ee 3561_unit_1(c)al-dhaifallah 14351 ee 3561 : - computational methods in electrical engineering...

EE 3561_Unit_1 (c)Al-Dhaifallah 1435 1

EE 3561 : - Computational Methods in Electrical Engineering

Unit 1: Introduction to Computational Methods and Taylor Series

Lectures 1-3:

Mujahed Al-Dhaifallah (Term 351)


Mujahed Al-Dhaifallahالله ضيف آل مجاهد

Office Hours Office Hours: SMT, 1:30 – 2:30 PM, by appointment

Office: Dean Office. Tel: 7842983 Email:

[email protected]

http://faculty.sau.edu.sa/m.aldhaifallah

Prerequisites Successful completion of Math 1070,

Math2040, CS1090


Course Description EE 3561 is a course on

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.


Textbooks “Numerical Methods for Engineers”.

By Steven C. Chapra and Raymond P. Canale


Attendance Regular lecture attendance is required.

There will be part of the grade on attendance


Student Evaluation

Exam 1 10Exam 2 10Computer Work 5Quizzes 5HW 5Attendance 5Final Exams 60Total 100


Quizzes Announced After each HW. From HW material


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.


Exams Attendance is mandatory. Make-up exam are not given unless

permission is obtained prior to exam day from the instructor

a valid, documented emergency has arisen


Academic Dishonesty

cheating fabrication falsification multiple submissions plagiarism complicity



Rules and RegulationsNo 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


Lecture 1Introduction to

Computational Methods

What are Computational METHODS? Why do we need them? Topics covered in EE3561.

Reading Assignment: pages 3-10 of text book


Analytical Solutions Analytical methods give exact solutions Example: Analytical method to evaluate the integral

Numerical methods are mathematical procedures to calculate approximate solution

(Trapezoid Method)

3

1

3

0

3

1

3

1

0

31

0

2 x

dxx

2

1)1()0(

2

01 221

0

2

dxx


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.


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,… )


Outlines of the Course Taylor Theorem Number

Representation Solution of nonlinear

Equations Interpolation Numerical

Differentiation Numerical Integration

Solution of linear Equations

Least Squares curve fitting

Solution of ordinary differential equations


Solution of Nonlinear Equations Some simple equations can be solved analytically

Many other equations have no analytical solution

31

)1(2

)3)(1(444 :solution Analytic

034

2

2

xandx

roots

xx

solution analytic No052 29

xex

xx


Methods for solving Nonlinear Equations

o Bisection Methodo Newton-Raphson Methodo Secant Method


Solution of Systems ofLinear Equations

unknowns? 1000 in equations 1000

have weif do What to

123,2

523,3

asit solve can We

52

3

12

2221

21

21

xx

xxxx

xx

xx


Cramer’s Rule is not practical

compute. toyears 10 than more needscomputer super A

needed. are tionsmultiplica102.3 system, 30by 30 a solve To

tions.multiplica

1)N!1)(N(N need weunknowns Nin equations N solve To

problems. largefor practicalnot is Rule sCramer'But

2

21

11

51

31

,1

21

11

25

13

system thesolve toused becan Rule sCramer'

20

35

21

xx


Methods for solving Systems of Linear Equations

o Naive Gaussian Eliminationo Gaussian Elimination with Scaled

Partial pivoting


Curve Fitting Given a set of data

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.

x 0 1 2

y 0.5 10.3 21.3


Interpolation Given a set of data

find a polynomial P(x) whose graph passes through all tabulated points.

xi 0 1 2

yi 0.5 10.3 15.3

tablein the is)( iii xifxPy


Methods for Curve Fitting o Least Squares

o Linear Regressiono Nonlinear least Squares Problems

o Interpolationo Newton polynomial interpolationo Lagrange interpolation


Integration Some functions can be integrated analytically

?

solutions analytical no have functionsmany But

dxe

xxdx

ax

0

3

1

23

1

2

42

1

2

9

2

1


Methods for Numerical Integration

o Upper and Lower Sumso Trapezoid Methodo Romberg Methodo Gauss Quadrature


Solution of Ordinary Differential Equations

only cases special

for available are solutions Analytical *

equations thesatisfies that function a is

0)0(;1)0(

0)(3)(3)(

equation aldifferenti theosolution tA

x(t)

xx

txtxtx


Summary 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.

Solution of nonlinear Equations Solution of linear Equations Curve fitting

Least Squares Interpolation

Numerical Integration Numerical Differentiation Solution of ordinary differential

equations

Topics Covered in the Course


Number Representation Normalized Floating Point Representation Significant Digits Accuracy and Precision Rounding and Chopping

Reading assignment: Chapter 2

Lecture 2 Number Representation and

accuracy


Representing Real Numbers You are familiar with the decimal system

Decimal System Base =10 , Digits(0,1,…9) Standard Representations

21012 10510410210110345.312

part part

fraction integralsign

54.213


Normalized Floating Point Representation Normalized Floating Point Representation

No integral part, Advantage Efficient in representing very small or very large numbers

integer:,0

exponent mantissa sign

10.0

1

4321

nd

dddd n


Binary System

Binary System Base=2, Digits{0,1}

exponent mantissasign

21.0 432nbbb

10 11 bb

1010321

2 )625.0()212021()101.0(


7-Bit Representation(sign: 1 bit, Mantissa 3bits,exponent 3 bits)


Fact Number that have finite expansion in one numbering

system may have an infinite expansion in another numbering system

You can never represent 0.1 exactly in any computer

210 ...)011000001100110.0()1.0(


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) (0.375= 01111) (1.5=00111)


Representation

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Gap near zero


Remarks

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)


Significant Digits Significant digits are those digits that can be

used with confidence.

0 1 2 3 4

Length of green rectangle = 3.45significant


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-02Subtracting nearly equal numbers causes loss of significance


Accuracy and Precision Accuracy is related to closeness to the true value

Precision is related to the closeness to other estimated

values


Better Precision

Better accuracy

Accuracy is related to closeness to the true value Precision is related to the closeness to other estimated values

Accuracy and Precision


Rounding and Chopping Rounding: Replace the number by the nearest machine number Chopping: Throw all extra digits

0 1 2

True 1.1681

Rounding (1.2) Chopping (1.1)


Error DefinitionsTrue Error

can be computed if the true value is known

100* valuetrue

ionapproximat valuetrue

Errorelative Percent RAbsolute

ionapproximat valuetrue

ErrorTrue Absolute

t

tE


Error DefinitionsEstimated error

Used when the true value is not known

100*estimatecurrent

estimate prevoius estimatecurrent

Errorelative Percent RAbsolute Estimated

estimate prevoius estimatecurrent

ErrorAbsolute Estimated

a

aE


Notation

We say the estimate is correct to n decimal digits if

We say the estimate is correct to n decimal digits rounded if

n10Error

n 102

1Error


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.


Summary Rounding Chopping Error Definitions:

Absolute true error True Percent relative error Estimated absolute error Estimated percent relative error

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