Numerical analysis

570 Pages·2004·9.235 MB·English

by  Steven T. Karris

Numerical
Analysis
Using MATLAB® and Spreadsheets
Second Edition
Steven T. Karris
Orchard Publications
www.orchardpublications.com
Students and working professionals will
Numerical Analysis
find Numerical Analysis Using MATLAB®
and Spreadsheets, Second Edition, to be a
Using MATLAB® and Spreadsheets concise and easy-to-learn text. It provides
complete, clear, and detailed explanations
Second Edition
of the principal numerical analysis meth-
ods and well known functions used in sci-
ence and engineering. These are illustrat-
ed with many real-world examples.
This text includes the following chapters:
• Introduction to MATLAB • Root Approximations • Sinusoids and Complex Numbers • Matrices
and Determinants • Review of Differential Equations • Fourier, Taylor, and Maclaurin Series
• Finite Differences and Interpolation • Linear and Parabolic Regression • Solution of Differential
Equations by Numerical Methods • Integration by Numerical Methods • Difference Equations
• Partial Fraction Expansion • The Gamma and Beta Functions • Orthogonal Functions and Matrix
Factorizations • Bessel, Legendre, and Chebyshev Polynomials • Optimization Methods
Each chapter contains numerous practical applications supplemented with detailed instructions
for using MATLAB and/or Microsoft Excel® to obtain quick solutions.
Steven T. Karris is the president and founder of Orchard Publications. He earned a bachelors
degree in electrical engineering at Christian Brothers University, Memphis, Tennessee, a masters
degree in electrical engineering at Florida Institute of Technology, Melbourne, Florida, and has
done post-master work at the latter. He is a registered professional engineer in California and
Florida. He has over 30 years of professional engineering experience in industry. In addition, he
has over 25 years of teaching experience that he acquired at several educational institutions as
an adjunct professor. He is currently with UC Berkeley Extension.
Orchard Publications
Visit us on the Internet
www.orchardpublications.com
or email us: [email protected]
ISBN 0-9744239-1-2
$39.95 U.S.A.
Numerical Analysis
Using MATLAB and Spreadsheets
Second Edition
Steven T. Karris
Orchard Publications
www.orchardpublications.com
Numerical Analysis Using MATLAB and Spreadsheets, Second Edition
Copyright ” 2004 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, [email protected]
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 (LCCN) 2003099336
Copyright TX 5-589-152
ISBN 0-9744239-1-2
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 13 and includes SIMULINK which is a
* These topics are discussed in Mathematics for Business, Science, and Technology by this author, Orchard Publications,
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 the documentation which also
includes demo models with detailed explanations. Additional information including purchasing
may be obtained from The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098.
Phone: [phone], Fax: [phone], e-mail: [email protected] 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 Second 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 13.
Orchard Publications
Fremont, California
www.orchardpublications.com
[email protected]
Table of Contents
Chapter 1
Introduction to MATLAB
Command Window.......................................................................................................................1-1
Roots of Polynomials.....................................................................................................................1-3
Polynomial Construction from Known Roots..............................................................................1-4
Evaluation of a Polynomial at Specified Values...........................................................................1-5
Rational Polynomials....................................................................................................................1-7
Using MATLAB to Make Plots....................................................................................................1-9
Subplots.......................................................................................................................................1-18
Multiplication, Division and Exponentiation.............................................................................1-18
Script and Function Files............................................................................................................1-25
Display Formats..........................................................................................................................1-29
Summary.....................................................................................................................................1-30
Exercises......................................................................................................................................1-35
Solutions to Exercises.................................................................................................................1-36
Chapter 2
Root Approximations
Newton’s Method for Root Approximation.................................................................................2-1
Approximations with Spreadsheets..............................................................................................2-7
The Bisection Method for Root Approximation........................................................................2-19
Summary.....................................................................................................................................2-27
Exercises......................................................................................................................................2-28
Solutions to Exercises.................................................................................................................2-29
Chapter 3
Sinusoids and Phasors
Alternating Voltages and Currents..............................................................................................3-1
Characteristics of Sinusoids..........................................................................................................3-2
Inverse Trigonometric Functions...............................................................................................3-10
Phasors........................................................................................................................................3-10
Addition and Subtraction of Phasors.........................................................................................3-11
Multiplication of Phasors............................................................................................................3-12
Division of Phasors.....................................................................................................................3-12
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Exponential and Polar Forms of Phasors....................................................................................3-13
Summary......................................................................................................................................3-18
Exercises......................................................................................................................................3-21
Solutions to Exercises..................................................................................................................3-22
Chapter 4
Matrices and Determinants
Matrix Definition.........................................................................................................................4-1
Matrix Operations........................................................................................................................4-2
Special Forms of Matrices............................................................................................................4-5
Determinants................................................................................................................................4-9
Minors and Cofactors.................................................................................................................4-12
Cramer’s Rule.............................................................................................................................4-16
Gaussian Elimination Method...................................................................................................4-18
The Adjoint of a Matrix.............................................................................................................4-19
Singular and Non-Singular Matrices.........................................................................................4-20
The Inverse of a Matrix..............................................................................................................4-21
Solution of Simultaneous Equations with Matrices...................................................................4-23
Summary.....................................................................................................................................4-29
Exercises.....................................................................................................................................4-33
Solutions to Exercises.................................................................................................................4-35
Chapter 5
Differential Equations, State Variables, and State Equations
Simple Differential Equations.......................................................................................................5-1
Classification.................................................................................................................................5-2
Solutions of Ordinary Differential Equations (ODE)...................................................................5-5
Solution of the Homogeneous ODE.............................................................................................5-8
Using the Method of Undetermined Coefficients for the Forced Response...............................5-10
Using the Method of Variation of Parameters for the Forced Response....................................5-19
Expressing Differential Equations in State Equation Form........................................................5-23
Solution of Single State Equations..............................................................................................5-27
The State Transition Matrix.......................................................................................................5-28
Computation of the State Transition Matrix..............................................................................5-30
Eigenvectors................................................................................................................................5-37
Summary......................................................................................................................................5-41
Exercises......................................................................................................................................5-46
Solutions to Exercises..................................................................................................................5-47
ii Numerical Analysis Using MATLAB and Spreadsheets, Second Edition
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Chapter 6
Fourier, Taylor, and Maclaurin Series
Wave Analysis..............................................................................................................................6-1
Evaluation of the Coefficients......................................................................................................6-2
Symmetry......................................................................................................................................6-7
Waveforms in Trigonometric Form of Fourier Series.................................................................6-12
Alternate Forms of the Trigonometric Fourier Series................................................................6-25
The Exponential Form of the Fourier Series..............................................................................6-28
Line Spectra................................................................................................................................6-33
Numerical Evaluation of Fourier Coefficients............................................................................6-36
Power Series Expansion of Functions.........................................................................................6-37
Taylor and Maclaurin Series.......................................................................................................6-40
Summary.....................................................................................................................................6-47
Exercises......................................................................................................................................6-50
Solutions to Exercises.................................................................................................................6-52
Chapter 7
Finite Differences and Interpolation
Divided Differences......................................................................................................................7-1
Factorial Polynomials....................................................................................................................7-6
Antidifferences............................................................................................................................7-11
Newton’s Divided Difference Interpolation Method.................................................................7-15
Lagrange’s Interpolation Method...............................................................................................7-18
Gregory-Newton Forward Interpolation Method.......................................................................7-19
Gregory-Newton Backward Interpolation Method....................................................................7-20
Interpolation with MATLAB.....................................................................................................7-23
Summary.....................................................................................................................................7-37
Exercises......................................................................................................................................7-42
Solutions to Exercises.................................................................................................................7-43
Chapter 8
Linear and Parabolic Regression
Curve Fitting.................................................................................................................................8-1
Linear Regression..........................................................................................................................8-2
Parabolic Regression.....................................................................................................................8-7
Regression with Power Series Approximations..........................................................................8-14
Summary.....................................................................................................................................8-24
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Exercises.....................................................................................................................................8-26
Solutions to Exercises.................................................................................................................8-28
Chapter 9
Solution of Differential Equations by Numerical Methods
Taylor Series Method...................................................................................................................9-1
Runge-Kutta Method...................................................................................................................9-5
Adams’ Method..........................................................................................................................9-13
Milne’s Method..........................................................................................................................9-16
Summary.....................................................................................................................................9-17
Exercises.....................................................................................................................................9-20
Solutions to Exercises.................................................................................................................9-21
Chapter 10
Integration by Numerical Methods
The Trapezoidal Rule.................................................................................................................10-1
Simpson’s Rule...........................................................................................................................10-6
Summary...................................................................................................................................10-13
Exercises...................................................................................................................................10-15
Solution to Exercises................................................................................................................10-16
Chapter 11
Difference Equations
Definition, Solutions, and Applications.....................................................................................11-1
Fibonacci Numbers....................................................................................................................11-7
Summary...................................................................................................................................11-10
Exercises...................................................................................................................................11-13
Solutions to Exercises...............................................................................................................11-14
Chapter 12
Partial Fraction Expansion
Partial Fraction Expansion.........................................................................................................12-1
Alternate Method of Partial Fraction Expansion....................................................................12-13
Summary...................................................................................................................................12-18
Exercises...................................................................................................................................12-21
Solutions to Exercises...............................................................................................................12-22
iv Numerical Analysis Using MATLAB and Spreadsheets, Second Edition
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