EE 3561 : - Computational Methods
in Electrical Engineering
Unit 1:
Introduction to Computational Methods and Taylor
Series
Lectures 1-3:
Mujahed Al-Dhaifallah
(Term 351)
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Mujahed Al-Dhaifallah
‫مجاهد آل ضيف هللا‬

Office Hours
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Office Hours: SMT, 1:30 – 2:30 PM,
by appointment
Office: Dean Office.
Tel:
7842983
Email:
 muja2007hed@gmail.com
 http://faculty.sau.edu.sa/m.aldhaifallah
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Prerequisites
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Successful completion of Math 1070,
Math2040, CS1090
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Course Description

EE 3561 is a course on
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Introduction to computational methods using
computer packages, e.g. Matlab, Mathcad or
IMSL.
Solution of non-linear equations.
Solution of large systems of linear equations.
Interpolation.
Function approximation.
Numerical differentiation and integration.
Solution of the initial value problem of ordinary
differential equations.
Applications on Electrical Engineering.
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Textbooks

“Numerical Methods for Engineers”.

By Steven C. Chapra and Raymond P. Canale
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Attendance

Regular lecture attendance is required.
There will be part of the grade on
attendance
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Student Evaluation
Exam 1
Exam 2
Computer Work
Quizzes
HW
Attendance
Final Exams
Total
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10
10
5
5
5
5
60
100
7
Quizzes
Announced
 After each HW. From HW material

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Assignment Requirements
Late assignments will not be accepted.
 assignments are due at the beginning of
lecture.
 Sloppy or disorganized work will
adversely affect your grade.

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Exams
Attendance is mandatory.
 Make-up exam are not given unless
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permission is obtained prior to exam day from
the instructor
a valid, documented emergency has arisen
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Academic Dishonesty
cheating
 fabrication
 falsification
 multiple submissions
 plagiarism
 complicity

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Rules and Regulations
No make up quizzes
 DN
grade --- (25%) 8 unexcused
absences
 Homework Assignments are due to
the beginning of the lectures.
 Absence is not an excuse for not
submitting the Homework.

Late submission may not be accepted
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Lecture 1
Introduction to Computational
Methods
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What are
Computational METHODS?
Why do we need them?
Topics covered in EE3561.
Reading Assignment: pages 3-10 of text book
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Analytical Solutions

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Analytical methods give exact solutions
Example: Analytical method to evaluate the integral
3 1
x
 0 x dx  3
1

2
0
1 0 1
  
3 3 3
Numerical methods are mathematical procedures to
calculate approximate solution
1 0
1
2
2
 0 x dx  2 (0)  (1)   2
1
2
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(Trapezoid Method)
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Computational Methods
Computational Methods:
Algorithms that are used to obtain
approximate solutions of a mathematical
problem.
Why do we need them?
1. No analytical solution exists,
2. An analytical solution is difficult to obtain
or not practical.
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What do we need
Basic Needs in the Computational Methods:
 Practical:
can be computed in a reasonable amount of time.

Accurate:
 Good approximate to the true value
 Information about the approximation
error (Bounds, error order,… )
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Outlines of the Course
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Taylor Theorem
Number
Representation
Solution of nonlinear
Equations
Interpolation
Numerical
Differentiation
Numerical Integration
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Solution of linear
Equations
Least Squares curve
fitting
Solution of ordinary
differential equations
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Solution of Nonlinear Equations

Some simple equations can be solved analytically
x2  4 x  3  0
Analytic solution : roots 
4
4 2  4(1)( 3)
2(1)
x  1 and x  3

Many other equations have no analytical solution
x 9  2 x 2  5  0

 No analytic solution
x

xe

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Methods for solving Nonlinear
Equations
o
o
o
Bisection Method
Newton-Raphson Method
Secant Method
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Solution of Systems of
Linear Equations
x1  x2  3
x1  2 x2  5
We can solve it as
x1  3  x2 ,
3  x2  2 x 2  5
 x2  2, x1  3  2  1
What to do if we have
1000 equations in 1000 unknowns?
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Cramer’s Rule is not practical
Cramer' s Rule can be used to solve the system
3
5
x1 
1
1
1
2
 1,
1
2
1
1
x2 
1
1
3
5
2
1
2
But Cramer' s Rule is not practical for large problems.
To solve N equations in N unknowns we need (N  1)(N  1)N!
multiplica tions.
To solve a 30 by 30 system, 2.3 1035 multiplica tions are needed.
A super computer needs more than 10 20 years to compute.
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Methods for solving Systems of Linear
Equations
o
o
Naive Gaussian Elimination
Gaussian Elimination with Scaled
Partial pivoting
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Curve Fitting

Given a set of data
x
0
1
2
y
0.5
10.3
21.3
Select a curve that best fit the data.
 One choice is find the curve so that the
sum of the square of the error is
minimized.

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Interpolation
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Given a set of data
xi
0
1
2
yi
0.5
10.3
15.3
find a polynomial P(x) whose graph
passes through all tabulated points.
yi  P( xi ) if xi is in the table
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Methods for Curve Fitting
o
Least Squares
o
o
o
Linear Regression
Nonlinear least Squares Problems
Interpolation
o
o
Newton polynomial interpolation
Lagrange interpolation
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Integration

Some functions can be integrated analytically
3
3
1 2
9 1
 xdx  2 x  2  2  4
1
1
But many functions have no analytical solutions
a
e
x2
dx  ?
0
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Methods for Numerical Integration
o
o
o
o
Upper and Lower Sums
Trapezoid Method
Romberg Method
Gauss Quadrature
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Solution of Ordinary Differential Equations
A solution t o the differenti al equation
x(t )  3 x (t )  3 x(t )  0
x (0)  1; x(0)  0
is a function x(t) that satisfies the equations
* Analytical solutions are available for
special cases only
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Summary
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Computational
Methods:
Algorithms that are
used to obtain
numerical solution of a
mathematical problem.
We need them when
No analytical solution
exist or it is difficult
to obtain.
Topics Covered in the Course
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Solution of nonlinear Equations
Solution of linear Equations
Curve fitting
Least Squares
Interpolation
Numerical Integration
Numerical Differentiation
Solution of ordinary differential
equations
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Lecture 2
Number Representation and accuracy
 Number Representation
 Normalized Floating Point Representation
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Significant Digits
Accuracy and Precision
Rounding and Chopping
 Reading assignment:
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Chapter 2
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Representing Real Numbers

You are familiar with the decimal system
312.45  3 10 2  1101  2 100  4 10 1  5 10 2

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Decimal System Base =10 , Digits(0,1,…9)
Standard Representations
 3 1 2 . 4 5
sign integral
fraction
part
part
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Normalized Floating Point
Representation

Normalized Floating Point Representation
 0. d1 d 2 d 3 d 4  10n
sign
mantissa
d1  0,
n : integer

No integral part,

Advantage
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exponent
Efficient in representing very small or very large numbers
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Binary System

Binary System
Base=2, Digits{0,1}
 0. 1 b2 b3 b4  2 n
sign
mantissa
exponent
b1  0  b1  1
1
2
3
(0.101) 2  (1 2  0  2  1 2 )10  (0.625)10
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7-Bit Representation
(sign: 1 bit, Mantissa 3bits,exponent 3 bits)
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Fact

Number that have finite expansion in one numbering
system may have an infinite expansion in another
numbering system
(0.1)10  (0.000110011001100...)2

You can never represent 0.1 exactly in any computer
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Representation
 Hypothetical Machine (real computers use ≥ 23 bit
mantissa)
 Example:
If a machine has 5 bits representation
distributed as follows
Mantissa 2 bits
exponent 2 bit
sign 1 bit
Possible machine numbers
(0.25=00001)
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(0.375= 01111) (1.5=00111)
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1
Representation
0.8
Gap near zero
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
0.2
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0.4
0.6
0.8
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1.4
1.6
1.8
2
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Remarks
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Numbers that can be exactly represented are called
machine numbers
Difference between machine numbers is not uniform. So,
sum of machine numbers is not necessarily a machine
number
0.25 + .375 =0.625 (not a machine number)
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Significant Digits

Significant digits are those digits that can be
used with confidence.
0
1
2
3
Length of green rectangle =
4
3.45
significant
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Loss of Significance

Mathematical operations may lead to
reducing the number of significant digits
0.123466 E+02
6 significant digits
─ 0.123445 E+02
6 significant digits
──────────────
0.000021E+02
2 significant digits
0. 210000E-02
Subtracting nearly equal numbers causes loss of significance
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Accuracy and Precision
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Accuracy is related to closeness to the true value

Precision is related to the closeness to other estimated
values
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Accuracy and Precision
Better
Precision
Accuracy is
related to
closeness to the
true value
Precision is
related to the
closeness to
other estimated
values
Better
accuracy
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Rounding and Chopping
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Rounding: Replace the number by the nearest
machine number
Chopping: Throw all extra digits
True 1.1681
0
1
Rounding
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(1.2)
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Chopping (1.1)
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Error Definitions
True Error
can be computed if the true value is known
Absolute True Error
Et  true value  approximat ion
Absolute Percent R elative Error
true value  approximat ion
t 
* 100
true value
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Error Definitions
Estimated error
Used when the true value is not known
Estimated Absolute Error
Ea  current estimate  prevoius estimate
Estimated Absolute Percent R elative Error
current estimate  prevoius estimate
a 
* 100
current estimate
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Notation
We say the estimate is correct to n decimal
digits if
n
Error  10
We say the estimate is correct to n decimal
digits rounded if
1
Error 
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 10
n
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Summary

Number Representation
Number that have finite expansion in one numbering system may
have an infinite expansion in another numbering system.

Normalized Floating Point Representation



Efficient in representing very small or very large numbers
Difference between machine numbers is not uniform
Representation error depends on the number of bits used in the
mantissa.
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Summary
Rounding
Chopping
 Error Definitions:



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
Absolute true error
True Percent relative error
Estimated absolute error
Estimated percent relative error
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