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TEKNIK KOMPUTASI

(TEI 116)3 SKS

Oleh: Husni Rois Ali, S.T., M.Eng.

Noor Akhmad Setiawan, S.T., M.T., Ph.D.

Email : husni.rois.ali@ugm.ac.id or

husni.rois.ali@gmail.com

Personal web: husniroisali.staff.ugm.ac.id

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Grading

First Half (tentative)

Midterm (80%)

Homework, quiz, etc. (20%)

Second Half (?)

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References:

Numerical Methods for Engineers, Sixth Edition

by Steven Chapra(Author), Raymond

Canale(Author)

http://numericalmethods.eng.usf.edu

http://www.amazon.com/s/ref=dp_byline_sr_book_1?ie=UTF8&field-author=Steven+Chapra&search-alias=books&text=Steven+Chapra&sort=relevancerankhttp://www.amazon.com/s/ref=dp_byline_sr_book_2?ie=UTF8&field-author=Raymond+Canale&search-alias=books&text=Raymond+Canale&sort=relevancerankhttp://www.amazon.com/s/ref=dp_byline_sr_book_2?ie=UTF8&field-author=Raymond+Canale&search-alias=books&text=Raymond+Canale&sort=relevancerankhttp://numericalmethods.eng.usf.edu/http://numericalmethods.eng.usf.edu/http://www.amazon.com/s/ref=dp_byline_sr_book_2?ie=UTF8&field-author=Raymond+Canale&search-alias=books&text=Raymond+Canale&sort=relevancerankhttp://www.amazon.com/s/ref=dp_byline_sr_book_2?ie=UTF8&field-author=Raymond+Canale&search-alias=books&text=Raymond+Canale&sort=relevancerankhttp://www.amazon.com/s/ref=dp_byline_sr_book_1?ie=UTF8&field-author=Steven+Chapra&search-alias=books&text=Steven+Chapra&sort=relevancerankhttp://www.amazon.com/s/ref=dp_byline_sr_book_1?ie=UTF8&field-author=Steven+Chapra&search-alias=books&text=Steven+Chapra&sort=relevancerank

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What are we going to learn?

Finding the root of the equations i.e. solve for

Linear algebraic equation i.e. given and solve

( ) 0f x x

2 3 2 0x x

'a s 'b s

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What are we going to learn? (2)

Curve fitting

Integration

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What are we going to learn? (3)

Ordinary differential equations (ODE)

Partial differential equation

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Analytic vs Numerical Approach

Analytic

You obtain the exact

solution

It is single step There is no error (since it is

exact)

Normally used for simple

problem This is how you work

Numerical

You obtain approximate

solution

It is an iterative approach The error is a crucial factor

Used for complex problems

This is how your computer

and all computationalsoftware works

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Analytic vs. Numerical Approach (2)suppose you want to find the roots of 2

Analytic

Formulate to mathematical

equation

Factorize

Then

Numerical

Formulate to mathematical

equation

Start from an arbitrary initial

value of , e.g.

You update the value of to

better approximation, until you

close enough

How you move from one point to

another depends on the theal orithm

2( ) 2 0f x x

( 2)( 2) 0x x

2x

2( ) 2 0f x x

x0 1x

x

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Introduction

Mathematical Modeling and Engineering Problem solving

Requires understanding of

engineering systemsBy observation and experimentTheoretical analysis and

generalization

Computers are great tools,

however, without fundamentalunderstanding of engineering

problems, they will be useless.

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A mathematical model is represented as a functional

relationship of the form

Dependent independent forcing

Variable =f variables, parameters, functions

It ranges from a simple equation to very complex ones

Dependent variable: Characteristic that usually reflects thestate of the system

Independent variables: Dimensions such as time and spacealong which the systems behavior is being determined

Parameters: reflect the systems properties or composition

Forcing functions: external influences acting upon the system

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Newtons 2ndlaw of Motion

The model is formulated as

F = m a

F=net force acting on the body (N)a forcing

function

m=mass of the object (kg)a parameter

a=its acceleration (m/s2) -> the dependent variable

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Formulation of Newtons 2ndlaw has several

characteristics that are typical of mathematical

models of the physical world:

It describes a natural process or system in

mathematical terms

It represents an idealization and simplification of

reality Finally, it yields reproducible results,

consequently, can be used for predictive purposes.

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Some mathematical models of physical phenomena

may be much more complex.

Complex models may not be solved exactly or

require more sophisticated mathematical techniques

than simple algebra for their solution

Example, modeling of a falling parachutist:

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with , andD U D U

dv F

dt m

F F F F mg F cv

dv mg cv

dt m

dv c

g vdt m

This is a differential equation and is written in

terms of the differential rate of change dv/dt

of the variable that we are interested in

predicting.

If the parachutist is initially at rest (v=0 at t=0),using calculus

tmcec

gmtv )/(1)(

Independent

variable

Dependent

variable

Parameters

Forcingfunction

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Write the velocity equation as

We may obtain

Rearranging yields

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Computer software

Excel

Matlab

Scilab Fortran 90 (IMSL)

C++

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Approximations Errors

For many engineering problems, we cannot obtain analyticalsolutions.

Numerical methods yield approximate results, results that areclose to the exact analytical solution. We cannot exactlycompute the errors associated with numerical methods.

Only rarely given data are exact, since they originate frommeasurements. Therefore there is probably error in the inputinformation.

Algorithm itself usually introduces errors as well, e.g., unavoidableround-offs, etc

The output information will then contain error from both of thesesources.

How confident we are in our approximate result? The question is how much error is present in our calculation

and is it tolerable?

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Accuracy.How close is a computed or

measured value to the true value

Precision (or reproducibility).How close is a

computed or measured value to previously

computed or measured values. Inaccuracy(or bias).A systematic deviation

from the actual value.

Imprecision(or uncertainty).Magnitude ofscatter.

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Significant Figures

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Significant Figures (2)

Number of significant figures indicates precision. Significant digits of anumber are those that can be usedwith confidence, e.g.,the number ofcertain digits plus one estimated digit.

53,800 How many significant figures?

5.38 x 104 3

5.380 x 104 4

5.3800 x 104 5

Zeros are sometimes used to locate the decimal point not significantfigures.

0.00001753 4

0.0001753 4

0.001753 4

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Significant Figures (3)

Implication :

1. Numerical methods yield approximate results. We must develop

criteria to specify how confident we are in our approximate result.

One way to do this is in terms of significant figures.

2. Although quantities such as , e, or 7 represent specific quantities,they cannot be expressed exactly by a limited number of digits.

Because computers retain only a finite number of significant figures,

such numbers can never be represented exactly. The omission of the

remaining significant figures is called round-off error.

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Error Definitions

True Value = Approximation + Error

Et= True valueApproximation (+/-)

valuetrue

errortrueerrorrelativefractionalTrue

%100valuetrue

errortrueerror,relativepercentTrue

t

True error

F i l th d th t l ill b

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For numerical methods, the true value will be

known only when we deal with functions that can

be solved analytically (simple systems). In real

world applications, we usually not know the answera priori. Then

Iterative approach, example Newtons method

%100

ionApproximat

erroreApproximat

a

%100ionapproximatCurrent

ionapproximatPrevious-ionapproximatCurrenta (+ / -)

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Use absolute value.

Computations are repeated until stopping criterion issatisfied.

If the following criterion is met

you can be sure that the result is correct to at least nsignificant figures.

sa Pre-specified % tolerance based on

the knowledge of your solution

)%10(0.5n)-(2

s

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Two sources of numerical error

1) Round off error

2) Truncation error

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Round-off Error

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Round off Error

Caused by representing a number approximately

It is due to the limited sources of computer in

representing the number

10.333333

3 2 1.4142...

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Problems created by round off error

28 Americans were killed on February 25,

1991 by an Iraqi Scud missile in Dhahran,

Saudi Arabia.

The patriot defense system failed to track and

intercept the Scud. Why?

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Problem with Patriot missile

Clock cycle of 1/10 seconds was

represented in 24-bit fixed point

register created an error of 9.5 x 10-8

seconds.

The battery was on for 100

consecutive hours, thus causing an

inaccuracy of

1hr

3600s100hr

0.1s

s109.5 8

s342.0

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Problem (cont.)

The shift calculated in the ranging system of

the missile was 687 meters.

The target was considered to be out of range

at a distance greater than 137 meters.

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Truncation Error

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Truncation error

Error caused by truncating or

approximating a mathematical procedure.

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Example of Truncation Error

Taking only a few terms of a Maclaurin series to

approximate

....................!3!2

132

xxxex

xe

If only 3 terms are used,

!21

2

xxeErrorTruncation x

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Another Example of Truncation Error

Using a finite x to approximate )(xf

x

xfxxfxf

)()()(

P

Q

secant line

tangent line

Figure 1. Approximate derivative using finite x

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Another Example of Truncation Error

Using finite rectangles to approximate an

integral.

y = x2

0

30

60

90

0 1.5 3 4.5 6 7.5 9 10.5 12

x

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Example 1 Maclaurin series

Calculate the value of2.1e with an absolute

relative approximate error of less than 1%.

...................!3

2.1

!2

2.12.11

322.1

e

n

1 1 __ ___

2 2.2 1.2 54.545

3 2.92 0.72 24.658

4 3.208 0.288 8.9776

5 3.2944 0.0864 2.6226

6 3.3151 0.020736 0.62550

aE %a2.1e

6 terms are required.

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Example 2 Differentiation

Find )3(f for2)( xxf using

x

xfxxfxf

)()()(

and 2.0x

2.0

)3()2.03(

)3(' ff

f

2.0

)3()2.3( ff

2.0

32.3 22

2.0

924.10

2.0

24.1 2.6

The actual value is

,2)(' xxf 632)3(' f

Truncation error is then, 2.02.66

Can you find the truncation error with 1.0x

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Example 3 Integration

Use two rectangles of equal width to approximate

the area under the curve for2)( xxf over the interval ]9,3[

y = x2

0

30

60

90

0 3 6 9 12

x

9

3

2dxx

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Integration example (cont.)

)69()()36()(6

2

3

2

9

3

2 xx xxdxx

3)6(3)3( 22 13510827

Choosing a width of 3, we have

Actual value is given by

9

3

2

dxx

9

3

3

3

x

2343

3933

Truncation error is then

99135234