numerical analysis 9th edition.pdf

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Cover Page

Title Page

CONTENTS

Preface

About the Text

Algorithms and Programs

New for This Edition

Supplements

Possible Course Suggestions

Acknowledgments

Chapter 1: Mathematical Preliminaries and Error Analysis

1.1: Review of Calculus

1.2: Round-off Errors and Computer Arithmetic

1.3: Algorithms and Convergence

1.4: Numerical Software

Chapter 2: Solutions of Equations in One Variable

2.1: The Bisection Method

2.2: Fixed-Point Iteration

2.3: Newton’s Method and Its Extensions

2.4: Error Analysis for Iterative Methods

2.5: Accelerating Convergence

2.6: Zeros of Polynomials and Müller’s Method

2.7: Survey of Methods and Software

Chapter 3: Interpolation and Polynomial Approximation

3.1: Interpolation and the Lagrange Polynomial

3.2: Data Approximation and Neville’s Method

3.3: Divided Differences

3.4: Hermite Interpolation

3.5: Cubic Spline Interpolation

3.6: Parametric Curves

3.7: Survey of Methods and Software

Chapter 4: Numerical Differentiation and Integration

4.1: Numerical Differentiation

4.2: Richardson’s Extrapolation

4.3: Elements of Numerical Integration

4.4: Composite Numerical Integration

4.5: Romberg Integration

4.6: Adaptive Quadrature Methods

4.7: Gaussian Quadrature

4.8: Multiple Integrals

4.9: Improper Integrals

4.10: Survey of Methods and Software

Chapter 5: Initial-Value Problems for Ordinary Differential Equations

5.1: The Elementary Theory of Initial-Value Problems

5.2: Euler’s Method

5.3: Higher-Order Taylor Methods

5.4: Runge-Kutta Methods

5.5: Error Control and the Runge-Kutta-Fehlberg Method

5.6: Multistep Methods

5.7: Variable Step-Size Multistep Methods

5.8: Extrapolation Methods

5.9: Higher-Order Equations and Systems of Differential Equations

5.10: Stability

5.11: Stiff Differential Equations

5.12: Survey of Methods and Software

Chapter 6: Direct Methods for Solving Linear Systems

6.1: Linear Systems of Equations

6.2: Pivoting Strategies

6.3: Linear Algebra and Matrix Inversion

6.4: The Determinant of a Matrix

6.5: Matrix Factorization

6.6: Special Types of Matrices

6.7: Survey of Methods and Software

Chapter 7: IterativeTechniques in Matrix Algebra

7.1: Norms of Vectors and Matrices

7.2: Eigenvalues and Eigenvectors

7.3: The Jacobi and Gauss-Siedel Iterative Techniques

7.4: Relaxation Techniques for Solving Linear Systems

7.5: Error Bounds and Iterative Refinement

7.6: The Conjugate Gradient Method

7.7: Survey of Methods and Software

Chapter 8: Approximation Theory

8.1: Discrete Least Squares Approximation

8.2: Orthogonal Polynomials and Least Squares Approximation

8.3: Chebyshev Polynomials and Economization of Power Series

8.4: Rational Function Approximation

8.5: Trigonometric Polynomial Approximation

8.6: Fast Fourier Transforms

8.7: Survey of Methods and Software

Chapter 9: Approximating Eigenvalues

9.1: Linear Algebra and Eigenvalues

9.2: Orthogonal Matrices and Similarity Transformations

9.3: The Power Method

9.4: Householder’s Method

9.5: The QR Algorithm

9.6: Singular Value Decomposition

9.7: Survey of Methods and Software

Chapter 10: Numerical Solutions of Nonlinear Systems of Equations

10.1: Fixed Points for Functions of Several Variables

10.2: Newton’s Method

10.3: Quasi-Newton Methods

10.4: Steepest Descent Techniques

10.5: Homotopy and Continuation Methods

10.6: Survey of Methods and Software

Chapter 11: Boundary-Value Problems for Ordinary Differential Equations

11.1: The Linear Shooting Method

11.2: The Shooting Method for Nonlinear Problems

11.3: Finite-Difference Methods for Linear Problems

11.4: Finite-Difference Methods for Nonlinear Problems

11.5: The Rayleigh-Ritz Method

11.6: Survey of Methods and Software

Chapter 12: Numerical Solutions to Partial Differential Equations

12.1: Elliptic Partial Differential Equations

12.2: Parabolic Partial Differential Equations

12.3: Hyperbolic Partial Differential Equations

12.4: An Introduction to the Finite-Element Method

12.5: Survey of Methods and Software

Bibliography

Answers to Selected Exercises

Exercise Set 1.1 (Page 14)

Exercise Set 1.2 (Page 28)

Exercise Set 1.3 (Page 39)

Exercise Set 2.1 (Page 54)

Exercise Set 2.2 (Page 64)

Exercise Set 2.3 (Page 75)

Exercise Set 2.4 (Page 85)

Exercise Set 2.5 (Page 90)

Exercise Set 2.6 (Page 100)

Exercise Set 3.1 (Page 114)

Exercise Set 3.2 (Page 123)

Exercise Set 3.3 (Page 133)

Exercise Set 3.4 (Page 142)

Exercise Set 3.5 (Page 161)

Exercise Set 3.6 (Page 170)

Exercise Set 4.1 (Page 182)

Exercise Set 4.2 (Page 191)

Exercise Set 4.3 (Page 202)

Exercise Set 4.4 (Page 210)

Exercise Set 4.5 (Page 218)

Exercise Set 4.6 (Page 227)

Exercise Set 4.7 (Page 234)

Exercise Set 4.8 (Page 248)

Exercise Set 4.9 (Page 254)

Exercise Set 5.1 (Page 264)

Exercise Set 5.2 (Page 273)

Exercise Set 5.3 (Page 281)

Exercise Set 5.4 (Page 291)

Exercise Set 5.5 (Page 300)

Exercise Set 5.6 (Page 314)

Exercise Set 5.7 (Page 320)

Exercise Set 5.8 (Page 327)

Exercise Set 5.9 (Page 337)

Exercise Set 5.10 (Page 347)

Exercise Set 5.11 (Page 354)

Exercise Set 6.1 (Page 368)

Exercise Set 6.2 (Page 379)

Exercise Set 6.3 (Page 390)

Exercise Set 6.4 (Page 399)

Exercise Set 6.5 (Page 409)

Exercise Set 6.6 (Page 425)

Exercise Set 7.1 (Page 441)

Exercise Set 7.2 (Page 449)

Exercise Set 7.3 (Page 459)

Exercise Set 7.4 (Page 467)

Exercise Set 7.5 (Page 476)

Exercise Set 7.6 (Page 492)

Exercise Set 8.1 (Page 506)

Exercise Set 8.2 (Page 518)

Exercise Set 8.3 (Page 527)

Exercise Set 8.4 (Page 537)

Exercise Set 8.5 (Page 546)

Exercise Set 8.6 (Page 557)

Exercise Set 9.1 (Page 568)

Exercise Set 9.2 (Page 573)

Exercise Set 9.3 (Page 590)

Exercise Set 9.4 (Page 600)

Exercise Set 9.5 (Page 611)

Exercise Set 9.6 (Page 625)

Exercise Set 10.1 (Page 636)

Exercise Set 10.2 (Page 644)

Exercise Set 10.3 (Page 652)

Exercise Set 10.4 (Page 659)

Exercise Set 10.5 (Page 666)

Exercise Set 11.1 (Page 677)

Exercise Set 11.2 (Page 684)

Exercise Set 11.3 (Page 689)

Exercise Set 11.4 (Page 696)

Exercise Set 11.5 (Page 710)

Exercise Set 12.1 (Page 723)

Exercise Set 12.2 (Page 736)

Exercise Set 12.3 (Page 744)

Exercise Set 12.4 (Page 758)

INDEX

Index of Algorithms

Glossary of Notation

Trigonometry

Common Series

The Greek Alphabet

Common Graphs

Numerical Analysis Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Numerical Analysis N I N T H E D I T I O N Richard L. Burden Youngstown State University J. Douglas Faires Youngstown State University Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

This is an electronic version of the print textbook. Due to electronic rights restrictions, some third party content may be suppressed. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it. For valuable information on pricing, previous editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Numerical Analysis, Ninth Edition Richard L. Burden and J. Douglas Faires Editor-in-Chief: Michelle Julet Publisher: Richard Stratton Senior Sponsoring Editor: Molly Taylor Associate Editor: Daniel Seibert Editorial Assistant: Shaylin Walsh Associate Media Editor: Andrew Coppola Senior Marketing Manager: Jennifer Pursley Jones Marketing Coordinator: Erica O’Connell Marketing Communications Manager: Mary Anne Payumo Content Project Manager: Jill Clark Art Director: Jill Ort Senior Manufacturing Buyer: Diane Gibbons Senior Rights Acquisition Specialist: Katie Huha Production Service: Cadmus Communications Text Designer: Jay Purcell Cover Designer: Wing Ngan Cover Image: Spiral Vortex Photographer: Akira Inoue Collection: Amana images, Gettyimages.com Compositor: Cadmus Communications © 2011, 2005, 2001 Brooks/Cole, Cengage Learning ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher. For product information and technology assistance, contact us at: Cengage Learning Customer & Sales Support, 1-800-354-9706 For permission to use material from this text or product, submit all requests online at www.cengage.com/permissions. Further permissions questions can be emailed to permissionrequest@cengage.com. Library of Congress Control Number: 2010922639 ISBN-13: 978-0-538-73351-9 ISBN-10: 0-538-73351-9 Brooks/Cole 20 Channel Center Street Boston, MA 02210 USA Cengage Learning is a leading provider of customized learning solutions with ofﬁce locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil and Japan. Locate your local ofﬁce at international.cengage.com/region. Cengage Learning products are represented in Canada by Nelson Education, Ltd. For your course and learning solutions, visit www.cengage.com. Purchase any of our products at your local college store or at our preferred online store www.cengagebrain.com. Printed in Canada 1 2 3 4 5 6 7 14 13 12 11 10 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Contents Preface ix 1 Mathematical Preliminaries and Error Analysis 1 2 1.1 Review of Calculus 1.2 Round-off Errors and Computer Arithmetic 1.3 Algorithms and Convergence 1.4 Numerical Software 32 41 17 2 Solutions of Equations in One Variable 47 56 48 2.1 The Bisection Method 2.2 Fixed-Point Iteration 2.3 Newton’s Method and Its Extensions 2.4 Error Analysis for Iterative Methods 2.5 Accelerating Convergence 2.6 Zeros of Polynomials and Müller’s Method 2.7 Survey of Methods and Software 67 79 101 86 3.1 Interpolation and the Lagrange Polynomial 3.2 Data Approximation and Neville’s Method 3.3 Divided Differences 3.4 Hermite Interpolation 3.5 Cubic Spline Interpolation 3.6 Parametric Curves 3.7 Survey of Methods and Software 124 164 171 136 144 3 Interpolation and Polynomial Approximation 105 91 106 117 4 Numerical Differentiation and Integration 173 174 4.1 Numerical Differentiation 185 4.2 Richardson’s Extrapolation 4.3 Elements of Numerical Integration 193 v Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

vi Contents 203 220 213 4.4 Composite Numerical Integration 4.5 Romberg Integration 4.6 Adaptive Quadrature Methods 4.7 Gaussian Quadrature 4.8 Multiple Integrals 4.9 Improper Integrals 4.10 Survey of Methods and Software 235 250 228 256 5 Initial-Value Problems for Ordinary Differential 282 276 260 Equations 259 5.1 The Elementary Theory of Initial-Value Problems 5.2 Euler’s Method 266 5.3 Higher-Order Taylor Methods 5.4 Runge-Kutta Methods 5.5 Error Control and the Runge-Kutta-Fehlberg Method 5.6 Multistep Methods 5.7 Variable Step-Size Multistep Methods 5.8 Extrapolation Methods 5.9 Higher-Order Equations and Systems of Differential Equations 5.10 Stability 339 5.11 Stiff Differential Equations 348 5.12 Survey of Methods and Software 302 315 321 293 355 328 6 Direct Methods for Solving Linear Systems 357 372 358 6.1 Linear Systems of Equations 6.2 Pivoting Strategies 6.3 Linear Algebra and Matrix Inversion 6.4 The Determinant of a Matrix 6.5 Matrix Factorization 400 6.6 Special Types of Matrices 411 6.7 Survey of Methods and Software 396 428 381 7 IterativeTechniques in Matrix Algebra 431 443 432 7.1 Norms of Vectors and Matrices 7.2 Eigenvalues and Eigenvectors 7.3 The Jacobi and Gauss-Siedel Iterative Techniques 7.4 Relaxation Techniques for Solving Linear Systems 7.5 Error Bounds and Iterative Reﬁnement 7.6 The Conjugate Gradient Method 479 7.7 Survey of Methods and Software 495 469 450 462 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

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