Numerical Methods. Numerical Methods. for Engineers. for Engineers. Steven C. Chapra Raymond P. Canale. Chapra Canale. Sixth Edition.

Transcription

1 Sixth Edition Features include: which are based on exciting new areas such as bioengineering. and differential equations. students using this text will be able to apply their new skills to their chosen field. Electronic Textbook Options an online resource where students can purchase the complete text in a digital format at almost half the cost of the traditional textbook. Students can access the text online for one year. learning, which include full text search, notes and highlighting, and tools for sharing contact your sales representative or visit Sixth Edition Numerical Methods for Engineers Chapra Canale Steven C. Chapra Raymond P. Canale MD DALIM # /12/09 CYAN MAG YELO BLK For more information, please visit for Engineers adaptive quadrature. Numerical Methods The sixth edition of Numerical Methods for Engineers offers an innovative and accessible presentation of numerical methods; the book has earned the Meriam-Wiley award, which is given by the American Society for Engineering Education for the best textbook. Because software packages are now regularly used for numerical analysis, this eagerly anticipated revision maintains its strong focus on appropriate use of computational tools.

2 cha01064_fm.qxd 3/25/09 10:51 AM Page i Numerical Methods for Engineers S I X TH ED I TION Steven C. Chapra Berger Chair in Computing and Engineering Tufts University Raymond P. Canale Professor Emeritus of Civil Engineering University of Michigan

3 cha01064_fm.qxd 3/25/09 10:51 AM Page ii NUMERICAL METHODS FOR ENGINEERS, SIXTH EDITION Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY Copyright 2010 by The McGraw-Hill Companies, Inc. All rights reserved. Previous editions 2006, 2002, and 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 The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning. Some ancillaries, including electronic and print components, may not be available to customers outside the United States. This book is printed on acid-free paper VNH/VNH 0 9 ISBN MHID Global Publisher: Raghothaman Srinivasan Sponsoring Editor: Debra B. Hash Director of Development: Kristine Tibbetts Developmental Editor: Lorraine K. Buczek Senior Marketing Manager: Curt Reynolds Project Manager: Joyce Watters Lead Production Supervisor: Sandy Ludovissy Associate Design Coordinator: Brenda A. Rolwes Cover Designer: Studio Montage, St. Louis, Missouri (USE) Cover Image: BrandX/JupiterImages Compositor: Macmillan Publishing Solutions Typeface: 10/12 Times Roman Printer: R. R. Donnelley Jefferson City, MO All credits appearing on page or at the end of the book are considered to be an extension of the copyright page. MATLAB is a registered trademark of The MathWorks, Inc. Library of Congress Cataloging-in-Publication Data Chapra, Steven C. Numerical methods for engineers / Steven C. Chapra, Raymond P. Canale. 6th ed. p. cm. Includes bibliographical references and index. ISBN ISBN (hard copy : alk. paper) 1. Engineering mathematics Data processing. 2. Numerical calculations Data processing 3. Microcomputers Programming. I. Canale, Raymond P. II. Title. TA345.C dc22

4 To Margaret and Gabriel Chapra Helen and Chester Canale

5 CONTENTS PREFACE xiv GUIDED TOUR xvi ABOUT THE AUTHORS xviii 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 A Simple Mathematical Model Conservation Laws and Engineering 18 Problems 21 CHAPTER 2 Programming and Software Packages and Programming Structured Programming Modular Programming Excel MATLAB Mathcad Other Languages and Libraries 46 Problems 47 CHAPTER 3 Approximations and Round-Off Errors Significant Figures Accuracy and Precision Error Definitions Round-Off Errors 62 Problems 76 iv

6 CONTENTS v CHAPTER 4 Truncation Errors and the Taylor Series The Taylor Series Error Propagation Total Numerical Error Blunders, Formulation Errors, and Data Uncertainty 103 Problems 105 EPILOGUE: PART ONE 107 PT1.4 Trade-Offs 107 PT1.5 Important Relationships and Formulas 110 PT1.6 Advanced Methods and Additional References 110 PART TWO ROOTS OF PT2.1 Motivation 113 EQUATIONS 113 PT2.2 Mathematical Background 115 PT2.3 Orientation 116 CHAPTER 5 Bracketing Methods Graphical Methods The Bisection Method The False-Position Method Incremental Searches and Determining Initial Guesses 138 Problems 139 CHAPTER 6 Open Methods Simple Fixed-Point Iteration The Newton-Raphson Method The Secant Method Brent s Method Multiple Roots Systems of Nonlinear Equations 167 Problems 171 CHAPTER 7 Roots of Polynomials Polynomials in Engineering and Science Computing with Polynomials Conventional Methods 180

7 vi CONTENTS 7.4 Müller s Method Bairstow s Method Other Methods Root Location with Software Packages 190 Problems 200 CHAPTER 8 Case Studies: Roots of Equations Ideal and Nonideal Gas Laws (Chemical/Bio Engineering) Greenhouse Gases and Rainwater (Civil/Environmental Engineering) Design of an Electric Circuit (Electrical Engineering) Pipe Friction (Mechanical/Aerospace Engineering) 209 Problems 213 EPILOGUE: PART TWO 223 PT2.4 Trade-Offs 223 PT2.5 Important Relationships and Formulas 224 PT2.6 Advanced Methods and Additional References 224 PART THREE LINEAR ALGEBRAIC PT3.1 Motivation 227 EQUATIONS 227 PT3.2 Mathematical Background 229 PT3.3 Orientation 237 CHAPTER 9 Gauss Elimination Solving Small Numbers of Equations Naive Gauss Elimination Pitfalls of Elimination Methods Techniques for Improving Solutions Complex Systems Nonlinear Systems of Equations Gauss-Jordan Summary 271 Problems 271 CHAPTER 10 LU Decomposition and Matrix Inversion LU Decomposition The Matrix Inverse Error Analysis and System Condition 287 Problems 293

8 CONTENTS vii CHAPTER 11 Special Matrices and Gauss-Seidel Special Matrices Gauss-Seidel Linear Algebraic Equations with Software Packages 307 Problems 312 CHAPTER 12 Case Studies: Linear Algebraic Equations Steady-State Analysis of a System of Reactors (Chemical/Bio Engineering) Analysis of a Statically Determinate Truss (Civil/Environmental Engineering) Currents and Voltages in Resistor Circuits (Electrical Engineering) Spring-Mass Systems (Mechanical/Aerospace Engineering) 324 Problems 327 EPILOGUE: PART THREE 337 PT3.4 Trade-Offs 337 PT3.5 Important Relationships and Formulas 338 PT3.6 Advanced Methods and Additional References 338 PART FOUR OPTIMIZATION 341 PT4.1 Motivation 341 PT4.2 Mathematical Background 346 PT4.3 Orientation 347 CHAPTER 13 One-Dimensional Unconstrained Optimization Golden-Section Search Parabolic Interpolation Newton s Method Brent s Method 364 Problems 364 CHAPTER 14 Multidimensional Unconstrained Optimization Direct Methods Gradient Methods 372 Problems 385

9 viii CONTENTS CHAPTER 15 Constrained Optimization Linear Programming Nonlinear Constrained Optimization Optimization with Software Packages 399 Problems 410 CHAPTER 16 Case Studies: Optimization Least-Cost Design of a Tank (Chemical/Bio Engineering) Least-Cost Treatment of Wastewater (Civil/Environmental Engineering) Maximum Power Transfer for a Circuit (Electrical Engineering) Equilibrium and Minimum Potential Energy (Mechanical/Aerospace Engineering) 426 Problems 428 EPILOGUE: PART FOUR 436 PT4.4 Trade-Offs 436 PT4.5 Additional References 437 PART FIVE CURVE FITTING 439 PT5.1 Motivation 439 PT5.2 Mathematical Background 441 PT5.3 Orientation 450 CHAPTER 17 Least-Squares Regression Linear Regression Polynomial Regression Multiple Linear Regression General Linear Least Squares Nonlinear Regression 481 Problems 484 CHAPTER 18 Interpolation Newton s Divided-Difference Interpolating Polynomials Lagrange Interpolating Polynomials Coefficients of an Interpolating Polynomial Inverse Interpolation Additional Comments Spline Interpolation Multidimensional Interpolation 519 Problems 522

10 CONTENTS ix CHAPTER 19 Fourier Approximation Curve Fitting with Sinusoidal Functions Continuous Fourier Series Frequency and Time Domains Fourier Integral and Transform Discrete Fourier Transform (DFT) Fast Fourier Transform (FFT) The Power Spectrum Curve Fitting with Software Packages 550 Problems 559 CHAPTER 20 Case Studies: Curve Fitting Linear Regression and Population Models (Chemical/Bio Engineering) Use of Splines to Estimate Heat Transfer (Civil/Environmental Engineering) Fourier Analysis (Electrical Engineering) Analysis of Experimental Data (Mechanical/Aerospace Engineering) 568 Problems 570 EPILOGUE: PART FIVE 580 PT5.4 Trade-Offs 580 PT5.5 Important Relationships and Formulas 581 PT5.6 Advanced Methods and Additional References 583 PART SIX NUMERICAL PT6.1 Motivation 585 DIFFERENTIATION PT6.2 Mathematical Background 595 AND PT6.3 Orientation 597 INTEGRATION 585 CHAPTER 21 Newton-Cotes Integration Formulas The Trapezoidal Rule Simpson s Rules Integration with Unequal Segments Open Integration Formulas Multiple Integrals 625 Problems 627

11 x CONTENTS CHAPTER 22 Integration of Equations Newton-Cotes Algorithms for Equations Romberg Integration Adaptive Quadrature Gauss Quadrature Improper Integrals 648 Problems 651 CHAPTER 23 Numerical Differentiation High-Accuracy Differentiation Formulas Richardson Extrapolation Derivatives of Unequally Spaced Data Derivatives and Integrals for Data with Errors Partial Derivatives Numerical Integration/Differentiation with Software Packages 661 Problems 668 CHAPTER 24 Case Studies: Numerical Integration and Differentiation Integration to Determine the Total Quantity of Heat (Chemical/Bio Engineering) Effective Force on the Mast of a Racing Sailboat (Civil/Environmental Engineering) Root-Mean-Square Current by Numerical Integration (Electrical Engineering) Numerical Integration to Compute Work (Mechanical/Aerospace Engineering) 678 Problems 682 EPILOGUE: PART SIX 692 PT6.4 Trade-Offs 692 PT6.5 Important Relationships and Formulas 693 PT6.6 Advanced Methods and Additional References 693 PART SEVEN ORDINARY PT7.1 Motivation 697 DIFFERENTIAL PT7.2 Mathematical Background 701 EQUATIONS 697 PT7.3 Orientation 703

12 CONTENTS xi CHAPTER 25 Runge-Kutta Methods Euler s Method Improvements of Euler s Method Runge-Kutta Methods Systems of Equations Adaptive Runge-Kutta Methods 742 Problems 750 CHAPTER 26 Stiffness and Multistep Methods Stiffness Multistep Methods 756 Problems 776 CHAPTER 27 Boundary-Value and Eigenvalue Problems General Methods for Boundary-Value Problems Eigenvalue Problems Odes and Eigenvalues with Software Packages 798 Problems 805 CHAPTER 28 Case Studies: Ordinary Differential Equations Using ODEs to Analyze the Transient Response of a Reactor (Chemical/Bio Engineering) Predator-Prey Models and Chaos (Civil/Environmental Engineering) Simulating Transient Current for an Electric Circuit (Electrical Engineering) The Swinging Pendulum (Mechanical/Aerospace Engineering) 824 Problems 828 EPILOGUE: PART SEVEN 838 PT7.4 Trade-Offs 838 PT7.5 Important Relationships and Formulas 839 PT7.6 Advanced Methods and Additional References 839 PART EIGHT PARTIAL PT8.1 Motivation 843 DIFFERENTIAL PT8.2 Orientation 846 EQUATIONS 843

13 xii CONTENTS CHAPTER 29 Finite Difference: Elliptic Equations The Laplace Equation Solution Technique Boundary Conditions The Control-Volume Approach Software to Solve Elliptic Equations 867 Problems 868 CHAPTER 30 Finite Difference: Parabolic Equations The Heat-Conduction Equation Explicit Methods A Simple Implicit Method The Crank-Nicolson Method Parabolic Equations in Two Spatial Dimensions 883 Problems 886 CHAPTER 31 Finite-Element Method The General Approach Finite-Element Application in One Dimension Two-Dimensional Problems Solving PDEs with Software Packages 906 Problems 910 CHAPTER 32 Case Studies: Partial Differential Equations One-Dimensional Mass Balance of a Reactor (Chemical/Bio Engineering) Deflections of a Plate (Civil/Environmental Engineering) Two-Dimensional Electrostatic Field Problems (Electrical Engineering) Finite-Element Solution of a Series of Springs (Mechanical/Aerospace Engineering) 922 Problems 926 EPILOGUE: PART EIGHT 929 PT8.3 Trade-Offs 929 PT8.4 Important Relationships and Formulas 929 PT8.5 Advanced Methods and Additional References 930

14 CONTENTS xiii APPENDIX A: THE FOURIER SERIES 931 APPENDIX B: GETTING STARTED WITH MATLAB 933 APPENDIX C: GETTING STARTED WITH MATHCAD 941 BIBLIOGRAPHY 952 INDEX 955