lec 1 -2 introduction

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Lecture 1 - Introduction

Dr. Nasir M Mirza

Numerical Methods

Email: [email protected]


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Lectures Goals

General Introduction to Computer Applications inEngineering and Sciences

Introduction to numerical analysis

Why you should be able to write and understand

computer programs for numerical methods?

Computer languages C & C++ and Matlab aMathematical Laboratory


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Books on Numerical Analysis

Text Books:

1. Numerical Analysis, By Burden & Faires, Recent

Edition.

2. Numerical methods for engineers and scientists by A.C. Bajpai, I.M. Calus and J.A.Fairley;

Softwares: C++ or Matlab & SIMULINK


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Introduction

Professor: Dr. Nasir M MirzaOffice : 104 BLOCK-A; PIEAS, P.O.

Nilore, Islamabad, 45650.

Email: [email protected]

URL: http://www.pieas.edu.pk
http://www.pieas.edu.pk/http://www.pieas.edu.pk/


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Topics

Matlab or C++ Computer Errors

Roots f(x) = 0

Linear Methods

Nonlinear Methods

Linear Systems

LU Decomposition

Eigenvalue Analysis

Fitting Data Interpolation

Curve Fitting

Numerical Integration

ODEs

Initial Value Problems

Systems of ODEs

Boundary ValueProblems


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Why numerical analysis and methods?

Integration becomes easy by use of numerical methods;

When you wish to solve simultaneous linear equations, you needto find inverse of a matrix A. Thou can is is easy when you havethree equations and three unknown. If say you have 50 equationsand fifty unknowns then with help of digital computers andnumerical analysis you can solve this.

Say you have to solve one equation exp(x)=10 x. it will be verytroublesome or impossible to solve it analytically. However,numerically it is very easy to solve it.

Complex differential equations can be solved very easily usingnumerical analysis. Event set of coupled differential equations

can be solved very quickly and easily using numerical methods. When Discrete data is given, one can differentiate and integrate

using numerical techniques.


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What you need for numerical analysis

A good scientific calculator for numericals. When you wish to solve simultaneous linear equations,

say 50 equations and fifty unknowns then you will needdigital computers (say a good PC).

You will also need good knowledge of one computerprogramming language (say C or C++ or MATLAB).

Good knowledge ofmathematics will also help.

Aneat software to plot results is also needed.


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Applications

Signal Processing

CFD (ComputationalFluid Dynamics)

Structural Analysis

Finite Element Analysis

Interpolation

Optimization

CAD (Computer Aided-Drafting)

Data Collection


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Numerical Accuracy and Errors

Whenever calculations are performed there aremany possible sources of errors. These include

Mistakes made by person carrying out calculations,

The use of inaccurate formula; The use of inaccurate data (or round-off errors).

The first type of error should not be there at all.

The second type is due to chopping off aninfinite series and this error is called truncationerror.


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Truncation ErrorsThis type is due to chopping off an infinite series. For example; let us

approximate the first derivative at a point x = a by following form:

The accuracy will increase when h is decreased. To show this, consider

f(x) = exp(2x) ; we also know, df/dx = 2exp(2x)

For x = 2; exact answer is 109.1963.

Now u sing above app roxim ate formula, we f ind values

h

afhaf

dx

dy )()(

2635.1342.0

5982.544509.812.0

)22exp()2.22exp()()( h

afhafdxdy

8412.11405.0

5982.543403.60

05.0

)22exp()05.22exp()()(

h

afhaf

dx

dy

2906.11001.0

5982.547011.55

01.0

)22exp()01.22exp()()(

h

afhaf

dx

dy


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Roundoff Errors

When we were computig exp(4) or exp(4.1) we were rounding off theresults. That is how many digits are written after the decimal places or

significant figures.

When rounding off one digit:

if digit lies in the range 0 4 , the previous digit is unchanged.

if the digit lies in the range 69, the previous digit is increased by

one.

For examp le, 7.4727 becomes 7.473

76.34 becom es 76.34

15.235 becom es 15.24

When rounding off two digit:

if digit lies in the range 0 49 , the previous digit is unchanged.

if the digit lies in the range 51 99, the previous digit is increased by

one.


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What is Error

Hence in rounding off 7.4727 becomes 7.473 and error is -0.0003

76.34 becomes 76.34 and error is 0.000

15.235 becomes 15.24 and error is -0.005

18.496 becomes 18.50 introducing error0.004

17.208 becomes 17.21 introducing error -0.002

valueeapproximatvalueexact The error, , in any quantity is given by

18.50 + 17.21 = 35.71 introducing error

0.006

18.50 - 17.21 = 1.29 introducing error -0.007

The individual errors may be positive or negative but when they are

added or subtracted they may reinforce each other.


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Numerical Errors

Precision Limits

Stability

Convergence

Divergence

Alaising

Round-off Errors

Truncation Errors

Machine Precision


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Example; first rounding

Let p = 0.54617 and q = 0.54601.

The exact value of r = p q

r = 0.54617 0.54601 = 0.00016

Here subtraction is performed using four-digit arithmetic.

Rounding p and q to four digits gives

p* = 0.5462 & q* - 0.5460,

r* = p* - q* = 0.0002

25.000016.0

0002.000016.0*

r

rrerrorrelative


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Example; now chopping

Let p = 0.54617 and q = 0.54601.

The exact value of r = p q

r = 0.54617 0.54601 = 0.00016

If chopping is used to obtain the four digits, then the four-digit approximations to p, q, and r are

p* = 0.5461,

q* = 0.5460, andr* = p* - q* = 0.0001.

375.000016.0

0001.000016.0*

r

rrerrorrelative


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Example 2

The loss of accuracy due to round-off error can often beavoided by a reformulation of the problem, as illustrated

in the next example.

The quadratic formula states that the roots ofax2 + bx +

c = 0, when a 0, are

a

acbbx

a

acbbx

2

4;

2

42

2

2

1

Using four-digit rounding arithmetic, consider this formulaapplied to the equation x2 + 62.10x + 1 = 0, whose roots

are approximately

x1 = -0.01610723 ; x2 = -62.08390


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Example 2

06.62.3852000.4.3856

)000.1)(000.1)(000.4()10.62(422

acb

0200.0

2

06.6210.62

2

4)1(

2

a

acbbxfl

In this equation, b2

is much larger than 4ac, so thenumerator in the calculation for x involves the subtraction

of nearly equal numbers. Since

We then have

Then relative error for x1= -0.01611 is

1104.2

01611.0

02000.001611.0

r


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Example 2

10.62

000.2

2.124

2

06.6210.62

2

4)1(

2

a

acbbxfl

On the other hand, the calculation for x2 involves theaddition of the nearly equal numbers. This presents no

problem since

Then relative error for x1= -0.01611 is

4

102.308.62

10.6208.62

r

This relative error is small.


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Example 2

acbbc

x

acbba

ac

acbb

acbb

a

acbbx

4

2

42

4

4

4

2

4

22

22

22

2

1109.108.62

00.5008.62

r

If we use this formula for x2, it will be a mistake

Using this we get

This relative error is large due to

subtraction of nearly equal

numbers and a division.

00.50

0400.0

000.2

06.6210.62

000.2)2(

xfl


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Example 3Accuracy loss due to roundoff error can also be reduced by rearranging

calculations, as shown in this example.

Evaluate: f(x) = x3 6.1x2 + 3.2x + 1.5 at x = 4.71

Use the three-digit arithmetic.

Table gives the intermediate results in the calculations.

Note that the three-digit chopping values simply retain the leading

three digits, with no rounding involved, and differ significantly from the

three-digit rounding values.

x x2

x3

6.1x2

3.2xExact 4.71 22.1841 104.487111 135.32301 15.072

(chopping) 4.71 22.1 104. 134. 15.0

(rounding) 4.71 22.2 105. 135. 15.1


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Example 3

05.0263899.14

5.13263899.14

r

Exact: f(4.71) = 104.487111 - 135.32301 + 15.072+ 1.5 = -14.263899;

Three-digit (chopping): f(4.71) = ((104. - 134.) + 15.0) + 1.5 = -13.5;

Three-digit (rounding): f(4.71) = ((105. - 135.) + 15.1) + 1.5 = -13.4.

The relative errors for the three-digit:

06.0263899.14

4.13263899.14

r

For chopping:

For rounding:

As an alternative approach, f(x) can be written in a nested manner.


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Example 3

As an alternative approach, f(x) can be written in a nested manner as

f(x) = x3 - 6.1x2 + 3.2x + 1.5 = ((x - 6.1)x + 3.2)x + 1.5.

This gives Three-digit (chopping):

f(4.71) = ((4.71 - 6.1)4.71 + 3.2)4.71 + 1.5 = -14.2

and a three-digit rounding answer of -14.3.

The new relative errors are

Three-digit (chopping): relative error = 0.0045;

Three-digit (rounding): relative error = 0.0025

Nesting has reduced the relative error for the chopping approximation

to less than 10% of that obtained initially. For the rounding

approximation the improvement has been even more dramatic; the

error in this case has been reduced by more than 95%.


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Software

Operating Systems

Windows - NT, ME, Windows

Unix VMS - VAX

Linux


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Software

Languages

Fundamental Assembler (Bit manipulations)

Engineering Languages

Fortran

Cobol

Pascal

C++

Basic

HTML and Java


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Software

Higher-Order Programming Maple - Mathematical Programming Language

Mathematica - Mathematical ProgrammingLanguage

Java - Internet Programming Language

Matlab - Matrix Laboratory


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Software

Tools Word Processors

Spreadsheets

Database Management Graphics

Mathematical Computer Codes


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Matlab -Matrix Laboratory

Currently Matlab 7.0 is available on This will be available on the network with a

SIMULINK tool box

Student Version is also available in the maket.


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What is a program?

Program consist of three main components:

Input

Main Program - Numerical methods andanalysis and/or evaluation.

Output - Results.


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Inputs

Numerical values Initialization of the variables

Conditions

Equations


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Main Program

Using flow charts, the programs can be designedto perform a task. Using:

Loops

Conditions

Error Convergence


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Output

Outputs are the results of the program. They can gothrough a series of post-processing methods.

Numerical Values

Decisions

Graphs and Plots


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MatLab

Variable Types Integers

Real Values (Float)

Complex Numbers (a + ib) a - real value

b - imaginary value (i is the square root of-1)


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Matlab

Data types Numerical

Scalars

Vectors

Matrices

Logic Types

Alpha/Numerical Types


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Matlab

A scalar value is the simple number, a, 2,3.14157,

A vector is a union of ax = (x1, x2, x3, x4)

Transpose vector xT = x1x2x3

x4


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Matlab

Matrix is a combination of vectors and scalars.Scalar and vectors are subsets of matrices.

Matlab uses matrix to do mathematicalmethods.


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Matlab

Set of computer functions Circular functions - sin(x),cos(x), tan(x), asin(x),

acos(x), atan(x)

Hyperbolic functions - sinh(x), cosh(x), tanh(x)

Logarithmic functions - ln(x), log(x), exp(x)

Logic functions - abs(x), real(x), imag(x)


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Matlab

Simple commands clc - clears window

clg - clear graphic window

clear - clears the workspace who - variable list

whos - variable list with size

help - when doubt use it!


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Matlab

Simple commands and symbols ^C - an escape from a loop

inf - infinity

NaN - No numerical value


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lec 1 -2 introduction

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