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MATLABBasics

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MATLAB Documentation

http://www.mathworks.com/access/helpdesk/help/techdoc/

http://www.sosmath.com/matrix/matrix.html

Matrix Algebra

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What is MATLAB?MATLAB (Matrix laboratory) is an

interactive software system. It integrates mathematical computing, visualization, and a powerful language to provide a flexible environment for technical computing. Typical uses include

• Math and computation

• Algorithm development

• Data acquisition

• Modeling, simulation, and prototyping

• Data analysis, exploration, and visualization

• Scientific and engineering graphics

• Application development, including graphical user interface building

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The MATLAB Product Family

The MathWorks offers a set of integrated products for data analysis, visualization, application development, simulation, design, and code generation. MATLAB is the foundation for all the MathWorks products.

Demos: http://www.mathworks.com/products/matlab/demos.html

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Using MATLAB in CUHK

• With Windows Version

• With Unix Version

• 200 concurrent licenses using throughout the Departments in CUHK

• Licenses controlled by a License Server

• Used by more than 10 Departments in Engineering and Science Faculties

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Starting MATLAB

• Windows double-click the MATLAB shortcut icon on your

Windows desktop.

• UNIXtype matlab at the operating system prompt.

• After starting MATLAB, the MATLAB desktop opens.

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Quitting MATLAB

• select Exit MATLAB from the File menu in the desktop, or type quit in the Command Window.

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MATLAB Desktop

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Command Window

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Command History

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Current Directory Browser

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»workspace

Command line variables saved in MATLAB workspace

Workspace Browser

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Window Preferences

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Getting help

• MATLAB Documentation

• >> helpdesk or doc– Online Reference (HTML / PDF)

– Solution Search Engine

– Link to The MathWorks (www.mathworks.com)• FTP site & latest documentation

• Submit Questions, Bugs & Requests

• MATLAB access - MATLAB Digest / Download upgrades

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Using Help

• The help command >> help• The help window >> helpwin

• The lookfor command >> lookfor» lookfor exampleDDEX1 Example 1 for DDE23.DDEX1DE Example of delay differential equations for solving with DDE23.DDEX2 Example 2 for DDE23.ODEEXAMPLES Browse ODE/DAE/BVP/PDE examples.....

» help lookforLOOKFOR Search all M-files for keyword. LOOKFOR XYZ looks for the string XYZ in the first comment line (the H1 line) of the HELP text in all M-files found on MATLABPATH. For all files in which a match occurs, LOOKFOR displays the H1 line.....

» lookfor exampleDDEX1 Example 1 for DDE23.DDEX1DE Example of delay differential equations for solving with DDE23.DDEX2 Example 2 for DDE23.ODEEXAMPLES Browse ODE/DAE/BVP/PDE examples.....

» help lookforLOOKFOR Search all M-files for keyword. LOOKFOR XYZ looks for the string XYZ in the first comment line (the H1 line) of the HELP text in all M-files found on MATLABPATH. For all files in which a match occurs, LOOKFOR displays the H1 line.....

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Calculations at the Command Line

» -5/(4.8+5.32)^2ans = -0.0488» (3+4i)*(3-4i)ans = 25» cos(pi/2)ans = 6.1230e-017» exp(acos(0.3))ans = 3.5470

» -5/(4.8+5.32)^2ans = -0.0488» (3+4i)*(3-4i)ans = 25» cos(pi/2)ans = 6.1230e-017» exp(acos(0.3))ans = 3.5470

» a = 2;

» b = 5;

» a^b

ans =

32

» x = 5/2*pi;

» y = sin(x)

y =

1

» z = asin(y)

z =

1.5708

» a = 2;

» b = 5;

» a^b

ans =

32

» x = 5/2*pi;

» y = sin(x)

y =

1

» z = asin(y)

z =

1.5708

Results assigned to “ans” if name not specified

() parentheses for function inputs

Semicolon suppresses screen output

MATLAB as a calculator Assigning Variables

Numbers stored in double-precision floating point format

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>> 2 +3 ( 5 ) >> 2 *3 ( 6 ) >> 1/2 ( 0.5000 )>> 2 ^3 ( 8 )>> 0/1 ( 0 )>> 1/0 ( Warning: Divide by zero. Inf )>> 0/0 ( NaN )

Up/Down arrow to recall previous commandsOr use Ctrl+C and Ctrl+V to reuse commands

Simple Mathematics

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cos(x), sin(x), tan(x), asinh(x), atan(x), atanh(x), …

ceil(x): smallest integer which exceeds x, e.g. ceil(-3.9) returns -3floor(x): largest integer not exceeding x, e.g. floor(3.8) returns 3

date, exp(x), log(x), log10(x), sqrt(x), abs(x)

max(x): maximum element of vector xmin(x): minimum element of vector xmean(x): mean value of elements of vector xsum(x): sum of elements of vector xsize(a): number of rows and columns of matrix a

Some Common Functions

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rand: random number in the interval [0, 1)

realmax: largest positive floating point numberrealmin: smallest positive floating point number

rem(x, y): remainder when x is divided by y, e.g. rem(19,5) returns 4

sign(x): returns -1, 0 or 1 depending on whether x is negative, zero or positive

sort(x): sort elements of vector x into ascending order (by column if x is a matrix)

Some Common Functions

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A Matlab program can be edited and saved (using Notepad) to a file with .m extension. It is also called a M-file, a script file or simply a script.

When the name of the file is entered in >>, Matlab (or right-click and then run) carries out each statement in the file as if it were entered at the prompt. You are encouraged to use this method.

The M-file

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Basic Concepts

a = 2;

b = 7;

c = a + b;

disp(c)

Variables such as a, b and c are called scalars; they are single-valued.

MATLAB also handles vectors and matrices, which are the key to many powerful features of the language.

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Vectors

A vector is a special type of matrix, having only one row, or one column.

x = [1 3 0 -1 5]

a = [5, 6, 8]

y = 1:10 (elements are the integers 1, 2, …, 10)

z = 1:0.5:4 (elements are the values 1, 1.5, …, 4 in increments of 0.5)

x’ is the transpose of x. Or you can do it directly: [1 3 0 -1 5]’.

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Working with Matrices

MATLAB == MATrix LABoratory

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The Matrix in MATLAB

4 10 1 6 2

8 1.2 9 4 25

7.2 5 7 1 11

0 0.5 4 5 56

23 83 13 0 10

1

2

Rows (m) 3

4

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Columns(n)

1 2 3 4 51 6 11 16 21

2 7 12 17 22

3 8 13 18 23

4 9 14 19 24

5 10 15 20 25

A = A (2,4)

A (17)

Rectangular Matrix:

Scalar: 1-by-1 arrayVector: m-by-1 array

1-by-n arrayMatrix: m-by-n array

where m, n can be 1, 2, 3, 4, …

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Any MATLAB expression can be entered as a matrix element

Entering Numeric Arrays

» a=[1 2;3 4]

a =

1 2

3 4

» b=[-2.8, sqrt(-7), (3+5+6)*3/4]

b =

-2.8000 0 + 2.6458i 10.5000

» b(2,5) = 23

b =

-2.8000 0 + 2.6458i 10.5000 0 0

0 0 0 0 23.0000

» a=[1 2;3 4]

a =

1 2

3 4

» b=[-2.8, sqrt(-7), (3+5+6)*3/4]

b =

-2.8000 0 + 2.6458i 10.5000

» b(2,5) = 23

b =

-2.8000 0 + 2.6458i 10.5000 0 0

0 0 0 0 23.0000

Row separator:semicolon (;)

Column separator:space / comma (,)

Use square brackets [ ]

Matrices must be rectangular. (Set undefined elements to zero)

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Entering Numeric Arrays - cont.

» w=[1 2;3 4] + 5w = 6 7 8 9» x = 1:5

x = 1 2 3 4 5» y = 2:-0.5:0

y = 2.0000 1.5000 1.0000 0.5000 0 » z = rand(2,4)

z =

0.9501 0.6068 0.8913 0.4565 0.2311 0.4860 0.7621 0.0185

» w=[1 2;3 4] + 5w = 6 7 8 9» x = 1:5

x = 1 2 3 4 5» y = 2:-0.5:0

y = 2.0000 1.5000 1.0000 0.5000 0 » z = rand(2,4)

z =

0.9501 0.6068 0.8913 0.4565 0.2311 0.4860 0.7621 0.0185

Scalar expansion

Creating sequences:colon operator (:)

Utility functions for creating matrices.(Ref: Utility Commands)

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Numerical Array Concatenation - [ ]

» a=[1 2;3 4]

a =

1 2

3 4

» cat_a=[a, 2*a; 3*a, 4*a; 5*a, 6*a]cat_a = 1 2 2 4 3 4 6 8 3 6 4 8 9 12 12 16 5 10 6 12 15 20 18 24

>> size(cat_a)ans =

6 4

» a=[1 2;3 4]

a =

1 2

3 4

» cat_a=[a, 2*a; 3*a, 4*a; 5*a, 6*a]cat_a = 1 2 2 4 3 4 6 8 3 6 4 8 9 12 12 16 5 10 6 12 15 20 18 24

>> size(cat_a)ans =

6 4

Use [ ] to combine existing arrays as matrix “elements”

Row separator:semicolon (;)

Column separator:space / comma (,)

Use square brackets [ ]

The resulting matrix must be rectangular.

4*a

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Array Subscripting / Indexing

4 10 1 6 2

8 1.2 9 4 25

7.2 5 7 1 11

0 0.5 4 5 56

23 83 13 0 10

1

2

3

4

5

1 2 3 4 51 6 11 16 21

2 7 12 17 22

3 8 13 18 23

4 9 14 19 24

5 10 15 20 25

A =

A(3,1)A(3)

A(1:5,5)A(:,5) A(21:25)

A(4:5,2:3)A([9 14;10 15])

• Use () parentheses to specify index• colon operator (:) specifies range / ALL• [ ] to create matrix of index subscripts• ‘end’ specifies maximum index value

A(1:end,end) A(:,end)A(21:end)’

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Matrix Multiplication

• Inner dimensions must be equal

• Dimension of resulting matrix = outermost dimensions of multiplied matrices

• Resulting elements = dot product of the rows of the 1st matrix with the columns of the 2nd matrix» a = [1 2 3 4; 5 6 7 8];

» b = ones(4,3);

» c = a*b

c =

10 10 10 26 26 26

» a = [1 2 3 4; 5 6 7 8];

» b = ones(4,3);

» c = a*b

c =

10 10 10 26 26 26

[2x4]

[4x3]

[2x4]*[4x3] [2x3]

a(2nd row).b(3rd column)

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Array Multiplication

• Matrices must have the same dimensions

• Dimensions of resulting matrix = dimensions of multiplied matrices

• Resulting elements = product of corresponding elements from the original matrices

Same rules apply for other array operations

» a = [1 2 3 4; 5 6 7 8];

» b = [1:4; 1:4];

» c = a.*b

c =

1 4 9 16 5 12 21 32

» a = [1 2 3 4; 5 6 7 8];

» b = [1:4; 1:4];

» c = a.*b

c =

1 4 9 16 5 12 21 32 c(2,4) = a(2,4)*b(2,4)

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bal = 15000 * rand;

if bal < 5000rate = 0.09;

elseif bal < 10000rate = 0.12;

elserate = 0.15;

end

newbal = bal + rate + bal;disp(’New balance is: ’)disp(newbal)

Deciding with if

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for index = j:k

statementsend

for index = j:m:k (m is the increment)statements

end

Repeating with for

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Create a program in newton.m file to calculate the square root of 2

%NEWTON Newton Method examplea = 2;x = a/2;for i = 1:6 x = (x+a/x)/2; disp (x)end

Square rooting with Newton Method

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>> newton 1.5000 1.4167 1.4142 1.4142 1.4142 1.4142>> format long>> newton 1.50000000000000 1.41666666666667 1.41421568627451 1.41421356237469 1.41421356237309 1.41421356237309

Running newton.m

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fprintf formats the output as specified by a format string.

fprintf ('format string', list of variables)fprintf ('filename', 'format string' , list of variables)

balance = 123.45678901;fprintf('New balance: %8.3f', balance)

%8.3f means fixed point over 8 columns altogether (including the decimal point and a possible minus sign), with 3 decimal places (spaces are filled in from the left if necessary).

Input / Output

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fprintf example (io_1.m)

balance = 12345;rate = 0.09;interest = rate * balance;balance = balance + interest;fprintf('Interest rate: %6.3f New balance: %8.2f\n', rate, balance);

>> io_1Interest rate: 0.090 New balance: 13456.05

>>

Input / Output Examples

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Input / Output

The input statement gives the user the prompt in the text string and then waits for input from the keyboard. It provides a more flexible way of getting data into a program than by assignment statements which need to be edited each time the data must be changed. It allows you to enter data while a script is running.

The general form of the input statement is:

variable = input(’prompt’);

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Interactive Input (io_2.m)

balance = input('Enter bank balance: ');rate = input('Enter interest rate: ');interest = rate * balance;balance = balance + interest;fprintf('New balance: %8.2f\n', balance);

>> io_2Enter bank balance: 2000Enter interest rate: 0.08New balance: 2160.00>>

Input / Output Examples

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2-D Plotting

• Specify x-data and/or y-data

• Specify color, line style and marker symbol (clm), default values used if ‘clm’ not specified)

• Syntax:– Plotting single line:

– Plotting multiple lines:plot(x1, y1, 'clm1', x2, y2, 'clm2', ...)

plot(xdata, ydata, 'clm')

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x = 0 : 10y = 2 * xplot (x, y)plot (x, sin(x))x = 0 : 0.1 :10;pauseplot (x, sin(x))plot (x, sin(x)), grid

2-D Plot – Examples

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Graphs may be labelled with the following statements:

gtext(’text’): writes a string in the graph window

grid: add/removes grid lines to/from the current graph

text(x, y, ’text’): writes the text at the point specified by x and y

title(’text’): writes the text as a title on top of the graph

xlabel(’text’): labels the x-axis

ylabel(’text’): labels the y-axis

2-D Plot – Labels

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The function plot3 is the 3-D version of plot. The command plot3(x,y,z) draws a 2-D projection of a line in 3-D through the points whose co-ordinates are the elements of the vectors x, y and z.

plot3(rand(1,10), rand(1,10), rand(1,10))

The above command generates 10 random points in 3-D space, and joins them with lines.

3D Plot - Examples

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MATLABExercise