Numerical Methods for Engineers and Scientists Using MATLAB®, Second Edition.

495 Pages·2017·19.747 MB·English

by  Esfandiari, Ramin S.

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• Demonstrates effective applications of MATLAB user-defined functions r
or script files, and built-in functions when available, for all methods s
discussed
• Employs a direct, pragmatic and easy-to-follow approach in presenting
numerical methods and their applications in engineering and science
• Shows readers meticulously worked-out examples using hand calcula-
tions and verification of results by executing user-defined functions, and
when applicable, the built-in MATLAB functions SECOND
• Provides exercises of various degrees of difficulty for each topic
EDITION
K29770
Ramin S. Esfandiari
6000 Broken Sound Parkway, NW
Suite 300, Boca Raton, FL 33487
711 Third Avenue
New York, NY 10017
2 Park Square, Milton Park
Abingdon, Oxon OX14 4RN, UK
Numerical Methods for
Engineers and Scientists
Using MATLAB®
Second Edition

Numerical Methods for
Engineers and Scientists
Using MATLAB®
Second Edition
Ramin S. Esfandiari, PhD
MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the
accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products
does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particu-
lar use of the MATLAB® software.
CRC Press
Taylor & Francis Group,
6000 Broken Sound Parkway NW, Suite 300,
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© 2017 by Taylor & Francis Group, LLC
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Library of Congress Cataloging–in–Publication Data
Names: Esfandiari, Ramin S., author.
Title: Numerical methods for engineers and scientists using MATLAB / Ramin
S. Esfandiari.
Description: Second edition. | Boca Raton : a CRC title, part of the Taylor &
Francis imprint, a member of the Taylor & Francis Group, the academic
division of T&F Informa, plc, [2017]
Identifiers: LCCN 2016039623 | ISBN 9781498777421 (hardback : alk. paper)
Subjects: LCSH: Engineering mathematics. | Numerical analysis.
Classification: LCC TA335 .E843 2017 | DDC 620.00285/53--dc23
LC record available at https://lccn.loc.gov/2016039623
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
To
My wife Haleh, my sisters Mandana and Roxana, and my parents to whom I owe everything

Contents
Preface .............................................................................................................................................xv
Acknowledgments ......................................................................................................................xix
Author ...........................................................................................................................................xxi
1. Background and Introduction ..............................................................................................1
Part 1: Background ..................................................................................................................1
1.1 Differential Equations ..................................................................................................1
1.1.1 Linear, First-Order ODEs ................................................................................1
1.1.2 Second-Order ODEs with Constant Coefficients ........................................2
1.1.2.1 Homogeneous Solution ...................................................................2
1.1.2.2 Particular Solution ...........................................................................3
1.1.3 Method of Undetermined Coefficients .........................................................3
1.2 Matrix Analysis .............................................................................................................4
1.2.1 Matrix Operations............................................................................................5
1.2.2 Matrix Transpose .............................................................................................5
1.2.3 Special Matrices ...............................................................................................6
1.2.4 Determinant of a Matrix .................................................................................6
1.2.5 Properties of Determinant ..............................................................................6
1.2.5.1 Cramer’s Rule ....................................................................................7
1.2.6 Inverse of a Matrix ...........................................................................................8
1.2.7 Properties of Inverse ........................................................................................9
1.2.8 Solving a Linear System of Equations ..........................................................9
1.3 Matrix Eigenvalue Problem .........................................................................................9
1.3.1 Solving the Eigenvalue Problem ..................................................................10
1.3.2 Similarity Transformation ............................................................................11
1.3.3 Matrix Diagonalization .................................................................................11
1.3.4 Eigenvalue Properties of Matrices ...............................................................12
Part 2: Introduction to Numerical Methods .......................................................................12
1.4 Errors and Approximations .......................................................................................12
1.4.1 Sources of Computational Error ..................................................................12
1.4.2 Binary and Hexadecimal Numbers ............................................................13
1.4.3 Floating Point and Rounding Errors ...........................................................13
1.4.4 Round-Off: Chopping and Rounding .........................................................14
1.4.5 Absolute and Relative Errors .......................................................................15
1.4.6 Error Bound ....................................................................................................16
1.4.7 Transmission of Error from a Source to the Final Result .........................16
1.4.8 Subtraction of Nearly Equal Numbers .......................................................17
1.5 Iterative Methods ........................................................................................................19
1.5.1 Fundamental Iterative Method ....................................................................20
1.5.2 Rate of Convergence of an Iterative Method ..............................................21
Problem Set (Chapter 1) ........................................................................................................22
2. Introduction to MATLAB® .................................................................................................27
2.1 MATLAB Built-In Functions .....................................................................................27
vii
viii Contents
2.1.1 Rounding Commands ...................................................................................27
2.1.2 Relational Operators ......................................................................................28
2.1.3 Format Options ..............................................................................................28
2.2 Vectors and Matrices ..................................................................................................29
2.2.1 Linspace...........................................................................................................30
2.2.2 Matrices ...........................................................................................................30
2.2.3 Determinant, Transpose, and Inverse .........................................................32
2.2.4 Slash Operators ..............................................................................................33
2.2.5 Element-by-Element Operations ..................................................................33
2.2.6 Diagonal Matrices and Diagonals of a Matrix ...........................................34
2.3 Symbolic Math Toolbox ..............................................................................................36
2.3.1 Anonymous Functions ..................................................................................38
2.3.2 MATLAB Function ........................................................................................38
2.3.3 Differentiation ................................................................................................39
2.3.4 Partial Derivatives .........................................................................................40
2.3.5 Integration .......................................................................................................40
2.4 Program Flow Control ................................................................................................41
2.4.1 for Loop .........................................................................................................41
2.4.2 The if Command ..........................................................................................42
2.4.3 while Loop ....................................................................................................43
2.5 Displaying Formatted Data .......................................................................................43
2.5.1 Differential Equations ...................................................................................44
2.6 Plotting .........................................................................................................................45
2.6.1 subplot ..........................................................................................................45
2.6.2 Plotting Analytical Expressions ..................................................................46
2.6.3 Multiple Plots ..................................................................................................46
2.7 User-Defined Functions and Script Files .................................................................47
2.7.1 Setting Default Values for Input Variables .................................................49
2.7.2 Creating Script Files ......................................................................................50
Problem Set (Chapter 2) ........................................................................................................51
3. Numerical Solution of Equations of a Single Variable .................................................55
3.1 Numerical Solution of Equations ..............................................................................55
3.2 Bisection Method ........................................................................................................55
3.2.1 MATLAB Built-In Function fzero .............................................................60
3.3 Regula Falsi Method (Method of False Position) ....................................................61
3.3.1 Modified Regula Falsi Method ....................................................................64
3.4 Fixed-Point Method ....................................................................................................65
3.4.1 Selection of a Suitable Iteration Function ...................................................66
3.4.2 A Note on Convergence ................................................................................67
3.4.3 Rate of Convergence of the Fixed-Point Iteration ......................................71
3.5 Newton’s Method (Newton–Raphson Method) .....................................................72
3.5.1 Rate of Convergence of Newton’s Method .................................................76
3.5.2 A Few Notes on Newton’s Method .............................................................77
3.5.3 Modified Newton’s Method for Roots with Multiplicity
2 or Higher ......................................................................................................78
3.6 Secant Method .............................................................................................................81
3.6.1 Rate of Convergence of Secant Method ......................................................83
3.6.2 A Few Notes on Secant Method ..................................................................83
Contents ix
3.7 Equations with Several Roots ....................................................................................83
3.7.1 Finding Roots to the Right of a Specified Point .........................................83
3.7.2 Finding Several Roots in an Interval Using fzero ..................................84
Problem Set (Chapter 3) ........................................................................................................88
4. Numerical Solution of Systems of Equations .................................................................95
4.1 Linear Systems of Equations .....................................................................................95
4.2 Numerical Solution of Linear Systems.....................................................................96
4.3 Gauss Elimination Method ........................................................................................96
4.3.1 Choosing the Pivot Row: Partial Pivoting with Row Scaling .................98
4.3.2 Permutation Matrices ....................................................................................99
4.3.3 Counting the Number of Operations ........................................................102
4.3.3.1 Elimination ....................................................................................102
4.3.3.2 Back Substitution ..........................................................................103
4.3.4 Tridiagonal Systems ....................................................................................103
4.3.4.1 Thomas Method ...........................................................................104
4.3.4.2 MATLAB Built-In Function "\" .................................................106
4.4 LU Factorization Methods .......................................................................................107
4.4.1 Doolittle Factorization .................................................................................107
4.4.2 Finding L and U Using Steps of Gauss Elimination ...............................108
4.4.3 Finding L and U Directly............................................................................108
4.4.3.1 Doolittle’s Method to Solve a Linear System ............................110
4.4.3.2 Operations Count .........................................................................112
4.4.4 Cholesky Factorization ................................................................................112
4.4.4.1 Cholesky’s Method to Solve a Linear System ...........................113
4.4.4.2 Operations Count .........................................................................115
4.4.4.3 MATLAB Built-In Functions lu and chol ...............................115
4.5 Iterative Solution of Linear Systems .......................................................................116
4.5.1 Vector Norms ................................................................................................116
4.5.2 Matrix Norms ...............................................................................................118
4.5.2.1 Compatibility of Vector and Matrix Norms .............................119
4.5.3 General Iterative Method ............................................................................120
4.5.3.1 Convergence of the General Iterative Method .........................120
4.5.4 Jacobi Iteration Method ...............................................................................121
4.5.4.1 Convergence of the Jacobi Iteration Method ............................122
4.5.5 Gauss–Seidel Iteration Method ..................................................................125
4.5.5.1 Convergence of the Gauss–Seidel Iteration Method ...............127
4.5.6 Indirect Methods versus Direct Methods for Large Systems ................130
4.6 Ill-Conditioning and Error Analysis ......................................................................131
4.6.1 Condition Number.......................................................................................131
4.6.2 Ill-Conditioning ...........................................................................................132
4.6.2.1 Indicators of Ill-Conditioning .....................................................133
4.6.3 Computational Error ...................................................................................133
4.6.3.1 Consequences of Ill-Conditioning .............................................135
4.6.4 Effects of Parameter Changes on the Solution ........................................136
4.7 Systems of Nonlinear Equations .............................................................................138
4.7.1 Newton’s Method for a System of Nonlinear Equations ........................138
4.7.1.1 Newton’s Method for Solving a System of
Two Nonlinear Equations ...........................................................138