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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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