Applied Numerical Analysis

620 Pages·2006·21.7 MB·English

Seventh Ed )n
bA,ani. "
merica
Patrick 0.W heatley
California Polytechnic State University
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Library of Congress Cataloging-in-Publication Data
Gerald, Curtis F., 1915-
Applied numerical analysis1Curtis F. Gerald, Patrick 0. Wheat1ey.-7th ed.
p. cm.
Includes bibliographical references and index.
ISBN 0-321-13304-8
1. Numerical analysis. I. Wheatley, Patrick 0.1 1. Title.
Copyright O 2004 Pearson Education, Inc.
All rights reserved. No part of this publication may be reproduced, stored in retrieval system, or transmitted,
in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior
written permission of the publisher. Printed in the United States of America.
Preface ix
0 Preliminaries 1
Contents of This Chapter 1
0.1 Analysis Versus Numerical Analysis 2
0.2 Computers and Numerical Analysis 4
0.3 An Illustrative Example 6
0.4 Kinds of Errors in Numerical Procedures 10
0.5 Interval Arithmetic 19
0.6 Parallel and Distributed Computing 21
0.7 Measuring the Efficiency of Numerical Procedures 26
Exercises 28
Applied Problems and Projects 30
'I C
1 Solving Nonlinear Equations 32
-
Contents of This Chapter 33
1.1 Interval Halving (Bisection) 33
1.2 Linear Interpolation Methods 38
1.3 Newton's Method 42
1.4 Muller's Method 50
Contents
1.5 Fixed-Point Iteration: x = g(x) Method 54
1.6 Multiple Roots 60
1.7 Nonlinear Systems 63
Exercises 67
Applied Problems and Projects 71
2 Solving Sets of Equations 76
Contents of This Chapter 76
2.1 Matrices and Vectors 77
2.2 Elimination Methods 88
2.3 The Inverse of a Matrix and Matrix Pathology 106
2.4 Ill-Conditioned Systems 110
2.5 Iterative Methods 121
2.6 Parallel Processing 129
Exercises 135
Applied Problems and Projects 141
3 Interpolation and Curve Fitting 147
Contents of This Chapter 148
Interpolating Polynomials 149
Divided Differences 157
Spline Curves 168
Bezier Curves and B-Splines Curves 179
Interpolating on a Surface 188
Least-Squares Approximations 199
Exercises 209
Applied Problems and Projects 215
Approximation of Functions 220
Contents of This Chapter 220
Contents vii
4.1 Chebyshev Polynomials and Chebyshev Series 221
4.2 Rational Function Approximations 232
4.3 Fourier Series 240
Exercises 252
Applied Problems and Projects 254
5 Numerical Differentiation and ntegration 256
Contents of This Chapter 257
Differentiation with a Computer 258
Numerical Integration-The Trapezoidal Rule 272
Simpson's Rules 280
An Application of Numerical Integration-Fourier Series and Fourier Transforms 285
Adaptive Integration 297
Gaussian Quadrature 301
Multiple Integrals 307
Applications of Cubic Splines 3 17
Exercises 321
Applied Problems and Projects 326
6 Numerical Solution of Ordinary
Differential Equations 329
Contents of This Chapter 330
The Taylor-Series Method 332
The Euler Method and Its Modifications 335
Runge- Kutta Methods 340
Multistep Methods 347
Higher-Order Equations and Systems 359
Stiff Equations 364
Boundary-Value Problems 366
Characteristic-ValueP roblems 38 1
Exercises 394
Applied Problems and Projects 399
viii Contents
7 Optimization 485
Contents of This Chapter 405
Finding the Minimum of y = f(x) 406
Minimizing a Function of Several Variables 417
Linear Programming 428
Nonlinear Programming 442
Other Optimizations 449
Exercises 453
Applied Problems and Projects 458
Partial-Differential Equations 461
Contents of This Chapter 463
8.1 Elliptic Equations 463
.2 Parabolic Equations 48 1
.3 Hyperbolic Equations 499
Exercises 509
Applied Problems and Projects 513
lement Analysis 517
Contents of This Chapter 5 18
9.1 Mathematical Background 5 18
9.2 Finite Elements for Ordinary-Differential Equations 526
9.3 Finite Elements for Partial-Differential Equations 535
Exercises 562
Applied Problems and Projects 564
Appendixes
A Some Basic Information from Calculus 567
B Software Resources 571
Answers to Selected Exercises 575
References 599
In this seventh edition, we continue on the path established in previous editions. Quoting
from the preface of the sixth edition, we "retain the same features that have made the book
popular: ease of reading so that the instructor does not have to 'interpret the book' for the
student, many illustrative examples that often solve the same problem with different pro-
cedures to clarify the comparison of methods, many exercises from which the instructor
may choose appropriately for the class, more challenging problems and projects that show
practical applications of the material."
We have made substantial improvements on the previous edition. These include:
Theoretical matters that previously were in a separa1:e section near the end of each chap-
ter have been merged with the description of the procedures.
Example computer programs that admittedly were not of professional quality have
been deleted, with the idea that this is not normallly a programming course anyway.
Easy-to-read algorithms have been retained so that students can write programs if they
desire.
There is greater emphasis on computer algebra systems; MATLAB is the predominant
system, but this is compared with Maple and Mathcmatica. The use of spreadsheets to
solve problems is covered as well.
A new chapter on optimization (Chapter 7) has been added that includes multivariable
cases as well as single-variable situations. Linear programming has been included, of
course, but the treatment is intended to provide a real understanding of the simplex
method rather than to merely give a recipe for solving the problem. Nonlinear program-
ming is treated to contrast this with the simpler linear case.
Boundary value problems for ordinary diffferential equations have been separated from
those for partial differential equations and are inclutded in the chapter on ordinary dif-
ferential equations. Partial differential equations that satisfy boundary conditions (ellip-
tic equations) are combined with the other types of partial differential equations in a
single chapter.
Preface
Many exercises have been modified or rewritten to provide an even greater variety. New
exercises and projects have been added and some of these are more challenging than in
the previous edition.
As in previous editions, this book is unique in its inclusion of a thorough survey of
numerical methods for solving partial differential equations and an introduction to the
finite element method.
Many suggestions from reviewers have allowed us to clarify and extend the treatment of
several topics and we have made editorial changes to make the book easier to read and
understand.
We again quote from the preface to the sixth edition:
Applied Numerical Analysis is written as a text for sophomores and juniors in engi-
neering, science, mathematics, and computer science. It should be a valuable source
book for practicing engineers. Because of its coverage of many numerical methods, the
text can serve as a valuable reference.
Although we assume that the student has a good knowledge of calculus, appropriate
topics are reviewed in the context of their use. An appendix gives a summary of
the most important items that are needed to develop and analyze numerical procedures.
We purposely keep the mathematical notation simple for clarity. Furthermore, the
answers to exercises marked with a b are found in the back of the text.
Acknowledgements
Many instructors have given valuable suggestions and constructive criticism. We mention
those whose thorough reviews have helped make this edition better:
Todd Arbogast, University of Texas at Austin
Neil Berger, University of Zllinois at Chicago
Barbara Bertram, Michigan Technological Sciences
Herman Gollwitzer, Drexel University
Chenyi Hu, University of Houston-Downtown
Tim Sauer, George Mason University
Daoqi Yang, Wayne State University
Kathie Yerion, Gonzaga University
We also want to express our thanks to those at Addison-Wesley who have
worked extensively with us to ensure the publication of another quality edition:
Greg Tobin, Bill Hoffman, RoseAnne Johnson, Cindy Cody, Pam Laskey, Heather Peck,
and Barbara Atkinson.
relirnina
This book teaches how a computer can be used to solve problems that may not be solvable
by the techniques that are taught in most calculus courses. It also shows how those prob-
lems that you may have solved before can be solved in a different way. Our emphasis is on
problems that exist in the real world, although these examples will be simplified. Many of
these simplified examples can be solved analytically, which allows a comparison with the
computer-derived solution.
Modern mathematics began when Isaac Newton found mathematical models that
matched the empirical laws that Johannes Kepler had reached after about 20 years of
observation of the planets. Today, most of applied m~athematicsi s a repetition of what
Newton did: to develop mathematical relationships that: can be used to simulate some real-
world situation and to predict its response to different external factors.
The beauty of mathematics is that it builds on simple cases to arrive at more complex
and useful ones. This is true for this book-we start with mathematical applications that
are easily understood but that become the basis for other, more important applications of
numerical analysis.
Contents o f fh,is, Chapter
'
We begin each chapter of this book with a list of the topics that are discussed in that
chapter.
0.1 Analysis Versus Numerical Analysis
Describes how numerical analysis differs from analytical analysis and shows
where each has special advantages. It briefly lists the topics that will be
covered in later chapters.