Computer Science

Numerical Methods for Engineers

SEVENTH EDITION

Steven C. Chapra Berger Chair in Computing and Engineering

Tufts University

Raymond P. Canale Professor Emeritus of Civil Engineering

University of Michigan

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NUMERICAL METHODS FOR ENGINEERS, SEVENTH EDITION

Published by McGraw-Hill Education, 2 Penn Plaza, New York, NY 10121. Copyright © 2015 by McGraw-Hill Education. All rights reserved. Printed in the United States of America. Previous editions © 2010, 2006, and 2002. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of McGraw-Hill Education, including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning.

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ISBN 978–0–07–339792–4 MHID 0–07–339792–x

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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

11.1 Special Matrices 300 11.2 Gauss-Seidel 304 11.3 Linear Algebraic Equations with Software Packages 311 Problems 316

CHAPTER 12 Case Studies: Linear Algebraic Equations 319

12.1 Steady-State Analysis of a System of Reactors (Chemical/Bio Engineering) 319 12.2 Analysis of a Statically Determinate Truss (Civil/Environmental Engineering) 322 12.3 Currents and Voltages in Resistor Circuits (Electrical Engineering) 326 12.4 Spring-Mass Systems (Mechanical/Aerospace Engineering) 328 Problems 331

EPILOGUE: PART THREE 341 PT3.4 Trade-Offs 341 PT3.5 Important Relationships and Formulas 342 PT3.6 Advanced Methods and Additional References 342

PART FOUR

OPTIMIZATION 345 PT4.1 Motivation 345 PT4.2 Mathematical Background 350 PT4.3 Orientation 351

CHAPTER 13 One-Dimensional Unconstrained Optimization 355

13.1 Golden-Section Search 356 13.2 Parabolic Interpolation 363 13.3 Newton’s Method 365 13.4 Brent’s Method 366 Problems 368

CHAPTER 14 Multidimensional Unconstrained Optimization 370

14.1 Direct Methods 371 14.2 Gradient Methods 375 Problems 388

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viii CONTENTS

CHAPTER 15 Constrained Optimization 390

15.1 Linear Programming 390 15.2 Nonlinear Constrained Optimization 401 15.3 Optimization with Software Packages 402 Problems 413

CHAPTER 16 Case Studies: Optimization 416

16.1 Least-Cost Design of a Tank (Chemical/Bio Engineering) 416 16.2 Least-Cost Treatment of Wastewater (Civil/Environmental Engineering) 421 16.3 Maximum Power Transfer for a Circuit (Electrical Engineering) 425 16.4 Equilibrium and Minimum Potential Energy (Mechanical/Aerospace Engineering) 429 Problems 431

EPILOGUE: PART FOUR 438 PT4.4 Trade-Offs 438 PT4.5 Additional References 439

PART FIVE

CURVE FITTING 441 PT5.1 Motivation 441 PT5.2 Mathematical Background 443 PT5.3 Orientation 452

CHAPTER 17 Least-Squares Regression 456

17.1 Linear Regression 456 17.2 Polynomial Regression 472 17.3 Multiple Linear Regression 476 17.4 General Linear Least Squares 479 17.5 Nonlinear Regression 483 Problems 487

CHAPTER 18 Interpolation 490

18.1 Newton’s Divided-Difference Interpolating Polynomials 491 18.2 Lagrange Interpolating Polynomials 502 18.3 Coeffi cients of an Interpolating Polynomial 507 18.4 Inverse Interpolation 507 18.5 Additional Comments 508 18.6 Spline Interpolation 511 18.7 Multidimensional Interpolation 521 Problems 524

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CONTENTS ix

CHAPTER 19 Fourier Approximation 526

19.1 Curve Fitting with Sinusoidal Functions 527 19.2 Continuous Fourier Series 533 19.3 Frequency and Time Domains 536 19.4 Fourier Integral and Transform 540 19.5 Discrete Fourier Transform (DFT) 542 19.6 Fast Fourier Transform (FFT) 544 19.7 The Power Spectrum 551 19.8 Curve Fitting with Software Packages 552 Problems 561

CHAPTER 20 Case Studies: Curve Fitting 563

20.1 Linear Regression and Population Models (Chemical/Bio Engineering) 563 20.2 Use of Splines to Estimate Heat Transfer (Civil/Environmental Engineering) 567 20.3 Fourier Analysis (Electrical Engineering) 569 20.4 Analysis of Experimental Data (Mechanical/Aerospace Engineering) 570 Problems 572

EPILOGUE: PART FIVE 582 PT5.4 Trade-Offs 582 PT5.5 Important Relationships and Formulas 583 PT5.6 Advanced Methods and Additional References 584

PART SIX

NUMERICAL PT6.1 Motivation 587 DIFFERENTIATION PT6.2 Mathematical Background 597 AND PT6.3 Orientation 599 INTEGRATION 587

CHAPTER 21 Newton-Cotes Integration Formulas 603

21.1 The Trapezoidal Rule 605 21.2 Simpson’s Rules 615 21.3 Integration with Unequal Segments 624 21.4 Open Integration Formulas 627 21.5 Multiple Integrals 627 Problems 629

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x CONTENTS

CHAPTER 22 Integration of Equations 633

22.1 Newton-Cotes Algorithms for Equations 633 22.2 Romberg Integration 634 22.3 Adaptive Quadrature 640 22.4 Gauss Quadrature 642 22.5 Improper Integrals 650 Problems 653

CHAPTER 23 Numerical Differentiation 655

23.1 High-Accuracy Differentiation Formulas 655 23.2 Richardson Extrapolation 658 23.3 Derivatives of Unequally Spaced Data 660 23.4 Derivatives and Integrals for Data with Errors 661 23.5 Partial Derivatives 662 23.6 Numerical Integration/Differentiation with Software Packages 663 Problems 670

CHAPTER 24 Case Studies: Numerical Integration and Differentiation 673

24.1 Integration to Determine the Total Quantity of Heat (Chemical/Bio Engineering) 673

24.2 Effective Force on the Mast of a Racing Sailboat (Civil/Environmental Engineering) 675

24.3 Root-Mean-Square Current by Numerical Integration (Electrical Engineering) 677

24.4 Numerical Integration to Compute Work (Mechanical/Aerospace Engineering) 680

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