Problems on Scientific Computing

NUMERICAL METHODS FOR ENGINEERS, SEVENTH EDITION

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Library of Congress Cataloging-in-Publication Data

Chapra, Steven C. Numerical methods for engineers / Steven C. Chapra, Berger chair in computing and engineering, Tufts University, Raymond P. Canale, professor emeritus of civil engineering, University of Michigan. — Seventh edition. pages cm Includes bibliographical references and index. ISBN 978-0-07-339792-4 (alk. paper) — ISBN 0-07-339792-X (alk. paper) 1. Engineering mathematics—Data processing. 2. Numerical calculations—Data processing 3. Microcomputers—Programming. I. Canale, Raymond P. II. Title. TA345.C47 2015 518.024’62—dc23 2013041704

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To

Margaret and Gabriel Chapra

Helen and Chester Canale

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iv

CONTENTS

PREFACE xiv

ABOUT THE AUTHORS xvi

PART ONE

MODELING, PT1.1 Motivation 3 COMPUTERS, AND PT1.2 Mathematical Background 5 ERROR ANALYSIS 3 PT1.3 Orientation 8

CHAPTER 1 Mathematical Modeling and Engineering Problem Solving 11

1.1 A Simple Mathematical Model 11 1.2 Conservation Laws and Engineering 18 Problems 21

CHAPTER 2 Programming and Software 27

2.1 Packages and Programming 27 2.2 Structured Programming 28 2.3 Modular Programming 37 2.4 Excel 39 2.5 MATLAB 43 2.6 Mathcad 47 2.7 Other Languages and Libraries 48 Problems 49

CHAPTER 3 Approximations and Round-Off Errors 55

3.1 Signifi cant Figures 56 3.2 Accuracy and Precision 58 3.3 Error Defi nitions 59 3.4 Round-Off Errors 65 Problems 79

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CONTENTS v

CHAPTER 4 Truncation Errors and the Taylor Series 81

4.1 The Taylor Series 81 4.2 Error Propagation 97 4.3 Total Numerical Error 101 4.4 Blunders, Formulation Errors, and Data Uncertainty 106 Problems 108

EPILOGUE: PART ONE 110 PT1.4 Trade-Offs 110 PT1.5 Important Relationships and Formulas 113 PT1.6 Advanced Methods and Additional References 113

PART TWO

ROOTS OF PT2.1 Motivation 117 EQUATIONS 117 PT2.2 Mathematical Background 119

PT2.3 Orientation 120

CHAPTER 5 Bracketing Methods 123

5.1 Graphical Methods 123 5.2 The Bisection Method 127 5.3 The False-Position Method 135 5.4 Incremental Searches and Determining Initial Guesses 141 Problems 142

CHAPTER 6 Open Methods 145

6.1 Simple Fixed-Point Iteration 146 6.2 The Newton-Raphson Method 151 6.3 The Secant Method 157 6.4 Brent’s Method 162 6.5 Multiple Roots 166 6.6 Systems of Nonlinear Equations 169 Problems 173

CHAPTER 7 Roots of Polynomials 176

7.1 Polynomials in Engineering and Science 176 7.2 Computing with Polynomials 179 7.3 Conventional Methods 182

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vi CONTENTS

7.4 Müller’s Method 183 7.5 Bairstow’s Method 187 7.6 Other Methods 192 7.7 Root Location with Software Packages 192 Problems 202

CHAPTER 8 Case Studies: Roots of Equations 204

8.1 Ideal and Nonideal Gas Laws (Chemical/Bio Engineering) 204 8.2 Greenhouse Gases and Rainwater (Civil/Environmental Engineering) 207 8.3 Design of an Electric Circuit (Electrical Engineering) 209 8.4 Pipe Friction (Mechanical/Aerospace Engineering) 212 Problems 215

EPILOGUE: PART TWO 226 PT2.4 Trade-Offs 226 PT2.5 Important Relationships and Formulas 227 PT2.6 Advanced Methods and Additional References 227

PART THREE

LINEAR ALGEBRAIC PT3.1 Motivation 231 EQUATIONS 231 PT3.2 Mathematical Background 233

PT3.3 Orientation 241

CHAPTER 9 Gauss Elimination 245

9.1 Solving Small Numbers of Equations 245 9.2 Naive Gauss Elimination 252 9.3 Pitfalls of Elimination Methods 258 9.4 Techniques for Improving Solutions 264 9.5 Complex Systems 271 9.6 Nonlinear Systems of Equations 271 9.7 Gauss-Jordan 273 9.8 Summary 275 Problems 275

CHAPTER 10 LU Decomposition and Matrix Inversion 278 10.1 LU Decomposition 278 10.2 The Matrix Inverse 287 10.3 Error Analysis and System Condition 291 Problems 297

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CONTENTS vii

CHAPTER 11 Special Matrices and Gauss-Seidel 300