Numerical Analysis Using MATLAB® and Excel®, Third Edition
Copyright ” 2007 Orchard Publications. All rights reserved. Printed in the United States of America. No part of this
publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system,
without the prior written permission of the publisher.
Direct all inquiries to Orchard Publications,
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Product and corporate names are trademarks or registered trademarks of the Microsoft™ Corporation and The
MathWorks™, Inc. They are used only for identification and explanation, without intent to infringe.
Library of Congress Cataloging-in-Publication Data
Library of Congress Control Number: 2007922100
Copyright TX 5-589-152
ISBN-13: 978-1-934404-04-1
ISBN-10: 1-934404-04-7
Disclaimer
The author has made every effort to make this text as complete and accurate as possible, but no warranty is implied.
The author and publisher shall have neither liability nor responsibility to any person or entity with respect to any loss
or damages arising from the information contained in this text.
Preface
Numerical analysis is the branch of mathematics that is used to find approximations to difficult
problems such as finding the roots of non−linear equations, integration involving complex
expressions and solving differential equations for which analytical solutions do not exist. It is
applied to a wide variety of disciplines such as business, all fields of engineering, computer science,
education, geology, meteorology, and others.
Years ago, high−speed computers did not exist, and if they did, the largest corporations could only
afford them; consequently, the manual computation required lots of time and hard work. But now
that computers have become indispensable for research work in science, engineering and other
fields, numerical analysis has become a much easier and more pleasant task.
This book is written primarily for students/readers who have a good background of high−school
algebra, geometry, trigonometry, and the fundamentals of differential and integral calculus.* A
prior knowledge of differential equations is desirable but not necessary; this topic is reviewed in
Chapter 5.
One can use Fortran, Pascal, C, or Visual Basic or even a spreadsheet to solve a difficult problem.
It is the opinion of this author that the best applications programs for solving engineering
problems are 1) MATLAB which is capable of performing advanced mathematical and
engineering computations, and 2) the Microsoft Excel spreadsheet since the versatility offered by
spreadsheets have revolutionized the personal computer industry. We will assume that the reader
has no prior knowledge of MATLAB and limited familiarity with Excel.
We intend to teach the student/reader how to use MATLAB via practical examples and for
detailed explanations he/she will be referred to an Excel reference book or the MATLAB User’s
Guide. The MATLAB commands, functions, and statements used in this text can be executed
with either MATLAB Student Version 12 or later. Our discussions are based on a PC with
Windows XP platforms but if you have another platform such as Macintosh, please refer to the
appropriate sections of the MATLAB’s User Guide that also contains instructions for installation.
MATLAB is an acronym for MATrix LABoratory and it is a very large computer application
which is divided to several special application fields referred to as toolboxes. In this book we will
be using the toolboxes furnished with the Student Edition of MATLAB. As of this writing, the
latest release is MATLAB Student Version Release 14 and includes SIMULINK which is a
* These topics are discussed in Mathematics for Business, Science, and Technology, Third Edition, ISBN 0−9709511−
0−8. This text includes probability and other advanced topics which are supplemented by many practical applications using
Microsoft Excel and MATLAB.
software package used for modeling, simulating, and analyzing dynamic systems. SIMULINK is
not discussed in this text; the interested reader may refer to Introduction to Simulink with
Engineering Applications, ISBN 0−9744239−7−1. Additional information including purchasing
the software may be obtained from The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA
01760−2098. Phone: 508 647−7000, Fax: 508 647−7001, e−mail:
info@mathwork.com and web
site
http://www.mathworks.com.
The author makes no claim to originality of content or of treatment, but has taken care to present
definitions, statements of physical laws, theorems, and problems.
Chapter 1 is an introduction to MATLAB. The discussion is based on MATLAB Student Version
5 and it is also applicable to Version 6. Chapter 2 discusses root approximations by numerical
methods. Chapter 3 is a review of sinusoids and complex numbers. Chapter 4 is an introduction to
matrices and methods of solving simultaneous algebraic equations using Excel and MATLAB.
Chapter 5 is an abbreviated, yet practical introduction to differential equations, state variables,
state equations, eigenvalues and eigenvectors. Chapter 6 discusses the Taylor and Maclaurin
series. Chapter 7 begins with finite differences and interpolation methods. It concludes with
applications using MATLAB. Chapter 8 is an introduction to linear and parabolic regression.
Chapters 9 and 10 discuss numerical methods for differentiation and integration respectively.
Chapter 11 is a brief introduction to difference equations with a few practical applications.
Chapters 12 is devoted to partial fraction expansion. Chapters 13, 14, and 15 discuss certain
interesting functions that find wide application in science, engineering, and probability. This text
concludes with Chapter 16 which discusses three popular optimization methods.
New to the Third Edition
This is an extensive revision of the first edition. The most notable changes are the inclusion of
Fourier series, orthogonal functions and factorization methods, and the solutions to all end−of−
chapter exercises. It is in response to many readers who expressed a desire to obtain the solutions
in order to check their solutions to those of the author and thereby enhancing their knowledge.
Another reason is that this text is written also for self−study by practicing engineers who need a
review before taking more advanced courses such as digital image processing. The author has
prepared more exercises and they are available with their solutions to those instructors who adopt
this text for their class.
Another change is the addition of a rather comprehensive summary at the end of each chapter.
Hopefully, this will be a valuable aid to instructors for preparation of view foils for presenting the
material to their class.
The last major change is the improvement of the plots generated by the latest revisions of the
MATLAB® Student Version, Release 14.
Orchard Publications
Fremont, California
www.orchardpublications.com
info@orchardpublications.com
Table of Contents
1 Introduction to MATLAB 1−1
1.1 Command Window.................................................................................................1−1
1.2 Roots of Polynomials...............................................................................................1−3
1.3 Polynomial Construction from Known Roots ........................................................1−4
1.4 Evaluation of a Polynomial at Specified Values .....................................................1−5
1.5 Rational Polynomials ..............................................................................................1−8
1.6 Using MATLAB to Make Plots..............................................................................1−9
1.7 Subplots.................................................................................................................1−18
1.8 Multiplication, Division and Exponentiation.......................................................1−19
1.9 Script and Function Files......................................................................................1−26
1.10 Display Formats ....................................................................................................1−31
1.11 Summary ...............................................................................................................1−33
1.12 Exercises................................................................................................................1−37
1.13 Solutions to End−of−Chapter Exercises ...............................................................1−38
MATLAB Computations: Entire chapter
2 Root Approximations 2−1
2.1 Newton’s Method for Root Approximation...........................................................2−1
2.2 Approximations with Spreadsheets........................................................................2−7
2.3 The Bisection Method for Root Approximation .................................................2−19
2.4 Summary...............................................................................................................2−27
2.5 Exercises ...............................................................................................................2−28
2.6 Solutions to End−of−Chapter Exercises...............................................................2−29
MATLAB Computations: Pages 2−2 through 2−7, 2−14, 2−21 through 2−23,
2−29 through 2−34
Excel Computations: Pages 2−8 through 2−19, 2−24 through 2−26
3 Sinusoids and Phasors 3−1
3.1 Alternating Voltages and Currents ........................................................................3−1
3.2 Characteristics of Sinusoids....................................................................................3−2
3.3 Inverse Trigonometric Functions .........................................................................3−10
3.4 Phasors..................................................................................................................3−10
3.5 Addition and Subtraction of Phasors ...................................................................3−11
3.6 Multiplication of Phasors......................................................................................3−12
3.7 Division of Phasors ...............................................................................................3−13
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3.8 Exponential and Polar Forms of Phasors ..............................................................3−13
3.9 Summary ...............................................................................................................3−24
3.10 Exercises................................................................................................................3−27
3.11 Solutions to End−of−Chapter Exercises................................................................3−28
MATLAB Computations: Pages 3−15 through 3−23, 3−28 through 3−31
Simulink Modeling: Pages 3−16 through 3−23
4 Matrices and Determinants 4−1
4.1 Matrix Definition.....................................................................................................4−1
4.2 Matrix Operations ...................................................................................................4−2
4.3 Special Forms of Matrices........................................................................................4−5
4.4 Determinants ...........................................................................................................4−9
4.5 Minors and Cofactors ............................................................................................4−13
4.6 Cramer’s Rule ........................................................................................................4−18
4.7 Gaussian Elimination Method...............................................................................4−20
4.8 The Adjoint of a Matrix ........................................................................................4−22
4.9 Singular and Non−Singular Matrices ....................................................................4−22
4.10 The Inverse of a Matrix.........................................................................................4−23
4.11 Solution of Simultaneous Equations with Matrices ..............................................4−25
4.12 Summary................................................................................................................4−32
4.13 Exercises ................................................................................................................4−36
4.14 Solutions to End−of−Chapter Exercises ................................................................4−38
MATLAB Computations: Pages 4−3, 4−5 through 4−8, 4−10, 4−12, 4−3, 4−5, 4−19
through 4−20, 4−24, 4−26, 4−28, 4−30, 4−38, 4−41, 4−43
Excel Computations: Pages 4−28 through 4−29, 4−42 through 4−43
5 Differential Equations, State Variables, and State Equations 5−1
5.1 Simple Differential Equations..................................................................................5−1
5.2 Classification............................................................................................................5−2
5.3 Solutions of Ordinary Differential Equations (ODE) .............................................5−6
5.4 Solution of the Homogeneous ODE ...................................................................... 5−8
5.5 Using the Method of Undetermined Coefficients for the Forced Response........ 5−10
5.6 Using the Method of Variation of Parameters for the Forced Response ............. 5−20
5.7 Expressing Differential Equations in State Equation Form.................................. 5−24
5.8 Solution of Single State Equations ....................................................................... 5−27
5.9 The State Transition Matrix ................................................................................ 5−28
5.10 Computation of the State Transition Matrix ...................................................... 5−30
5.11 Eigenvectors.......................................................................................................... 5−38
5.12 Summary.............................................................................................................. 5−42
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5.13 Exercises ............................................................................................................... 5−47
5.14 Solutions to End−of−Chapter Exercises............................................................... 5−49
MATLAB Computations: Pages 5−11, 5−13 through 5−14, 5−16 through 5−17,
5−19, 5−23, 5−33 through 5−35, 5−37,
5−49 through 5−53, 5−55
6 Fourier, Taylor, and Maclaurin Series 6−1
6.1 Wave Analysis ........................................................................................................6−1
6.2 Evaluation of the Coefficients ...............................................................................6−2
6.3 Symmetry ...............................................................................................................6−7
6.4 Waveforms in Trigonometric Form of Fourier Series .........................................6−12
6.5 Alternate Forms of the Trigonometric Fourier Series .........................................6−25
6.6 The Exponential Form of the Fourier Series .......................................................6−29
6.7 Line Spectra .........................................................................................................6−33
6.8 Numerical Evaluation of Fourier Coefficients .....................................................6−36
6.9 Power Series Expansion of Functions ..................................................................6−40
6.10 Taylor and Maclaurin Series ................................................................................6−41
6.11 Summary ..............................................................................................................6−48
6.12 Exercises ..............................................................................................................6−51
6.13 Solutions to End−of−Chapter Exercises ..............................................................6−53
MATLAB Computations: Pages 6−35, 6−45, 6−58 through 6−61
Excel Computations: Pages 6−37 through 6−39
7 Finite Differences and Interpolation 7−1
7.1 Divided Differences ...............................................................................................7−1
7.2 Factorial Polynomials .............................................................................................7−6
7.3 Antidifferences ...................................................................................................7−12
7.4 Newton’s Divided Difference Interpolation Method .........................................7−15
7.5 Lagrange’s Interpolation Method ........................................................................7−17
7.6 Gregory−Newton Forward Interpolation Method ..............................................7−19
7.7 Gregory−Newton Backward Interpolation Method ...........................................7−21
7.8 Interpolation with MATLAB .............................................................................7−24
7.9 Summary .............................................................................................................7−39
7.10 Exercises .............................................................................................................7−44
7.11 Solutions to End−of−Chapter Exercises .............................................................7−45
MATLAB Computations: Pages 7−8 through 7−9, 7−13 through 7−15,
7−26 through 7−38, 7−45 through 7−46,
7−48, 7−50, 7−52
Excel Computations: Pages 7−17 through 7−19, 7−22 through 7−25, 7−49, 7−52
iv Numerical Analysis Using MATLAB® and Excel®, Third Edition
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8 Linear and Parabolic Regression 8−1
8.1 Curve Fitting ..........................................................................................................8−1
8.2 Linear Regression ...................................................................................................8−2
8.3 Parabolic Regression ..............................................................................................8−7
8.4 Regression with Power Series Approximations ....................................................8−14
8.5 Summary ..............................................................................................................8−24
8.6 Exercises ...............................................................................................................8−26
8.7 Solutions to End−of−Chapter Exercises ...............................................................8−28
MATLAB Computations: Pages 8−11 through 8−14, 8−17 through 8−23,
8−30 through 8−34
Excel Computations: Pages 8−5 through 8−10, 8−15 through 8−19, 8−28 through 8−32
9 Solution of Differential Equations by Numerical Methods 9−1
9.1 Taylor Series Method ............................................................................................ 9−1
9.2 Runge−Kutta Method ............................................................................................ 9−5
9.3 Adams’ Method ................................................................................................... 9−13
9.4 Milne’s Method .................................................................................................... 9−15
9.5 Summary .............................................................................................................. 9−17
9.6 Exercises .............................................................................................................. 9−20
9.7 Solutions to End−of−Chapter Exercises .............................................................. 9−21
MATLAB Computations: Pages 9−5, 9−9 through 9−12, 9−21 through 9−23
Excel Computations: Page 9−2, 9−14, 9−22 through 9−26
10 Integration by Numerical Methods 10−1
10.1 The Trapezoidal Rule .......................................................................................... 10−1
10.2 Simpson’s Rule ..................................................................................................... 10−6
10.3 Summary ............................................................................................................ 10−14
10.4 Exercises ............................................................................................................ 10−15
10.5 Solution to End−of−Chapter Exercises .............................................................. 10−16
MATLAB Computations: Pages 10−3 through 10−6, 10−9 through 10−13,
10−16, 10−18 through 10−21
Excel Computations: Pages 10−10, 10−19 through 10−21
11 Difference Equations 11−1
11.1 Introduction ......................................................................................................... 11−1
11.2 Definition, Solutions, and Applications .............................................................. 11−1
11.3 Fibonacci Numbers .............................................................................................. 11−7
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11.4 Summary .............................................................................................................11−11
11.5 Exercises ............................................................................................................. 11−13
11.6 Solutions to End−of−Chapter Exercises .............................................................11−14
12 Partial Fraction Expansion 12−1
12.1 Partial Fraction Expansion ..................................................................................12−1
12.2 Alternate Method of Partial Fraction Expansion ..............................................12−13
12.3 Summary ............................................................................................................12−19
12.4 Exercises ............................................................................................................12−22
12.5 Solutions to End−of−Chapter Exercises ............................................................12−23
MATLAB Computations: Pages 12−3 through 12−5, 12−9 through 12−12,
12−16 through 12-18, 12−23 through 12−28
13 The Gamma and Beta Functions and Distributions 13−1
13.1 The Gamma Function .........................................................................................13−1
13.2 The Gamma Distribution ..................................................................................13−16
13.3 The Beta Function .............................................................................................13−17
13.4 The Beta Distribution ........................................................................................13−20
13.5 Summary ............................................................................................................13−22
13.6 Exercises ............................................................................................................13−24
13.7 Solutions to End−of−Chapter Exercises ............................................................13−25
MATLAB Computations: Pages 13−3, 13−5, 13−10, 13−19, 13−25
Excel Computations: Pages 13−5, 13−10, 13−16 through 13−17, 13−19, 13−21
14 Orthogonal Functions and Matrix Factorizations 14−1
14.1 Orthogonal Functions ......................................................................................14−1
14.2 Orthogonal Trajectories ...................................................................................14−2
14.3 Orthogonal Vectors ..........................................................................................14−4
14.4 The Gram−Schmidt Orthogonalization Procedure ..........................................14−7
14.5 The LU Factorization .......................................................................................14−9
14.6 The Cholesky Factorization ............................................................................14−23
14.7 The QR Factorization .....................................................................................14−25
14.8 Singular Value Decomposition .......................................................................14−28
14.9 Summary .........................................................................................................14−30
14.10 Exercises .........................................................................................................14−32
14.11 Solutions to End−of−Chapter Exercises .........................................................14−34
MATLAB Computations: Pages 14−8 through 14−9, 14−11 through 14−29,
14−36, 14−38 through 14−39
vi Numerical Analysis Using MATLAB® and Excel®, Third Edition
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15 Bessel, Legendre, and Chebyshev Functions 15−1
15.1 The Bessel Function ............................................................................................15−1
15.2 Legendre Functions ...........................................................................................15−10
15.3 Laguerre Polynomials .........................................................................................15−21
15.4 Chebyshev Polynomials .....................................................................................15−22
15.5 Summary ............................................................................................................15−27
15.6 Exercises .............................................................................................................15−32
15.7 Solutions to End−of−Chapter Exercises ............................................................15−33
MATLAB Computations: Pages 15−3 through 15−4, 15−6, 15−9,
14−19 through 15−22, 15−25, 15−33, 15−35 through 15−37
Excel Computations: Pages 15−5, 15−9
16 Optimization Methods 16−1
16.1 Linear Programming ........................................................................................... 16−1
16.2 Dynamic Programming ........................................................................................16−4
16.3 Network Analysis ...............................................................................................16−14
16.4 Summary ............................................................................................................16−19
16.5 Exercises .............................................................................................................15−20
16.6 Solutions to End−of−Chapter Exercises ............................................................15−22
MATLAB Computations: Pages 16−3
Excel Computations: Pages 16−4, 16−23, 16−25 through 16−27
A Difference Equations in Discrete−Time Systems A−1
A.1 Recursive Method for Solving Difference Equations........................................... A−1
A.2 Method of Undetermined Coefficients ................................................................A−1
MATLAB Computations: Pages A−4, A−7, A−9
B Introduction to Simulink® B−1
B.1 Simulink and its Relation to MATLAB ...............................................................B−1
B.2 Simulink Demos ..................................................................................................B−20
MATLAB Computations and Simulink Modeling: Entire Appendix B
C Ill-Conditioned Matrices C−1
C.1 The Norm of a Matrix ...........................................................................................C−1
C.2 Condition Number of a Matrix .............................................................................C−2
C.3 Hilbert Matrices ....................................................................................................C−3
Numerical Analysis Using MATLAB® and Excel®, Third Edition vii
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MATLAB Computations: Pages C−1, C−4 through C−5
References R−1
Index IN−1
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