superscriptcharacter (^) to the string These characters modify the characteror substring defined in braces immediately following For examplex 2 gives x2 while e^2x gives e2x
Note in Example B7 the commands xlabel and ylabel has beenxlabelylabel
used to set the axes labels in Figure B5 Instead of using thesethree commands the title and the axes labels could have been setusing the single xtitle command (see help xtitle for details)xtitle
In the example we have also customised the font size to make thetext more readable by a) passing an option to the title xlabelylabel commands and b) by using the command xset to set thexset
default font size before executing the xstring command Finallythe xstring command adds the text string to plot at the locationxstring
specified by variables b and c In the example the coordinates wewant to add the text are obtained by executing the xclick functionxclick
and clicking on the point required in the plot window
5 Additional details can be found in books covering Mathematics in LATEX
B6 Visualising data in Scilab 247
Table B10 Command sequences for the common mathematicalfunctions in Scilabarccos arcsin arctan cos cot
csc exp lim ln log
max min sec sin tan
B63 Plotting numeric data stored in an ASCII file
As data can be stored in a wide variety of ways in external filesScilab has a wide variety of functions which can be used to importthe data stored in these files (see help for the commands read
fscanf and xlsread) Since data can be stored in such a wideread fscanf and xlsread
range of formats we will confine our discussions here to the situa-tion where numeric data is stored in a ASCII (text) file where eachrow of the file contains the same number of entries
To illustrate the procedure for plotting data stored in ASCII file wewill consider the data obtained measuring the temperature of a cupof coffee every minute for a total of 100 mins The data recorded isstored in the a file called coffeedatatxt6 which simply containsthe two columns of numbers shown in Table B11 As an examplethe first 10 lines of the text file contains the following numbers7
00000 1000000
10000 950000
20000 903125
30000 859180
40000 817981
50000 779357
60000 743147
70000 709201
80000 677376
90000 647540
Note the current version of Scilab has a builtin text editor whichcan be used to view such text files To access this editor simplytype the command edit at the command prompt Once open youedit
can use the graphical interface to open the required file
The first issue is to get the data into Scilab so it is available forplot command In Scilab the command read can used to importread
the data and assign it to a variable In the current example wewould use the following command to import the data and assignit to a variable called cdata assuming the data is in the currentdirectory8
6 The file coffeedatatxt has been attached and should be available for savingprovided you are using Adobe Reader To save the file right click on thepurple Windows version or the red MacOSLinux version in the margin andselect save file from the options Ensure you save the file as coffeedatatxt
7 normally you will have received the data file in this format8 You can check what directory Scilab is in by typing the command pwd
Figure B6 Temperature of coffee based on the data recorded inthe file coffeedatatxt
cdata=read(rsquocoffeedatatxtrsquo-12)
The -12 following the name of the file tells Scilab that you donrsquotknow how many rows you have but you have two columns of dataSince we know that we have 101 rows of data we could have alsoused the following command to load the data
cdata=read(rsquocoffeedatatxtrsquo1012)
We now have a variable called cdata which has two columns ofdata The first column contains the time while the second containsthe Temperature records Hence we can plot it using the followingcommand
plot(cdata(1)cdata(2))
This gives the plot shown in Figure B6 after appropriate labelshave been added as described above
B7 Script files
Executing commands entered at the keyboard or keypad is themost primitive use of a calculating device It is only when se-quences of operations (to perform complex tasks) are stored awayand re-called for repeated application with different inputs that wehave a programmed operation Complex programs must necessarilyhave an interface with the user The basic interface tool in Scilabcapable of executing sequences of commands that are stored in a fileis the script The file names must have an extension of sce forexample program1sce assignment1sce These files are calledScript Sci-files or simply scripts
A script file is a file of standard keyboard charactersthat define commands to drive a program A script filecan be generated using any text editor or word proces-sor
To invoke a script in Scilab type the commandexec
exec filenamesce
where filename denotes the actual name of the file containing thescript When the script runs it is exactly the same as if you typedthe commands into the command window from the keyboard Thismeans that the Sci-file can use any variables you have defined priorto invoking the script Any variables created in the script are alsoavailable when the script is finished and any variable whose namewas assigned a value in the script will be over-written
If you give a script the same name as any variable that it computesin the running of the script then Scilab will give precedence tothe workspace variable over a script As stated in sectB3 the variablename will hide the script name after the script has run Scilab willtake the name as referring to the variable and not to the definedscript
Scilab also needs to know where to find the script to executedIf Scilab cannot find the script it will report that you have anundefined function or variable
The exact procedure for creating and saving Sci-files depends onthe Scilab version the operating system (MS Windows Macin-tosh or UnixLinux ) and the text editor you use The currentversion of Scilab includes a built-in editor which can be used tocreate and edit Sci-files and text files
To create a new Sci-file under the current version of Scilab simplytype edit newfilenamesce This will open up the Scilab editoredit
with a blank document called newfilename
B71 Script input and output
A script is a named file (with the extension sce) containing asequence of Scilab instructions which when its name is executedin a session performs those instructions in order as though theywere entered at the console
There are two common uses for scripts Pragmatically we maywant to save a particular computational sequence for later use orreference If the tasks involved are to be repeated often thenthe nature of the tasks should be analysed formulated into usefulfunctions (which we will discuss later) and the whole procedurepresented in a single function But if the tasks are only likely toget occasional use then the script may be adequately commentedand stored as a script file It is a quick and dirty way of saving
B7 Script files 251
the instructions for doing a particular job when we do not wantto go to the trouble of writing the functions for the job It is nota highly recommended modus operandi but useful all the same
The second use is for the script to act like a storyboard presentinginformation to a user in a structured and controlled way permit-ting the user to see the steps of a computational procedure andtheir outputs up on the screen At a more sophisticated level thescript takes on the proportions of an application controlling a largeand complex interaction with advanced interface features We willnot take our study of Scilab programming to this level but workmostly at the storyboard level using the simplest of Scilabrsquos in-terface tools
All the variables created in a script become global variables definedin the workspace and available for perusal As an application itis assumed that a script controls the entire workspace and thevariables it creates are lsquothe datarsquo of the application When scriptsare used as storyboards on a small scale then it cannot be assumedthat the variables they create deserve global recognition and it iscourteous to erase these variables when the script is finished unlessthey have some possible further use An unfortunate side-effect isthat if variables with the same name already exist in the workspacebefore a script is run then these variables will be over-written andlost
Used as storyboards scripts allow the user to cut paste and re-trythe various steps of a procedure and to modify the script to suithis own needs Storyboards may be used for demonstrations oftoolbox utilities or extended documentationexplanation of func-tions However if there is any danger that variables defined bythe script may over-write workspace variables then the demo ordocumentation should be re-fashioned into a function that leavesthe workspace variables untouched (more on this later)
Scripts will often read data from pre-prepared files and we will dis-cuss the functions provided by Scilab to perform this task below
Gathering data entered by the user really falls into the realm ofinterface design but we only briefly consider the simplest interfacefunction provided by Scilab Similarly we briefly consider thefunctions for providing the simplest forms of output presentation
Keyboard input
Scilab has a built-in function input that can be used to promptinput
the user to enter the value of a variable from the keyboard Forexample
Tc = input(rsquoPlease enter the temperature in degrees Celsiusrsquo)
will print the message
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
246 Appendix B Programming with Scilab
- 1 0
- 0 8
- 0 6
- 0 4
- 0 2
0 0
0 2
0 4
0 6
0 8
1 0
0 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 0
Figure B5 Plot of y = sin(2πt+ π
4
)illustrating the use of math-
ematical notation in labels
There are a few key concepts to note about the Example B7 Thefirst can be seen in the string being passed to the title commandtitle
In this string sin is prefixed with a backslash () This is doneso that the letters for sin are not typeset in italic which shouldonly be used for variables and not common mathematical func-tions Table B10 shows the command sequences to use for themore common mathematical functions5 To get the fraction in thetitle the command fracab has been used to produce a
b Ifsuperscripts and subscripts are required in a title or label these aresuperscripts amp subscripts
obtained by adding the subscript character ( ) and the superscriptcharacter (^) to the string These characters modify the characteror substring defined in braces immediately following For examplex 2 gives x2 while e^2x gives e2x
Note in Example B7 the commands xlabel and ylabel has beenxlabelylabel
used to set the axes labels in Figure B5 Instead of using thesethree commands the title and the axes labels could have been setusing the single xtitle command (see help xtitle for details)xtitle
In the example we have also customised the font size to make thetext more readable by a) passing an option to the title xlabelylabel commands and b) by using the command xset to set thexset
default font size before executing the xstring command Finallythe xstring command adds the text string to plot at the locationxstring
specified by variables b and c In the example the coordinates wewant to add the text are obtained by executing the xclick functionxclick
and clicking on the point required in the plot window
5 Additional details can be found in books covering Mathematics in LATEX
B6 Visualising data in Scilab 247
Table B10 Command sequences for the common mathematicalfunctions in Scilabarccos arcsin arctan cos cot
csc exp lim ln log
max min sec sin tan
B63 Plotting numeric data stored in an ASCII file
As data can be stored in a wide variety of ways in external filesScilab has a wide variety of functions which can be used to importthe data stored in these files (see help for the commands read
fscanf and xlsread) Since data can be stored in such a wideread fscanf and xlsread
range of formats we will confine our discussions here to the situa-tion where numeric data is stored in a ASCII (text) file where eachrow of the file contains the same number of entries
To illustrate the procedure for plotting data stored in ASCII file wewill consider the data obtained measuring the temperature of a cupof coffee every minute for a total of 100 mins The data recorded isstored in the a file called coffeedatatxt6 which simply containsthe two columns of numbers shown in Table B11 As an examplethe first 10 lines of the text file contains the following numbers7
00000 1000000
10000 950000
20000 903125
30000 859180
40000 817981
50000 779357
60000 743147
70000 709201
80000 677376
90000 647540
Note the current version of Scilab has a builtin text editor whichcan be used to view such text files To access this editor simplytype the command edit at the command prompt Once open youedit
can use the graphical interface to open the required file
The first issue is to get the data into Scilab so it is available forplot command In Scilab the command read can used to importread
the data and assign it to a variable In the current example wewould use the following command to import the data and assignit to a variable called cdata assuming the data is in the currentdirectory8
6 The file coffeedatatxt has been attached and should be available for savingprovided you are using Adobe Reader To save the file right click on thepurple Windows version or the red MacOSLinux version in the margin andselect save file from the options Ensure you save the file as coffeedatatxt
7 normally you will have received the data file in this format8 You can check what directory Scilab is in by typing the command pwd
Figure B6 Temperature of coffee based on the data recorded inthe file coffeedatatxt
cdata=read(rsquocoffeedatatxtrsquo-12)
The -12 following the name of the file tells Scilab that you donrsquotknow how many rows you have but you have two columns of dataSince we know that we have 101 rows of data we could have alsoused the following command to load the data
cdata=read(rsquocoffeedatatxtrsquo1012)
We now have a variable called cdata which has two columns ofdata The first column contains the time while the second containsthe Temperature records Hence we can plot it using the followingcommand
plot(cdata(1)cdata(2))
This gives the plot shown in Figure B6 after appropriate labelshave been added as described above
B7 Script files
Executing commands entered at the keyboard or keypad is themost primitive use of a calculating device It is only when se-quences of operations (to perform complex tasks) are stored awayand re-called for repeated application with different inputs that wehave a programmed operation Complex programs must necessarilyhave an interface with the user The basic interface tool in Scilabcapable of executing sequences of commands that are stored in a fileis the script The file names must have an extension of sce forexample program1sce assignment1sce These files are calledScript Sci-files or simply scripts
A script file is a file of standard keyboard charactersthat define commands to drive a program A script filecan be generated using any text editor or word proces-sor
To invoke a script in Scilab type the commandexec
exec filenamesce
where filename denotes the actual name of the file containing thescript When the script runs it is exactly the same as if you typedthe commands into the command window from the keyboard Thismeans that the Sci-file can use any variables you have defined priorto invoking the script Any variables created in the script are alsoavailable when the script is finished and any variable whose namewas assigned a value in the script will be over-written
If you give a script the same name as any variable that it computesin the running of the script then Scilab will give precedence tothe workspace variable over a script As stated in sectB3 the variablename will hide the script name after the script has run Scilab willtake the name as referring to the variable and not to the definedscript
Scilab also needs to know where to find the script to executedIf Scilab cannot find the script it will report that you have anundefined function or variable
The exact procedure for creating and saving Sci-files depends onthe Scilab version the operating system (MS Windows Macin-tosh or UnixLinux ) and the text editor you use The currentversion of Scilab includes a built-in editor which can be used tocreate and edit Sci-files and text files
To create a new Sci-file under the current version of Scilab simplytype edit newfilenamesce This will open up the Scilab editoredit
with a blank document called newfilename
B71 Script input and output
A script is a named file (with the extension sce) containing asequence of Scilab instructions which when its name is executedin a session performs those instructions in order as though theywere entered at the console
There are two common uses for scripts Pragmatically we maywant to save a particular computational sequence for later use orreference If the tasks involved are to be repeated often thenthe nature of the tasks should be analysed formulated into usefulfunctions (which we will discuss later) and the whole procedurepresented in a single function But if the tasks are only likely toget occasional use then the script may be adequately commentedand stored as a script file It is a quick and dirty way of saving
B7 Script files 251
the instructions for doing a particular job when we do not wantto go to the trouble of writing the functions for the job It is nota highly recommended modus operandi but useful all the same
The second use is for the script to act like a storyboard presentinginformation to a user in a structured and controlled way permit-ting the user to see the steps of a computational procedure andtheir outputs up on the screen At a more sophisticated level thescript takes on the proportions of an application controlling a largeand complex interaction with advanced interface features We willnot take our study of Scilab programming to this level but workmostly at the storyboard level using the simplest of Scilabrsquos in-terface tools
All the variables created in a script become global variables definedin the workspace and available for perusal As an application itis assumed that a script controls the entire workspace and thevariables it creates are lsquothe datarsquo of the application When scriptsare used as storyboards on a small scale then it cannot be assumedthat the variables they create deserve global recognition and it iscourteous to erase these variables when the script is finished unlessthey have some possible further use An unfortunate side-effect isthat if variables with the same name already exist in the workspacebefore a script is run then these variables will be over-written andlost
Used as storyboards scripts allow the user to cut paste and re-trythe various steps of a procedure and to modify the script to suithis own needs Storyboards may be used for demonstrations oftoolbox utilities or extended documentationexplanation of func-tions However if there is any danger that variables defined bythe script may over-write workspace variables then the demo ordocumentation should be re-fashioned into a function that leavesthe workspace variables untouched (more on this later)
Scripts will often read data from pre-prepared files and we will dis-cuss the functions provided by Scilab to perform this task below
Gathering data entered by the user really falls into the realm ofinterface design but we only briefly consider the simplest interfacefunction provided by Scilab Similarly we briefly consider thefunctions for providing the simplest forms of output presentation
Keyboard input
Scilab has a built-in function input that can be used to promptinput
the user to enter the value of a variable from the keyboard Forexample
Tc = input(rsquoPlease enter the temperature in degrees Celsiusrsquo)
will print the message
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
B6 Visualising data in Scilab 247
Table B10 Command sequences for the common mathematicalfunctions in Scilabarccos arcsin arctan cos cot
csc exp lim ln log
max min sec sin tan
B63 Plotting numeric data stored in an ASCII file
As data can be stored in a wide variety of ways in external filesScilab has a wide variety of functions which can be used to importthe data stored in these files (see help for the commands read
fscanf and xlsread) Since data can be stored in such a wideread fscanf and xlsread
range of formats we will confine our discussions here to the situa-tion where numeric data is stored in a ASCII (text) file where eachrow of the file contains the same number of entries
To illustrate the procedure for plotting data stored in ASCII file wewill consider the data obtained measuring the temperature of a cupof coffee every minute for a total of 100 mins The data recorded isstored in the a file called coffeedatatxt6 which simply containsthe two columns of numbers shown in Table B11 As an examplethe first 10 lines of the text file contains the following numbers7
00000 1000000
10000 950000
20000 903125
30000 859180
40000 817981
50000 779357
60000 743147
70000 709201
80000 677376
90000 647540
Note the current version of Scilab has a builtin text editor whichcan be used to view such text files To access this editor simplytype the command edit at the command prompt Once open youedit
can use the graphical interface to open the required file
The first issue is to get the data into Scilab so it is available forplot command In Scilab the command read can used to importread
the data and assign it to a variable In the current example wewould use the following command to import the data and assignit to a variable called cdata assuming the data is in the currentdirectory8
6 The file coffeedatatxt has been attached and should be available for savingprovided you are using Adobe Reader To save the file right click on thepurple Windows version or the red MacOSLinux version in the margin andselect save file from the options Ensure you save the file as coffeedatatxt
7 normally you will have received the data file in this format8 You can check what directory Scilab is in by typing the command pwd
Figure B6 Temperature of coffee based on the data recorded inthe file coffeedatatxt
cdata=read(rsquocoffeedatatxtrsquo-12)
The -12 following the name of the file tells Scilab that you donrsquotknow how many rows you have but you have two columns of dataSince we know that we have 101 rows of data we could have alsoused the following command to load the data
cdata=read(rsquocoffeedatatxtrsquo1012)
We now have a variable called cdata which has two columns ofdata The first column contains the time while the second containsthe Temperature records Hence we can plot it using the followingcommand
plot(cdata(1)cdata(2))
This gives the plot shown in Figure B6 after appropriate labelshave been added as described above
B7 Script files
Executing commands entered at the keyboard or keypad is themost primitive use of a calculating device It is only when se-quences of operations (to perform complex tasks) are stored awayand re-called for repeated application with different inputs that wehave a programmed operation Complex programs must necessarilyhave an interface with the user The basic interface tool in Scilabcapable of executing sequences of commands that are stored in a fileis the script The file names must have an extension of sce forexample program1sce assignment1sce These files are calledScript Sci-files or simply scripts
A script file is a file of standard keyboard charactersthat define commands to drive a program A script filecan be generated using any text editor or word proces-sor
To invoke a script in Scilab type the commandexec
exec filenamesce
where filename denotes the actual name of the file containing thescript When the script runs it is exactly the same as if you typedthe commands into the command window from the keyboard Thismeans that the Sci-file can use any variables you have defined priorto invoking the script Any variables created in the script are alsoavailable when the script is finished and any variable whose namewas assigned a value in the script will be over-written
If you give a script the same name as any variable that it computesin the running of the script then Scilab will give precedence tothe workspace variable over a script As stated in sectB3 the variablename will hide the script name after the script has run Scilab willtake the name as referring to the variable and not to the definedscript
Scilab also needs to know where to find the script to executedIf Scilab cannot find the script it will report that you have anundefined function or variable
The exact procedure for creating and saving Sci-files depends onthe Scilab version the operating system (MS Windows Macin-tosh or UnixLinux ) and the text editor you use The currentversion of Scilab includes a built-in editor which can be used tocreate and edit Sci-files and text files
To create a new Sci-file under the current version of Scilab simplytype edit newfilenamesce This will open up the Scilab editoredit
with a blank document called newfilename
B71 Script input and output
A script is a named file (with the extension sce) containing asequence of Scilab instructions which when its name is executedin a session performs those instructions in order as though theywere entered at the console
There are two common uses for scripts Pragmatically we maywant to save a particular computational sequence for later use orreference If the tasks involved are to be repeated often thenthe nature of the tasks should be analysed formulated into usefulfunctions (which we will discuss later) and the whole procedurepresented in a single function But if the tasks are only likely toget occasional use then the script may be adequately commentedand stored as a script file It is a quick and dirty way of saving
B7 Script files 251
the instructions for doing a particular job when we do not wantto go to the trouble of writing the functions for the job It is nota highly recommended modus operandi but useful all the same
The second use is for the script to act like a storyboard presentinginformation to a user in a structured and controlled way permit-ting the user to see the steps of a computational procedure andtheir outputs up on the screen At a more sophisticated level thescript takes on the proportions of an application controlling a largeand complex interaction with advanced interface features We willnot take our study of Scilab programming to this level but workmostly at the storyboard level using the simplest of Scilabrsquos in-terface tools
All the variables created in a script become global variables definedin the workspace and available for perusal As an application itis assumed that a script controls the entire workspace and thevariables it creates are lsquothe datarsquo of the application When scriptsare used as storyboards on a small scale then it cannot be assumedthat the variables they create deserve global recognition and it iscourteous to erase these variables when the script is finished unlessthey have some possible further use An unfortunate side-effect isthat if variables with the same name already exist in the workspacebefore a script is run then these variables will be over-written andlost
Used as storyboards scripts allow the user to cut paste and re-trythe various steps of a procedure and to modify the script to suithis own needs Storyboards may be used for demonstrations oftoolbox utilities or extended documentationexplanation of func-tions However if there is any danger that variables defined bythe script may over-write workspace variables then the demo ordocumentation should be re-fashioned into a function that leavesthe workspace variables untouched (more on this later)
Scripts will often read data from pre-prepared files and we will dis-cuss the functions provided by Scilab to perform this task below
Gathering data entered by the user really falls into the realm ofinterface design but we only briefly consider the simplest interfacefunction provided by Scilab Similarly we briefly consider thefunctions for providing the simplest forms of output presentation
Keyboard input
Scilab has a built-in function input that can be used to promptinput
the user to enter the value of a variable from the keyboard Forexample
Tc = input(rsquoPlease enter the temperature in degrees Celsiusrsquo)
will print the message
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
248 Appendix B Programming with Scilab
20
30
40
50
60
70
80
90
100
0 10 20 30 40 50 60 70 80 90 100
t ime (mins)
Figure B6 Temperature of coffee based on the data recorded inthe file coffeedatatxt
cdata=read(rsquocoffeedatatxtrsquo-12)
The -12 following the name of the file tells Scilab that you donrsquotknow how many rows you have but you have two columns of dataSince we know that we have 101 rows of data we could have alsoused the following command to load the data
cdata=read(rsquocoffeedatatxtrsquo1012)
We now have a variable called cdata which has two columns ofdata The first column contains the time while the second containsthe Temperature records Hence we can plot it using the followingcommand
plot(cdata(1)cdata(2))
This gives the plot shown in Figure B6 after appropriate labelshave been added as described above
B7 Script files
Executing commands entered at the keyboard or keypad is themost primitive use of a calculating device It is only when se-quences of operations (to perform complex tasks) are stored awayand re-called for repeated application with different inputs that wehave a programmed operation Complex programs must necessarilyhave an interface with the user The basic interface tool in Scilabcapable of executing sequences of commands that are stored in a fileis the script The file names must have an extension of sce forexample program1sce assignment1sce These files are calledScript Sci-files or simply scripts
A script file is a file of standard keyboard charactersthat define commands to drive a program A script filecan be generated using any text editor or word proces-sor
To invoke a script in Scilab type the commandexec
exec filenamesce
where filename denotes the actual name of the file containing thescript When the script runs it is exactly the same as if you typedthe commands into the command window from the keyboard Thismeans that the Sci-file can use any variables you have defined priorto invoking the script Any variables created in the script are alsoavailable when the script is finished and any variable whose namewas assigned a value in the script will be over-written
If you give a script the same name as any variable that it computesin the running of the script then Scilab will give precedence tothe workspace variable over a script As stated in sectB3 the variablename will hide the script name after the script has run Scilab willtake the name as referring to the variable and not to the definedscript
Scilab also needs to know where to find the script to executedIf Scilab cannot find the script it will report that you have anundefined function or variable
The exact procedure for creating and saving Sci-files depends onthe Scilab version the operating system (MS Windows Macin-tosh or UnixLinux ) and the text editor you use The currentversion of Scilab includes a built-in editor which can be used tocreate and edit Sci-files and text files
To create a new Sci-file under the current version of Scilab simplytype edit newfilenamesce This will open up the Scilab editoredit
with a blank document called newfilename
B71 Script input and output
A script is a named file (with the extension sce) containing asequence of Scilab instructions which when its name is executedin a session performs those instructions in order as though theywere entered at the console
There are two common uses for scripts Pragmatically we maywant to save a particular computational sequence for later use orreference If the tasks involved are to be repeated often thenthe nature of the tasks should be analysed formulated into usefulfunctions (which we will discuss later) and the whole procedurepresented in a single function But if the tasks are only likely toget occasional use then the script may be adequately commentedand stored as a script file It is a quick and dirty way of saving
B7 Script files 251
the instructions for doing a particular job when we do not wantto go to the trouble of writing the functions for the job It is nota highly recommended modus operandi but useful all the same
The second use is for the script to act like a storyboard presentinginformation to a user in a structured and controlled way permit-ting the user to see the steps of a computational procedure andtheir outputs up on the screen At a more sophisticated level thescript takes on the proportions of an application controlling a largeand complex interaction with advanced interface features We willnot take our study of Scilab programming to this level but workmostly at the storyboard level using the simplest of Scilabrsquos in-terface tools
All the variables created in a script become global variables definedin the workspace and available for perusal As an application itis assumed that a script controls the entire workspace and thevariables it creates are lsquothe datarsquo of the application When scriptsare used as storyboards on a small scale then it cannot be assumedthat the variables they create deserve global recognition and it iscourteous to erase these variables when the script is finished unlessthey have some possible further use An unfortunate side-effect isthat if variables with the same name already exist in the workspacebefore a script is run then these variables will be over-written andlost
Used as storyboards scripts allow the user to cut paste and re-trythe various steps of a procedure and to modify the script to suithis own needs Storyboards may be used for demonstrations oftoolbox utilities or extended documentationexplanation of func-tions However if there is any danger that variables defined bythe script may over-write workspace variables then the demo ordocumentation should be re-fashioned into a function that leavesthe workspace variables untouched (more on this later)
Scripts will often read data from pre-prepared files and we will dis-cuss the functions provided by Scilab to perform this task below
Gathering data entered by the user really falls into the realm ofinterface design but we only briefly consider the simplest interfacefunction provided by Scilab Similarly we briefly consider thefunctions for providing the simplest forms of output presentation
Keyboard input
Scilab has a built-in function input that can be used to promptinput
the user to enter the value of a variable from the keyboard Forexample
Tc = input(rsquoPlease enter the temperature in degrees Celsiusrsquo)
will print the message
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
A script file is a file of standard keyboard charactersthat define commands to drive a program A script filecan be generated using any text editor or word proces-sor
To invoke a script in Scilab type the commandexec
exec filenamesce
where filename denotes the actual name of the file containing thescript When the script runs it is exactly the same as if you typedthe commands into the command window from the keyboard Thismeans that the Sci-file can use any variables you have defined priorto invoking the script Any variables created in the script are alsoavailable when the script is finished and any variable whose namewas assigned a value in the script will be over-written
If you give a script the same name as any variable that it computesin the running of the script then Scilab will give precedence tothe workspace variable over a script As stated in sectB3 the variablename will hide the script name after the script has run Scilab willtake the name as referring to the variable and not to the definedscript
Scilab also needs to know where to find the script to executedIf Scilab cannot find the script it will report that you have anundefined function or variable
The exact procedure for creating and saving Sci-files depends onthe Scilab version the operating system (MS Windows Macin-tosh or UnixLinux ) and the text editor you use The currentversion of Scilab includes a built-in editor which can be used tocreate and edit Sci-files and text files
To create a new Sci-file under the current version of Scilab simplytype edit newfilenamesce This will open up the Scilab editoredit
with a blank document called newfilename
B71 Script input and output
A script is a named file (with the extension sce) containing asequence of Scilab instructions which when its name is executedin a session performs those instructions in order as though theywere entered at the console
There are two common uses for scripts Pragmatically we maywant to save a particular computational sequence for later use orreference If the tasks involved are to be repeated often thenthe nature of the tasks should be analysed formulated into usefulfunctions (which we will discuss later) and the whole procedurepresented in a single function But if the tasks are only likely toget occasional use then the script may be adequately commentedand stored as a script file It is a quick and dirty way of saving
B7 Script files 251
the instructions for doing a particular job when we do not wantto go to the trouble of writing the functions for the job It is nota highly recommended modus operandi but useful all the same
The second use is for the script to act like a storyboard presentinginformation to a user in a structured and controlled way permit-ting the user to see the steps of a computational procedure andtheir outputs up on the screen At a more sophisticated level thescript takes on the proportions of an application controlling a largeand complex interaction with advanced interface features We willnot take our study of Scilab programming to this level but workmostly at the storyboard level using the simplest of Scilabrsquos in-terface tools
All the variables created in a script become global variables definedin the workspace and available for perusal As an application itis assumed that a script controls the entire workspace and thevariables it creates are lsquothe datarsquo of the application When scriptsare used as storyboards on a small scale then it cannot be assumedthat the variables they create deserve global recognition and it iscourteous to erase these variables when the script is finished unlessthey have some possible further use An unfortunate side-effect isthat if variables with the same name already exist in the workspacebefore a script is run then these variables will be over-written andlost
Used as storyboards scripts allow the user to cut paste and re-trythe various steps of a procedure and to modify the script to suithis own needs Storyboards may be used for demonstrations oftoolbox utilities or extended documentationexplanation of func-tions However if there is any danger that variables defined bythe script may over-write workspace variables then the demo ordocumentation should be re-fashioned into a function that leavesthe workspace variables untouched (more on this later)
Scripts will often read data from pre-prepared files and we will dis-cuss the functions provided by Scilab to perform this task below
Gathering data entered by the user really falls into the realm ofinterface design but we only briefly consider the simplest interfacefunction provided by Scilab Similarly we briefly consider thefunctions for providing the simplest forms of output presentation
Keyboard input
Scilab has a built-in function input that can be used to promptinput
the user to enter the value of a variable from the keyboard Forexample
Tc = input(rsquoPlease enter the temperature in degrees Celsiusrsquo)
will print the message
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
250 Appendix B Programming with Scilab
A script file is a file of standard keyboard charactersthat define commands to drive a program A script filecan be generated using any text editor or word proces-sor
To invoke a script in Scilab type the commandexec
exec filenamesce
where filename denotes the actual name of the file containing thescript When the script runs it is exactly the same as if you typedthe commands into the command window from the keyboard Thismeans that the Sci-file can use any variables you have defined priorto invoking the script Any variables created in the script are alsoavailable when the script is finished and any variable whose namewas assigned a value in the script will be over-written
If you give a script the same name as any variable that it computesin the running of the script then Scilab will give precedence tothe workspace variable over a script As stated in sectB3 the variablename will hide the script name after the script has run Scilab willtake the name as referring to the variable and not to the definedscript
Scilab also needs to know where to find the script to executedIf Scilab cannot find the script it will report that you have anundefined function or variable
The exact procedure for creating and saving Sci-files depends onthe Scilab version the operating system (MS Windows Macin-tosh or UnixLinux ) and the text editor you use The currentversion of Scilab includes a built-in editor which can be used tocreate and edit Sci-files and text files
To create a new Sci-file under the current version of Scilab simplytype edit newfilenamesce This will open up the Scilab editoredit
with a blank document called newfilename
B71 Script input and output
A script is a named file (with the extension sce) containing asequence of Scilab instructions which when its name is executedin a session performs those instructions in order as though theywere entered at the console
There are two common uses for scripts Pragmatically we maywant to save a particular computational sequence for later use orreference If the tasks involved are to be repeated often thenthe nature of the tasks should be analysed formulated into usefulfunctions (which we will discuss later) and the whole procedurepresented in a single function But if the tasks are only likely toget occasional use then the script may be adequately commentedand stored as a script file It is a quick and dirty way of saving
B7 Script files 251
the instructions for doing a particular job when we do not wantto go to the trouble of writing the functions for the job It is nota highly recommended modus operandi but useful all the same
The second use is for the script to act like a storyboard presentinginformation to a user in a structured and controlled way permit-ting the user to see the steps of a computational procedure andtheir outputs up on the screen At a more sophisticated level thescript takes on the proportions of an application controlling a largeand complex interaction with advanced interface features We willnot take our study of Scilab programming to this level but workmostly at the storyboard level using the simplest of Scilabrsquos in-terface tools
All the variables created in a script become global variables definedin the workspace and available for perusal As an application itis assumed that a script controls the entire workspace and thevariables it creates are lsquothe datarsquo of the application When scriptsare used as storyboards on a small scale then it cannot be assumedthat the variables they create deserve global recognition and it iscourteous to erase these variables when the script is finished unlessthey have some possible further use An unfortunate side-effect isthat if variables with the same name already exist in the workspacebefore a script is run then these variables will be over-written andlost
Used as storyboards scripts allow the user to cut paste and re-trythe various steps of a procedure and to modify the script to suithis own needs Storyboards may be used for demonstrations oftoolbox utilities or extended documentationexplanation of func-tions However if there is any danger that variables defined bythe script may over-write workspace variables then the demo ordocumentation should be re-fashioned into a function that leavesthe workspace variables untouched (more on this later)
Scripts will often read data from pre-prepared files and we will dis-cuss the functions provided by Scilab to perform this task below
Gathering data entered by the user really falls into the realm ofinterface design but we only briefly consider the simplest interfacefunction provided by Scilab Similarly we briefly consider thefunctions for providing the simplest forms of output presentation
Keyboard input
Scilab has a built-in function input that can be used to promptinput
the user to enter the value of a variable from the keyboard Forexample
Tc = input(rsquoPlease enter the temperature in degrees Celsiusrsquo)
will print the message
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
B7 Script files 251
the instructions for doing a particular job when we do not wantto go to the trouble of writing the functions for the job It is nota highly recommended modus operandi but useful all the same
The second use is for the script to act like a storyboard presentinginformation to a user in a structured and controlled way permit-ting the user to see the steps of a computational procedure andtheir outputs up on the screen At a more sophisticated level thescript takes on the proportions of an application controlling a largeand complex interaction with advanced interface features We willnot take our study of Scilab programming to this level but workmostly at the storyboard level using the simplest of Scilabrsquos in-terface tools
All the variables created in a script become global variables definedin the workspace and available for perusal As an application itis assumed that a script controls the entire workspace and thevariables it creates are lsquothe datarsquo of the application When scriptsare used as storyboards on a small scale then it cannot be assumedthat the variables they create deserve global recognition and it iscourteous to erase these variables when the script is finished unlessthey have some possible further use An unfortunate side-effect isthat if variables with the same name already exist in the workspacebefore a script is run then these variables will be over-written andlost
Used as storyboards scripts allow the user to cut paste and re-trythe various steps of a procedure and to modify the script to suithis own needs Storyboards may be used for demonstrations oftoolbox utilities or extended documentationexplanation of func-tions However if there is any danger that variables defined bythe script may over-write workspace variables then the demo ordocumentation should be re-fashioned into a function that leavesthe workspace variables untouched (more on this later)
Scripts will often read data from pre-prepared files and we will dis-cuss the functions provided by Scilab to perform this task below
Gathering data entered by the user really falls into the realm ofinterface design but we only briefly consider the simplest interfacefunction provided by Scilab Similarly we briefly consider thefunctions for providing the simplest forms of output presentation
Keyboard input
Scilab has a built-in function input that can be used to promptinput
the user to enter the value of a variable from the keyboard Forexample
Tc = input(rsquoPlease enter the temperature in degrees Celsiusrsquo)
will print the message
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
252 Appendix B Programming with Scilab
Please enter the temperature in degrees Celsius
and Scilab will wait for the input The input is terminated bypressing the Enter (or Return) key
The input entered can be any valid Scilab expression which willbe evaluated and the answer assigned to the variable (in this caseTc) in the input statement For example
Please enter the temperature in degrees Celsius [025100]
where the [025100] has been entered by the user in response tothe input prompt The variable Tc will be assigned the row vector[0 25 50 75 100]
Example B8
Script quad_0s
Variables createdoverwritten
A B C D x1 x2 x y a b w thezeros
Purpose To illustrate the use of the
quadratic formula for solving
a quadratic equation of the form
Ax^2 + Bx + C = 0
with the user to enter ABC
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
clc
disp([
rsquo rsquo
rsquo Calculating the solutions rsquo
rsquo of the equation Ax^2 + Bx + C = 0rsquo
rsquo rsquo
])
input(rsquoPress Enter to startrsquo)
A = input(rsquoEnter value of A in Ax^2+Bx+C rsquo)
B = input(rsquoEnter value of B in Ax^2+Bx+C rsquo)
C = input(rsquoEnter value of C in Ax^2+Bx+C rsquo)
disp(rsquoFirst we compute the discriminantrsquo)
D = B^2 - 4AC
input(rsquoPress Enter to continuersquo)
The quadratic formula gives
x1 = (-B + sqrt(D))(2A)
x2 = (-B - sqrt(D))(2A)
thezeros=sort([x1x2])
input(rsquoPress Enter to continuersquo)
Check this from a graph of the function
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
B7 Script files 253
if Dlt0
a=-B(2A) b=-B(2A) w=4abs(a)
x1=a x2=a
else
a=min([x1x2]) b=max([x1x2]) w=b-a
end
x=linspace((a-w4)(b+w4)100)
y=Ax^2+Bx+C
plot(xyx10rsquorsquox20rsquorsquo)
clear x y a b w
There are several things to note about this script
(a) Echoing the instructions on the screen is turned off whenthe input function is to be used and comments arethen put onto the screen with the disp function Thismethod of displaying comments could be kept through-out the script
(b) The input function may be used to punctuate the pre-sentation so that it does not all just scroll past theviewer without his having the opportunity to read howthe computation progresses
(c) The graphing of the function involves a few considera-tions that we do not wish to trouble you with since wewant you to concentrate of the quadratic formula Thevariables used in this part of the script are erased withclearmdashbut note that if any variable in the workspaceis called a b x y or w then the result will be that it iserased
(d) If you write a similar script for solving say two equationsin two unknowns then you will see that most of the timewill be spent on perfecting the layout of the presentationrather than on the solution process
Formatted output
The semicolon at the end of a line in Scilab suppresses the displayof output from a computation but allowing the output to be dis-played in Scilabrsquos default mode is sometimes not very convenient
Scilab provides the function printf (a direct implementation ofprintf
the C function of nearly the same name) for formatted output Theform the command takes is
printf(rsquoformatrsquoAB)
The string rsquoformatrsquo describes how to display the variables ABformat
The format string contains text to be displayed plus special com-mands that describe how to display each number The specialcommands takes the general form
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
254 Appendix B Programming with Scilab
[-][number1number2]C
where anything in square brackets is optional
The sign tells Scilab to interpret the following characters ascodes for formatting a variable The number1 specifies the mini-mum field width9 If the number to be printed will not fit withinthis width then the field will be expanded The number2 specifiesthe number of digits to print to the right of the decimal pointThat is the precision of the output If the precision of the variableis larger than the precision of the output field then the last digit isrounded before printing Normally the numbers printed are rightjustified the minus sign indicates that they should be left justified
The final character in the format specifier rsquoCrsquo is the format codeThis is not optional it must appear to specify what form the num-ber is to be printed Table B12 lists the most useful format codes
Example B9 Some examples of using the printf command
printf(rsquoThe speed is (kph) 31frsquoSpeed)
if Speed = 6234 this produces the output
The speed is (kph) 623
If A = 3141 and B = 6282 then the command
printf(rsquoPi=53f TwoPi=53frsquoAB)
produces the output
Pi=3141 TwoPi=6282
If M=[12 15 06 03] then the command
printf(rsquo31f rsquoM)
produces the output
12 06 15 03
The printf command prints by column when a matrix is tobe printed
Example B10 An example of the rounding that can occur whenusing printf The output from the command
printf(rsquoPi=75f Pi=64frsquopipi)
is
Pi=314159 Pi=31416
The format string can also contain ordinary text which will be out-put unchanged Escape sequences can be embedded in the format
9 this is the minimum number of characters the number will occupy includingthe decimal point and plusmn signs For example the field width of minus2345 is 6
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
B7 Script files 255
Table B12 The printf format string number typesFormat code Description
f Decimal formate Exponential format with lowercase eE Exponential format with uppercase Eg e or f whichever is shorterG E or f whichever is shorter
Table B13 The printf format string escape sequencesEscape Sequence Description
n start a new line in the outputb Backspacet Tab Print a backslash Print a percentrsquorsquo Print an apostrophe
string that have special meaning Table B13 lists the most usefulescape sequences
File input and output
Variables can be defined from information that has been storedin a data file Scilab recognizes two different type of data filesmat-files and ascii-files A mat-file contains data stored in aspace-efficient binary format Only a Scilab program can readmat-files If data is to be shared with other programs then usethe ascii file type An ascii (American Standard Code for In-formation Interchange) file contains only the standard keyboardcharacters and is readable and editable on any platform using anytext editor
To save variables to a mat-file use the command save For exam-ple
save(rsquodata1matrsquoxy)
will save the matrices x and y in the file data1mat To restorethese matrices in a Scilab program use the command10
load(rsquodata1matrsquo)
This command will define the variables x and y and load them withthe values stored in the mat-file The advantage of saving data ina mat-file is that when it is loaded the variables are restored Thedisadvantage is that the file is a binary file and is only readable byScilab
10 Remember this file must be in the Scilab search path
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
256 Appendix B Programming with Scilab
An ascii data file that is going to be loaded into a Scilab programmust only contain numeric information and each row of the filemust contain the same number of data values The file can begenerated using a spread sheet program a text editor or by anyother program It can also be generated by a Scilab programusing the fprintfMat command To save the variable y to anascii file called data1dat the command is
fprintfMat(rsquodata1datrsquoy)
Each row of the matrix y will be written to a separate line in thedata file The mat extension is not added to an ascii file How-ever it is good practice as illustrated here to add the extensiondat to all ascii data files so that they can be distinguished fromother Scilab file types and are easily recognisable as data files
Exercise B11 Use the Scilab help command to discover moreoptions for fprintfMat
The fscanfMat command reads an ascii file of numbers For ex-ample
x=fscanfMat(rsquodata1datrsquo)
assigns to variable x the matrix of values in the text files data1datrsquo
Example B12 If the file data3dat contains the following data
000 000
001 001255
002 02507
003 037815
The commanddata3=fscanfMat(rsquodata3datrsquo)
will create a variable data3 with 2 columns and 4 rows thatwill contain the data from the file Then typing data3 at theprompt will give
gt data3=fscanfMat(rsquodata3datrsquo)
data3 =
0 0
00100 00126
00200 02507
00300 03781
Then the columns of data in the file can be stored in the vari-ables x and y by simply doing the following x=data3(1)and y=data3(2)
Exercise B13 Create the ascii data file xyzdat containing thefollowing data
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
B7 Script files 257
Table B14 Some of Scilabrsquos data inputoutput functionsFunction Description
input Request input from the keyboard Anyvalid Scilab expression is allowed in re-ply
disp Display variables in free formatfprintf Displayprint variables in a specified for-
matsave Save variables to a fileload Load data from a file
fscanfMat Load data from an ascii filefprinfMat Write ascii data to a file
000 001 002 003012 032 045 087134 121 178 147
Use the Scilab editor or a text editor of your choice
Load the data file into Scilab so that the three variables xy and z each contain a row of data from the file
Note that other more advanced commands exist in Scilab andcan be used to readwrite ascii files (see Table B14) Howeverthese are only used when a specific need arises and are thereforenot covered in detail here (see Scilab Help for more details)
B72 Formatting script files
An important aspect of writing script files is to ensure that thescript is readable and easily understood This ensures that anyonereading the script file (this includes the author of the file) canunderstand the purpose of the script
To help make the script understandable it should be commentedthroughout describing all aspects of the script In Scilab all texton a line following two characters is treated as a comment Thefirst comment line before any executable statement in a script is theline that is searched by the apropos command This line shouldapropos
contain the name of the function and keywords that help locatethe script The help command displays all the comment lines upto the first blank line or executable line that occur at the start ofa script A description of the script should be placed here 11
Lines in a script file should not be too long The lines should be
11 Note help files will only be available online in Scilab once the required xmlhelp files have been generated If you use the above structure these can begenerated from your Sci-file by running the command help from sci Foradditional details see Scilab help for this command
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
258 Appendix B Programming with Scilab
visible in the editorrsquos window(printout) without wrapping or hori-zontal scrolling This applies to both comment lines and statementlines
Statements can be broken across multiple lines using the continua-tion symbol rsquorsquo three dots in a row If three dots in a row appearContinuation symbol()
at the end of a line it means that the statement is continued onthe next line
Example B14 is an example of a script file that explains how tomake various temperature conversions
Example B14
Script temperatures
Variables createdoverwritten
Tc Tf Tk Tr T_all
Purpose To illustrate the conversion of
Celsius temperatures (Tc) to
Fahrenheit (Tf) Kelvin (Tk)
and Rankine (Tr) and to produce
a table of equivalents to Tc values
running from 0 to 100 in steps of 10
Source Leigh Brookshaw Mar 15 2000
Last Modified Tony Roberts 2007
-----------------------------------------------
disp(rsquoWe create the list of Celsius valuesrsquo)
Tc = [010100] input(rsquopausersquo)
disp(rsquoCalculate the Fahrenheit equivalentsrsquo)
Tf = 9Tc5 + 32 input(rsquopausersquo)
disp(rsquoCalculate the Kelvin equivalentsrsquo)
Tk = 27316 + Tc input(rsquopausersquo)
disp(rsquoCalculate the Rankine equivalentsrsquo)
Tr = 9Tk5 input(rsquopausersquo)
disp(rsquoCreate a matrix with equivalent temperaturesrsquo)
disp(rsquoin corresponding rowsrsquo)
T_all = [Tc Tf Tk Tr] input(rsquopausersquo)
Turn echoing off as the rest is instructional
printf(rsquoTemperature Conversion Tablenrsquo)
printf(rsquo Tc Tf Tk Trnrsquo)
printf(rsquo62f 62f 62f 62fnrsquoT_allrsquo)
The print instruction works because printf
prints a row at a time
n starts a new line in the output
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
B7 Script files 259
As a courtesy clear out the
introducedover-written variables
clear Tc Tf Tk Tr T_all
Exercise B15 Code the Example B14 and name the script filetemperaturessce Test that your script file is giving thecorrect answers Test the answers by calculating a few ofthe values in the output table using the Scilab commandwindow
Run your script with disp as shown then comment out thesedisplays when you are satisfied the script works
Exercise B16 The growth of bacteria in a colony can be mod-eled with the following exponential equation
ynew = yolde1386 t
where yold is the initial number of bacteria in the colonyynew is the new number of bacteria in the colony and t is theelapsed time in hours
Write a script that uses this equation to predict the numberof bacteria in the colony after 6 hours if the initial populationis 1 Print a table that shows the number of bacteria everyhalf hour up to 6 hours Also plot the answer
Exercise B17 Modify the script of Exercise B16 so that theuser enters the elapsed time in hours Use the input com-mand to ask the user for the input
Exercise B18 Modify the script of Exercise B17 so that theuser enters the initial population of the colony
Exercise B19 Carbon dating is a method for estimating the ageof organic substances such as bones wooden artifacts shellsand seeds The technique compares the amount of carbon 14a radioactive isotope of carbon contained in the object withthe amount it would have had when it was alive While livingthe object would have the same amount of carbon 14 as itssurrounding environment For the last tens of thousands ofyears the ratio of carbon 14 to ordinary carbon has remainedrelatively constant in the atmosphere this allows an estimateto be made of age based on the amount of surviving carbon14 in an object The equation is
If the proportion of carbon 14 remaining is 05 then the ageis 5700 years Which as you would expect is approximatelythe half life of Carbon 14
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
260 Appendix B Programming with Scilab
Write a script that allows the user to enter the proportion ofcarbon 14 remaining Print the age in years
What happens if the user enters a proportion le 0 Whathappens if the user enters a proportion ge 1
Exercise B20 Modify the script of Exercise B19 so that theage is rounded to the nearest year
B8 Function files
While a script is intended to be either the controlling unit for anentire application or a storyboard for a minor lsquoapplicationrsquo thecomputational work the various specialised tasks that make up anapplication are done through the use of functions We have seennumerous examples of functions alreadymdashthe built-in or primi-tive functions of Scilab such as input floor and ceil thearithmetic functions + - ^ and elementary functions ofmathematics such as sin cos tan exp log But in most appli-cations there are special tasks that recur and for which it wouldbe useful to have a function Scilab permits the user to define hisown functions and to store them in separate Sci-files called functionfiles
Functions must have a particular syntax Only pre-fix notation isallowed and the first executable line of a function Sci-file must bepreceded by a line of the form
function [uvw] = Fname(xyzt)
where Fname is the unique name of the function which requires oneor more (or no) inputs xyzt and returns one of more (or no)outputs uvw The following lines of the function file list in orderthe steps to be executed on whatever particular inputs are givenin the function call (they need not be called xy ) and theoutputs can be stored under any variable names whatsoever
Unlike a script file all the variables used in a function are local tothat function This means that the values of these variables areonly used within the function and do not over-write variables in theworkspace For example if you assign the variable pi a value of 3in the algorithm for some function (the sequence of steps executedwhen the function is called) then you need not be concerned thatit might over-write the standard value of pi in the workspace Thevariables used to identify the inputs and outputs of a function arealso local to the function
If a function is called without assigning its outputs to variablesthen the first of the outputs only is displayed on the screen
Functions are the building blocks of larger programs They feedtheir results back to other functions and ultimately to the program
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
B8 Function files 261
that interfaces with the user that is to a script file The inter-face tools (or functions) used by the script file may be more orless sophisticated but the hard computational work is done in thecomponent functions
The fact that variables defined within a function are local to thatfunction means that functions can be used in any program withoutfear that the variables defined within the function will clash withvariables defined in the main part of the program This also meansthe programmer need not know the contents of a function onlythe information it needs and the information it will return This isexactly the case with built-in (primitive) functionsmdashthe algorithmused is not open to the user and in fact may be proprietory infor-mation (information that might be thought to give this softwarea commercial edge over competitors) Many of the functions in asophisticated package like Mathematica fall into this category
Some important points
bull The first executable line in a function file must be preceededby a syntax line that lists all the input variables and all theoutput variables This line distinguishes a function Sci-filefrom a script Sci-file The form of the syntax line is
function [output_vars] = function_name(input_vars)
bull The output variables are enclosed in square brackets as shown
bull The input variables are enclosed with parenthesis as shown
bull A function does not have to have either input variables oroutput variables but most functions will both have inputsand deliver outputs
bull The function name should be the same as the Sci-file filenameIf the function name is Temp then the Sci-file name should beTempsce
bull Scilab is case sensitive by default but your operating sys-tem may not be For example while Scilab will recognizethat Temp and temp are two different variables your operat-ing system might not recognize that Tempsce and tempsce
are two different files
bull Functions are executed by calling the function with the cor-rect syntax in exactly the same way as built-in functions areused (that is functions such as sin ceil)
bull values needed by the function should be passed as input vari-ables to the function The use of the input command in func-tions should be avoided unless it is a user interface function
bull the last statement of the function is the last statement in theSci-file When the last statement is executed the function
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
262 Appendix B Programming with Scilab
returns and execution precedes from where the function wascalled
bull the return statement can be used to force an early returnfrom a function
bull It is a good practice to place an endfunction on the lastline of a function This makes it clear to other users exactlywhere the function ends
Example B21 The following function returns the distance trav-eled and the velocity of an object that is dropped The inputvariable is the elapsed time of falling
Function fall
Syntax [dist vel] = fall(tv0)
Input t = list of elapsed times in seconds since
body began a fall under gravity from
rest with no resistance
v0 = initial speed at the start of the fall
Outputs dist = list of distances fallen in metres
from beginning of fall to times t
vel = speed in metressec at time t
Algorithm Assumes vertical fall under constant gravity
with no air or other resistance forces
Source Leigh Brookshaw 2232000
Last Modified Tony Roberts 2007
function [dist vel] = fall(tv0)
The gravitational acceleration is assumed to be
g = 98 metressecondsecond
Under constant acceleration (g) speed t is given by
vel = gt + v0
and distances travelled by
dist = 05gt^2 + v0t
endfunction
Put this text in a text file say fallsce and load intoScilab with the command exec(rsquofallscersquo) after everytime you edit the text file
The following gives examples of using this function
gt [drop velocity] = fall(50)
drop =
1225000
velocity =
49
The output variables drop and velocity are scalars thathave been assigned the return values from the function
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
B8 Function files 263
gt sec = 5
gt [drop velocity] = fall(sec0)
drop =
1225000
velocity =
49
The input can be variables or explicit constants
gt secs=[15]
gt [drop velocity] = fall(secs0)
drop =
49000 196000 441000 784000 1225000
velocity =
98000 196000 294000 392000 490000
Input variable t can be a list The output variables are thenvectors of corresponding values for distance and speed (andconsequently of the same length
gt fall(50)
ans =
1225000
Here the result is not assigned to a variable Note that inthis case the only answer returned is the value of the firstvariable in the function definition namely dist This meansto get all output from a function you must assign all outputvariables
Exercise B22 Use the above function to plot (on the one graph)the distances travelled over a given number of seconds forbodies starting their falls with different speeds by applyingthe command plot(dropsecs) after each execution of thefunction
Example B23 The following code rewrites the script of Exam-ple B8 so that the roots of a quadratic are calculated withina function call
Function quadroot
Syntax [x] = quadroot(ABC)
Input A = coefficient of x^2
B = coefficient of x
C = constant coefficient
in a quadratic polynomial
Outputs x = a list of the solutions
of the equation Ax^2 + Bx + C = 0
Algorithm Uses the quadratic formula
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
264 Appendix B Programming with Scilab
function [x] = quadroot(ABC)
Calculate the square root of the discriminant
disc = sqrt(B^2- 4AC)
Calculate and display the two roots
x = (-B + disc[1-1])(2A)
endfunction
Put this text in a text file say quadrootsce and load intoScilab with the command exec(rsquoquadrootscersquo) after ev-ery time you edit the text file The result of running thefunction might be stored in a variable name and passed toother functions for further use
Exercise B24 Use the function of Example B23 to assist you indeciding on the plotting window for various quadratic poly-nomials
Since functions represent often repeated tasks that may be usefulin maintaining an application they are chosen and written withsome care and are well documented In Scilab the first block ofcomments in a function can be accessed via the help commandcommand once the help files are built (see help help from sci)For more sophisticated documentation we would add keywords orphrases that might be used by a person searching for a function todo a particular task Once (s)he has found the appropriate functionname (s)he can then use help to get more detail on its use
1 The first comment line should contain the functionrsquos nameand other keyword descriptors that might help someone lo-cate this function when they have a particular task to do butdonrsquot know the name of the function that does it
2 the next line should be the syntax line showing how thisfunction is to be called
3 then the input variables should be listed with units if appli-cable
4 followed by the output variables with units if applicable
5 input and output listings should be followed (or preceded)by a description of what the function does if this is notmade clear by the description to the input and output vari-ables this statement should also include a description of thealgorithm that is used if this has any special or unusualfeaturesmdashfor example rsquoiterative solution by Jacobirsquos methodrsquoor rsquoiterative solution by GuasssndashSeidel methodrsquo
6 divide the function code into recognisable blocks such asrsquoData Checkingrsquo rsquoSpecial Casesrsquo etc by the use of commentsif appropriate and also add comments to highlight where anyspecial features are implemented Do not add inane com-ments like
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
B9 Control structures 265
v= v+2 adds 2 to the value of v
but do not assume that even you will be able to follow yourown brilliant logic when you have long forgotten about howyou wrote the function let alone someone else trying to readthe code for the first time
B81 Local variables
The name of the input variables given in the function definitionline are local to that function This means that when the func-tion is called the input variables used in the call statement canhave different names All variables used in the function are clearedfrom memory after the function finishes executing The functionin Example B21 can be called using the statement
[drop velocity] = fall(sec)
The variable sec is the elapsed time in seconds but inside thefunction this is called t If inside the function the variable t
was changed it would not affect the variable sec in the callingprogram This is what is meant by the functionrsquos variables beinglocal to the function This feature means that functions can bewritten without concern that the calling program uses the samevariable names and therefore function files are portable and neednot be rewritten every time they are used in a different program
B9 Control structures
In the execution of the function control passes from one line tothe next However there are a number of situations in which itis natural to want control to pass to different lines depending onthe value of computed variables For example input data maymaybe be treated differently depending on when it is received (latepayment assignments etc)
Decisions to execute different lines of code in different circum-stances are called branching decisions and most computer lan-guages will provide a branching control structure for implementingsuch decisions Scilab provides the if then and the selectbranching control structures The tests that are made to determinethe blocks of code that should be executed often involve more thanone consideration and Scilab provides both relational and logicaloperators to simplify the expression of these conditions
Statements that alter the linear flow of the script are called controlstructures
A control structure is also required when blocks of code are to berepeated until a certain condion is met for example the computa-tion of interest on a loan balance until such time as the loan balanceis paid off or repeated approximation to the solution of a problem
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
lt less thanlt= less than or equalgt greater thangt= greater than or equal== equal~= not equal
until sufficient accuracy is achieved The repeated execution of ablock of code may be achieved by an iterative control structure orloop Scilab provides a while loop which continues repeatingthe block until a certain condition is met and a for loop whichrepeats the block a given number of times
The flow of control may also be disrupted by a recursive functiondefinitionmdashthat is a function definition which includes a call to thefunction itself (with dfifferent inputs of course) In this case thecurrent function execution is suspended until a result is returnedfrom the within-function function call Suspension will stack uponsuspension until a within-function function call produces a resultEvery recursively defined function must ensure that after somefinite number of recursions there will be a branch to an executableline rather than to a further recursion Such functions will bevery slow in executing and while they are very useful for solvingproblems theoretically they are never as efficient as an equivalentiterative solution and we will not use them much in this course
B91 Relational operations
To be able to create branching code we need to be able to askquestions For example typical sorts of questions a code may askare
bull Is the temperature greater than 100C
bull Is the height of the building less then 700m
bull Is that a car and is itrsquos speed equal to 60 kmh
Questions that can be asked in a program only allow for two possi-ble answers True or False Scilab represents True with the valueT and False with the value F
Relational operators are operators that compare two matrices ofequal size The operator does an element by element comparisonbetween the two matrices and returns a matrix of the same size withthe elements set to T if the comparison is true and to F otherwise
An expression that contains a relational operator is a logical ex-pression because the result is a matrix containing ones and zeros
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
B9 Control structures 267
Table B16 Scilabrsquos logical operatorsLogical operator Symbol
and amp
or |
not ~
that can be interpreted as true or false values respectively
Table B15 contains a list of Scilabrsquos relational operators
Example B25 Consider the following Scilab code
a = [3 4 6]
b = [3 5 1]
c = a lt b
d = a ~= b
e = a gt b
The result of this code is that the variables c d and e havethe values
c = [F T F]
d = [F T T]
e = [F F T]
B92 Logical operations
The logical expressions can be combined using the logical opera-tors The logical operators defined in Scilab are and or and notand are represented by the symbols in Table B16
Logical operators are used to combine logical statements createdusing the relational operators into more complex logical expres-sions for example
altb amp bltc
This operation is only valid if the two resultant matrices (repre-sented by altb and bltc) are the same size The entry in the matrixrepresented by this logical expression is T (true) if the values inthe corresponding entries in a b and c are such that altbltc oth-erwise the entry is F (false)
When two logical expressions are joined by or entries in the re-sulting matrix are F (false) if both expressions are false otherwiseit is T (true) If two expressions are joined by and entries in theresulting matrix are T (true) if both expressions are true other-wise it is F (false) Logical expression preceded by not reversesthe value of the relational operation For example if agtb is truethen ~(agtb) is false In this example the relational operation is
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
268 Appendix B Programming with Scilab
Table B17 Precedence of operatorsOperator Precedence Order of evaluation
() 1 Left to Right Innermost first~ 2 Right to Left^ 3 Left to Right 4 Left to Right 5 Right to Left+ - 6 Left to Right
lt gt lt= gt= == ~= 7 Left to Rightamp | 8 Left to Right
surrounded by brackets to modify the default precedence See Ta-ble B17 for a list of the order of logical operations
As it does with mathematical expressions Scilab evaluates logicalexpressions are evaluated from left to write Unlike mathematicalexpressions if at any point in the calculation the logical answeris known then the rest of the logical expression is ignored Forexample if x=4 then the statement
(xgt3) |(cos(z)lt0)
must evaluated to true as the first conditional part is true Thismeans that the value of cos(z) need not be evaluated and it isnot Once a logical statement evaluates to a known result extraconditionals are ignored The statement
(xlt3) amp (exp(z)lt10e4)
is false irrespective of the value of exp(z) Scilab will not botherto check whether exp(z)lt10e4 because it is not needed
Scilabrsquos logical operators can also be applied to variables not justused with relational operations exclusively For example if the twomatrices A and B are the same size then the result of A amp B willbe a matrix with elements set to one where both A and B havenonzero elements and zeros otherwise The logical operators orand not behave similarly
Example B26 If A = [6 4 0 1] B = [0 2 6 2] then
AampB = [F T F T]
~A = [F F T F]
A|B = [T T T T]
AltBampA = [F F F T]
Due to operator precedence the last example is equivalent to(AltB)ampA
Exercise B27 Let A = [3 2 7 0] B = [4 0 3 1] C = [8 1 0 5] Without using Scilab solve the following using the prece-dence table and the definitions for the operators
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
B9 Control structures 269
Table B18 Some of Scilabrsquos logical functionsFunction Description
find(x) Returns the indices of the nonzero elements ofthe vector x If x is a matrix it returns the rowand column indices of the nonzero elements
isnan(x) Returns a matrix with ones where the elementsof x are Nans zeros otherwise
isempty(x) Returns a one if x is an empty matrix zero oth-erwise
~A A | C A lt= B A+B gt C
~C+B C-4 gt B A ~= 7
Check your answers using Scilab
Exercise B28 Scilab also has many logical functions That isfunctions that return logical matrices Using Table B18 andthe Scilab help command work out the result of the fol-lowing commands Do not use Scilab to solve the problemsonly to check your answers
Let A=[3 2 7 0] B=[4 0 3 1] C=[8 1 0 5] and D=[]What is the result of
find(B) isempty(D) ~isempty(A) find(AgtB)
B93 Branch controls (conditional statements)
The if statement
The if statement is the most widely used conditional statementThe form of a simple if statement is
if logical expressionstatement group
end
If the logical expression is true the statements between the if
statement and the end statement are executed If the logical ex-pression is false then Scilab immediately jumps to the statementfollowing the end statement
It is important to indent the statements within an if structure inorder to see the control flow easily
Example B29 One important use of an if statement is to en-sure that only valid parameters are passed to functions
if x gt 0
y = log(x)
end
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
270 Appendix B Programming with Scilab
if (x gt= -1) amp (x lt= 1)
y = acos(x)180pi
end
If statements can be nested when required For example
if logical expression 1
statement group Aif logical expression 2
statement group Bend
statement group Cend
statement group D
Example B30 The following function extends the code of Ex-ample B23 to allow the coefficients of a number of quadraticswith real coefficients to be entered at once and to return notonly the zeros but also the vertices of the parabolas they rep-resent Checks are made for error conditions For exampleThe coefficient of the x2 term cannot be zero All coefficientvectors must be the same length
Function quadroots
Syntax [x v M] = quadroots(ABC)
Input A = coefficients of x^2
B = coefficients of x
C = constant coefficients
in the quadratic polynomials
Outputs x = the solutions of Ax^2 + Bx + C = 0
v = the vertex coordinates
both returned as 2 column matrices
with one row for the results
for each quadratic
M = the original coefficients returned
in the form of a matrix with one row
for each quadratic
Source Leigh Brookshaw 1532000
Last Modified Tony Roberts 2007
function [x v] = quadroots(ABC)
If function fails output is empty
x = [] v = []
Data checking
Check that all the coefficients have the same length
if (length(A) ~= length(B)) | (length(B) ~= length(C))
error(rsquoNot all coefficient arrays are of equal lengthrsquo)
end
Check that all the coefficients are real Tests such as
these are madein case the coefficients have been passed
to this function from some other computation
if ~isreal(A)
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
B9 Control structures 271
error(rsquoNot all elements of A are realrsquo)
end
if ~isreal(B)
error(rsquoNot all elements of B are realrsquo)
end
if ~isreal(C)
error(rsquoNot all elements of C are realrsquo)
end
Check that no value of A is zero
if find(A==0)
error(rsquoNo elements of A are allowed to be zerorsquo)
end
Computing the Output
M = [ABC]rsquo the coefficients in a matrix
xs = -B(2A) vertex abscissae
ords = Axs^2 + Bxs + C ordinates of vertices
v = [xsords]rsquo vertex coordinates
disc = BB - 4AC the value of the discriminant
x = ([xsxs]+([1-1]rsquosqrt(disc))[(2A)(2A)])rsquo zeros
endfunction
The if-else statement
The else clause added to an if statement means that two blocksof code can be specified One to be executed if the the logicalexpression is true the other if it is false For example
if logical expression
statement group Aelse
statement group Bend
If the logical expression is true then statement group A is exe-cuted If the logical expression is false then statement group Bis executed The statement groups can also contain if andorif-else statements to provide a nested structure
Example B31 The velocity of a cable car between two towersAssume that the scalar r is the distance from the nearesttower If the cable car is within 30 metres of the tower thevelocity is calculated using the equation velocity = 0425 +000175 times r2 If the cable car is further away from thetower the velocity is calculated using the equation velocity =0625+ 012times rminus 000025times r2
The correct velocity is calculated with the statements
if r lt= 30
velocity = 0425 + 000175r^2
else
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
272 Appendix B Programming with Scilab
velocity = 0625 + 012r - 000025r^2
end
Notice that control structures can be executed in the inter-active environment besides being used within functions
The if-elseif-else statement
When several levels of if-else statements are nested it may bedifficult to determine which logical expressions must be true (orfalse) to execute each set of statements In this case the elseif
clause is often used to clarify the program logic
The else and elseif clauses are optional in an if statement Thegeneral form of the if statement is
if logical expression 1
statement group Aelseif logical expression 2
statement group Belseif logical expression 3
statement group C
else
statement group Dend
If logical expression 1 is true then only statement group A is ex-ecuted If logical expression 1 is false and logical expression 2is true then only statement group B is executed If logical ex-pressions 1 and 2 are false and logical expression 3 is true thenstatement group C is executed If none of the logical expressionsare true then statement group D is executed If more than onelogical expression is true the first logical expression that is truedetermines which statement group is executed
The select statement
The select statement provides an alternative to a if-elseif-elseconstruct Though anything programmed using a select state-ment can be programmed using an if structure often the code ismore readable using the former
The structure of the switch statement is
switch input expression
case value 1 then
statement group Acase value 2 then
statement group B
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
B9 Control structures 273
else
statement group Dend
The input expression of the select statement must evaluate to ascalar or a string The value of the input expression is compared toeach case value If a match is found then the statements followingthe case statement are executed
At most only one case statement group is executed then executionresumes with the statement after the end statement
If no case statement matches the input expression then the state-ments following the else statement are executed The else state-ment is optional
Below is an example of a case statement where the select variabledirection contains a character string (remember that convstr(arsquolrsquo)is a Scilab primitive function that takes a text as input and re-turns it all in lower case)
select constr(directionrsquolrsquo)
case rsquonorthrsquo then
angle = 0
disp(rsquoDirection Northrsquo)
case rsquosouthrsquo then
angle = pi
disp(rsquoDirection Southrsquo)
case rsquoeastrsquo then
angle = 05pi
disp(rsquoDirection Eastrsquo)
case rsquowestrsquo then
angle = 15pi
disp(rsquoDirection Westrsquo)
else
angle = nan
disp(rsquoDirection Unknownrsquo)
end
B94 Iteration controls (explicit loops)
Scilab contains two constructs for explicitly looping the for state-ment and the while statement The for statement is used whenthe number of times the block of code must be looped over is knownThe while statement is used when the number of times the blockof code must be looped over is unknown The looping is terminatedwhen a specified condition is satisfied
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
274 Appendix B Programming with Scilab
The for statement
The general form of the for statement in Scilab is as follows
for index = expression matrix
statement groupend
The statements in the statement group are repeated as many timesas there are columns in the expression matrix (which could also bea scalar or a vector) Each time through the loop the index hasthe value of one of the columns in the expression matrix
Some examples of the for statement
bull calculating the factorial of 20
factorial = 1
for i=220
factorial = factoriali
end
Although this computation is far better done by simply prod(120)
bull Multiply the elements of an array
product = 1
A = [6 8 9 10 2 -5 1 7]
for i=A
product = producti
end
This is also much better done by prod(A)
The rules for a for loop are the following
bull The index of the for loop must be a variable
bull If the expression matrix is the empty matrix the loop willnot be executed Control will pass to the first statementfollowing the end statement
bull If the expression matrix is a scalar then the for loop willbe executed once only The index variable will contain thevalue of the scalar
bull If the expression matrix is a row vector then each timethrough the loop the index will contain the next value inthe the vector
bull If the expression matrix is a matrix then each time throughthe loop the index will contain the next column in the thematrix
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
B9 Control structures 275
bull On exiting the for loop the index contains the last valueused
bull The colon operator can be used to define the expression ma-trix The syntax is
for index=initialincrementlimit
The number of times the loop will be executed can be calcu-lated from the equation
floor
[limitminus initial
increment
]+ 1
If this value is negative the loop is not executed If incrementis omitted it is assumed to be one
bull Lastly and the most important rule of all a for loop mustnever never ever be the innermost loop in a computationThe innermost loop must always always always be a naturalvectormatrix command Two examples are the two exam-ples given above the for loop to compute a product mustbe replaced by the prod function See further discussion inSection B95
Example B32 Suppose the cable car distance from the pylon rof Example B31 is a vector Then Example B31 will haveto modified to calculate a vector of velocities
Initialize the velocity array to same size as rsquorrsquo
velocity = zeros(r)
Loop over the distance array and calculate velocities
for k = 1length(r)
if r(k) lt= 30
velocity(k) = 0425 + 000175r(k)^2
else
velocity(k) = 0625 + 012r(k) - 000025r(k)^2
end
end
However the above code has a for loop as the innermost loopand so violates the last criterion above Rewrite it using nat-ural matrixvector modes of computation After initialisingthe velocity to a correct size matrix simply
k=find(rlt=30)
velocity(k) = 0425+000175r(k)^2
k=find(rgt30)
velocity(k) = 0625+012r(k)-000025r(k)^2
Use the for loop only when the inside of the loop is lesstrivial
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
276 Appendix B Programming with Scilab
The while statement
The while loop is a structure for repeating a set of statements aslong as a specified condition is true The general form of the whileloop is
while expression
statement group
end
If the expression is true or all elements nonzero then the statementgroup is executed The expression is normally a logical expressionAfter the statements are executed the condition is re-tested If thecondition is still true the group of statements are executed againWhen the condition is false control jumps to the first statementfollowing the end statement If the expression is always true (or isa value that is nonzero) the loop becomes an infinite loop
Example B33
To sum the positive elements of the vector x
use the rsquolengthrsquo command to find the
number of elements in x
sum = 0 k = 1
while (k lt= length(x)) amp (x(k) gt= 0)
sum = sum + x(k)
k = k + 1
end
Of course again this violates our last criteria Instead inpractise code simply
sumx=sum(x(find(xgt=0)))
Use while loops when the computation inside the loop is notso simple
Exercise B34 Write a function containing the code of Exam-ple B33 Test the function and also modify it so that thecondition statement of the while loop is reversed that is
while (x(k) gt= 0) amp (k lt= length(x))
Now test the function again Explain why is there now anerror in the script
A danger of while loops is that they may never terminate Thisobviously occurs if the condition is always true A loop countercan be added to the condition to ensure that the loop terminatesFor example in the code below the termination condition we areinterested in is (agt=0) this is assumed to occur within 100 loops if
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
B9 Control structures 277
it does not then an error has occurred The variable count ensuresthe loop will not be infinite
count = 1
while (a lt 0)
count = count + 1
if count gt 100
error(rsquoLoop failed to terminatersquo)
end
end
To break out of an infinite loop and terminate execution of a scriptor function you can press Control C on the keyboard That ishold down the Ctrl key and press C simultaneously
Exercise B35 Timber management needs to determine howmuch of an area to leave uncut so that the harvested areais re-forested in a certain period of time Assume that thereforestation takes place at a known rate per year dependingon climate and soil conditions If the reforestation rate is ηthen the reforestation equation is (Forested area this year) =(Forested area last year) times (1 + η) Assume that there are11000 hectares total with 4500 hectares uncut and the re-forestation rate is 003 hectaresyear Write a function thatreturns a reforestation vector with the followinghelp lines inits definition
Function reforest
Syntax hectares = reforest(years rfrate inhec)
Input
years - the number of years to calculate
the reforestation
rfrate - the reforestation rate in hectaresyear
inhec - the initial forest area in hectares
Output
hectares - a vector that lists the forest area
in hectares for each year The length
of the vector is the same as the
number of years input parameter
function hectares = reforest(years rfrate inhec)
Exercise B36 Write a script to print a table showing the num-ber of hectares re-forested at the end of each year after askingthe user to enter the number of years to be used in the table
Exercise B37 Modify the program of Exercise B35 so that theuser can enter the number of hectares and the program willdetermine how many years are required for the number ofhectares to be re-forested
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
278 Appendix B Programming with Scilab
The break statement
Use the break statement to break out of a for loop or a while
loop before it has completed This statement is useful if an errorcondition is detected within the loop This statement is also use-ful for terminating an iteration when the desired result has beencomputed The loop is terminated at the point where the break
command is executed
B95 Implicit loops
If you are familiar with more traditional programming languagessuch as fortran basic or C the discussion on looping above willbe familiar The for and while looping discussed has its coun-terparts in many languages However since Scilab was designedto make matrix manipulations easy one would not expect to haveto use loops to implement operations like matrix multiplication orvector addition but to find them encapsulated in primitive func-tions It turns out that these primitives find surprising uses andthat in many situations where a classical scalar language needsloops to complete a calculation array processing primitives handlethe computation without any apparent looping
See Examples B32 and B33
These codes use the find function to create a vector k that con-tains the indices of all the elements in x that are greater than zeroFor example consider a vector x that is defined as follows
x = 7minus3 2 4minus5minus6
The command k=find(xgt0) returns the following
gt k=find(xgt0)
k =
1 3 4
This means that elements x1 x3 and x4 are the only positive valuesin vector x
In the second line the statement xp=x(k) is a vector that onlycontains the positive elements of x For our example we should geta vector contain the elements x1 x3 and x4 of x Thus xp shouldcontain the values 13 and 4 (see output below to verify this)
gt xp=x(k)
xp =
7 2 4
The Scilab function sum simple sums all of the elements of thisvector (see output below)
gt sumx=sum(xp)
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
B9 Control structures 279
sumx =
13
Scilab was designed to process matrix operations rapidly Themore traditional for and while loops have to be interpreted whichtakes an enormous amount of time Implied loops utilize speciallydesigned operators within Scilab This means the processing isextremely rapid An implied loop in Scilab executes up to ahundred times faster than the equivalent explicit for or while
loop
A disadvantage of the implied loop is that Scilab code can becometerse and hard to understand If you develop your skill in thinkingin vector matrix and array terms then this brevity will soon appearas elegance of expression and the scalar version will seem clumsyand pedestrian You should make every effort to think in theseterms as it constitutes rsquoparallel processingrsquo at the software level
B96 Using functions and scripts together
As a mini-application a script can be used as a driver for a par-ticular function As such it can interrogate a user for the requiredinputs and make the function call with the correct syntax This useis particularly attractive when the syntax for a function is rathertedious to find or remember Because of the danger of over-writingworkspace variables when one uses a script if scripts are going tobe used in this way to drive functions then it is a wise practice toeither give the variables long and unlikely names or as the first in-teraction with the user to check that the names used can be safelyover-written For example in a workspace for calculus computa-tions the names d and S might very well be used for functions thatcompute differences between list elements and sequences of partialsums of series Such names will be very convenient in that settingso that one could write for example d(y)d(x) for derivativeapproximations and S(yd(x)) for approximations to the inte-gral (as we shall do in other modules of this course) Of coursethese simple single character names are in grave danger of beingover-written by careless assignments in the interactive session or ascript
Exercise B38 A small rocket is being designed to make windshear measurements in the atmosphere Before testing be-gins the designers have developed a model of the rocketrsquostrajectory The height in metres above the ground at time tin seconds is
Write a function that takes the time in seconds and returnsthe height It should return a vector of heights when the timeis given as a vector
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
280 Appendix B Programming with Scilab
Write a driver script to run your function
Exercise B39 Using the function from Exercise B38 write ascript to compute the height of the rocket from t = 0 to thetime it hits the ground Use increments of one second Ifthe rocket has not hit the ground after 80 seconds stop theprogram and print out an error message Have the scriptprint out a table of times and heights of a size determine bythe user and on request plot the results
Exercise B40 Using the function from Exercise B38 write aprogram to compute the time at which the rocket reaches thehighest part of its trajectory and the time at which it impactsthe earth Both should be accurate to one second
B10 Numerical programming
B101 Modular Programs
Any problem that can be solved by a computational program canbe solved in one monolithic program contained within one file Infact since a program is just a string of bits we could convert thisbit string into a number to base 10 and present this single numberas an solution to the problem This is not particularly helpful ifyou have the job of maintaining the program (that is modifying itto meet current needs)
There are a number of reasons for avoiding a monolithic program
bull the project as a whole and it could be a large one has to beunderstood in its entirety to write a monolithic program Ifit is not understood in its entirety then many rewrites willbe necessary to incorporate all aspects of the project
bull large cumbersome files are difficult to work with
bull large monolithic programs are difficult to understand for theauthor and anyone else
Sensible programming attempts to identify the key operations re-quired to solve the problem and the tools that will help in puttingthe solution together These tasks are of course solved by runningfunctions The basic rule about functions is that each functionshould perform one task and one task only The project analy-sis should be task oriented each task coded and tested separatelyThe main program pulls together these disparate pieces with ascript that mainly contains calls to functions The advantage ofthis approach is
bull many programmers can work on the problem simultaneously
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
B10 Numerical programming 281
bull a monolithic project is broken down into smaller manageablechunks
bull if each function performs one task then the final code is easierto read and understand
bull it is easier to work with smaller files
Functions themselves should also be written in a block like man-ner That is similar tasks should appear together in the functionInitialization of all variables should appear together at the start ofthe function all checks on input variables should appear togetherFor example
1 Help comments name and keywords syntax inputs and out-puts
2 function [output variables] = fname(input variables)
3 Test that the input variables are valid
4 Initialize all variables that need it
5 Load any data that might be needed
6 Computational algorithm for function
The above is an outline for the design of a function Particularcircumstances will dictate the design of any individual functionbut the steps above should be borne in mind whenever a functionis written
B102 Debugging
Debugging a program is the process of finding and removing thebugs or errors Such errors usually fall into one of the followingcategories
bull Syntax errors such as omitting a parenthesis or comma orspelling a command name incorrectly Scilab usually detectsthe more obvious errors and displays a message describing theerror and its location
bull Runtime errors occur while the program is running and fallinto a number of categories
ndash Errors due to incorrect logic They do not necessarilyoccur every time the program is executed the occur-rence may depend on the type of input data A commonexample is division by zero A logic error is caused bythe author not anticipating all possible eventualities
ndash Errors due to incorrect method This can be a very sub-tle error recognising that the numerical method beingused is incorrect for the problem being solved Normallythis can only be fixed by rewriting the program using a
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
Adding titleslabels and legends to the plot
Plotting numeric data stored in an ASCII file
Script files
Script input and output
Formatting script files
Function files
Local variables
Control structures
Relational operations
Logical operations
Branch controls (conditional statements)
Iteration controls (explicit loops)
Implicit loops
Using functions and scripts together
Numerical programming
Modular Programs
Debugging
282 Appendix B Programming with Scilab
better method for the problem Unfortunately this er-ror might only be recognised after an immense amountof effort has been expended writing the program It isimportant when deciding on which numerical methodto use to solve a problem that the limitations of themethod are well understood
The Scilab error messages usually allow you to find syntax errorsHowever runtime errors are more difficult to locate and are moredifficult to spot It is important to check all answers produced byprograms Never trust the answer from a program Some teststhat should always be employed with computer solutions are
bull always check the order of magnitude of the solution Is itreasonable
bull if the solution has units check that the order of magnitudeof the solution is reasonable for those units
bull plot the answers and see if the plot looks reasonable
If you have tested the solution and think there is an error then tolocate the error try the following
bull Always test your program with a simple version of the prob-lem whose answer can be checked by hand calculations Ifyour program uses functions test the functions individuallyby writing test scripts for each function
bull Display any intermediate calculations by removing semicolonsat the end of statements
bull Use the Sci-file Debugger The Scilab Debugger is a set ofcommands that allow you to stop a script during executionand examine or modify variables
Programming with Scilab
Introduction
Getting started with Scilab
Variables
Arithmetic expressions
Precedence
Matrix arithmetic
Array arithmetic
Builtndashin functions
Arithmetic functions
Utility functions
Visualising data in Scilab
Plotting mathematical functions from within Scilab
