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Numerical.Methods.for.Engineers.pdf
Cover Page
Title Page
Copyright Page
Dedication
Contents
Preface
Guided Tour
About the Authors
PART ONE: MODELING, COMPUTERS, AND ERROR ANALYSIS
PT1.1 Motivation
PT1.2 Mathematical Background
PT1.3 Orientation
CHAPTER 1: Mathematical Modeling and Engineering Problem Solving
1.1 A Simple Mathematical Model
1.2 Conservation Laws and Engineering
Problems
CHAPTER 2: Programming and Software
2.1 Packages and Programming
2.2 Structured Programming
2.3 Modular Programming
2.4 Excel
2.5 MATLAB
2.6 Mathcad
2.7 Other Languages and Libraries
Problems
CHAPTER 3: Approximations and Round-Off Errors
3.1 Significant Figures
3.2 Accuracy and Precision
3.3 Error Definitions
3.4 Round-Off Errors
Problems
CHAPTER 4: Truncation Errors and the Taylor Series
4.1 The Taylor Series
4.2 Error Propagation
4.3 Total Numerical Error
4.4 Blunders, Formulation Errors, and Data Uncertainty
Problems
EPILOGUE: PART ONE
PT1.4 Trade-Offs
PT1.5 Important Relationships and Formulas
PT1.6 Advanced Methods and Additional References
PART TWO: ROOTS OF EQUATIONS
PT2.1 Motivation
PT2.2 Mathematical Background
PT2.3 Orientation
CHAPTER 5: Bracketing Methods
5.1 Graphical Methods
5.2 The Bisection Method
5.3 The False-Position Method
5.4 Incremental Searches and Determining Initial Guesses
Problems
CHAPTER 6: Open Methods
6.1 Simple Fixed-Point Iteration
6.2 The Newton-Raphson Method
6.3 The Secant Method
6.4 Brent’s Method
6.5 Multiple Roots
6.6 Systems of Nonlinear Equations
Problems
CHAPTER 7: Roots of Polynomials
7.1 Polynomials in Engineering and Science
7.2 Computing with Polynomials
7.3 Conventional Methods
7.4 Müller’s Method
7.5 Bairstow’s Method
7.6 Other Methods
7.7 Root Location with Software Packages
Problems
CHAPTER 8: Case Studies: Roots of Equations
8.1 Ideal and Nonideal Gas Laws (Chemical/Bio Engineering)
8.2 Greenhouse Gases and Rainwater (Civil/Environmental Engineering)
8.3 Design of an Electric Circuit (Electrical Engineering)
8.4 Pipe Friction (Mechanical/Aerospace Engineering)
Problems
EPILOGUE: PART TWO
PT2.4 Trade-Offs
PT2.5 Important Relationships and Formulas
PT2.6 Advanced Methods and Additional References
PART THREE: LINEAR ALGEBRAIC EQUATIONS
PT3.1 Motivation
PT3.2 Mathematical Background
PT3.3 Orientation
CHAPTER 9: Gauss Elimination
9.1 Solving Small Numbers of Equations
9.2 Naive Gauss Elimination
9.3 Pitfalls of Elimination Methods
9.4 Techniques for Improving Solutions
9.5 Complex Systems
9.6 Nonlinear Systems of Equations
9.7 Gauss-Jordan
9.8 Summary
Problems
CHAPTER 10: LU Decomposition and Matrix Inversion
10.1 LU Decomposition
10.2 The Matrix Inverse
10.3 Error Analysis and System Condition
Problems
CHAPTER 11: Special Matrices and Gauss-Seidel
11.1 Special Matrices
11.2 Gauss-Seidel
11.3 Linear Algebraic Equations with Software Packages
Problems
CHAPTER 12: Case Studies: Linear Algebraic Equations
12.1 Steady-State Analysis of a System of Reactors (Chemical/Bio Engineering)
12.2 Analysis of a Statically Determinate Truss (Civil/Environmental Engineering)
12.3 Currents and Voltages in Resistor Circuits (Electrical Engineering)
12.4 Spring-Mass Systems (Mechanical/Aerospace Engineering)
Problems
EPILOGUE: PART THREE
PT3.4 Trade-Offs
PT3.5 Important Relationships and Formulas
PT3.6 Advanced Methods and Additional References
PART FOUR: OPTIMIZATION
PT4.1 Motivation
PT4.2 Mathematical Background
PT4.3 Orientation
CHAPTER 13: One-Dimensional Unconstrained Optimization
13.1 Golden-Section Search
13.2 Parabolic Interpolation
13.3 Newton’s Method
13.4 Brent’s Method
Problems
CHAPTER 14: Multidimensional Unconstrained Optimization
14.1 Direct Methods
14.2 Gradient Methods
Problems
CHAPTER 15: Constrained Optimization
15.1 Linear Programming
15.2 Nonlinear Constrained Optimization
15.3 Optimization with Software Packages
Problems
CHAPTER 16: Case Studies: Optimization
16.1 Least-Cost Design of a Tank (Chemical/Bio Engineering)
16.2 Least-Cost Treatment of Wastewater (Civil/Environmental Engineering)
16.3 Maximum Power Transfer for a Circuit (Electrical Engineering)
16.4 Equilibrium and Minimum Potential Energy (Mechanical/Aerospace Engineering)
Problems
EPILOGUE: PART FOUR
PT4.4 Trade-Offs
PT4.5 Additional References
PART FIVE: CURVE FITTING
PT5.1 Motivation
PT5.2 Mathematical Background
PT5.3 Orientation
CHAPTER 17: Least-Squares Regression
17.1 Linear Regression
17.2 Polynomial Regression
17.3 Multiple Linear Regression
17.4 General Linear Least Squares
17.5 Nonlinear Regression
Problems
CHAPTER 18: Interpolation
18.1 Newton’s Divided-Difference Interpolating Polynomials
18.2 Lagrange Interpolating Polynomials
18.3 Coefficients of an Interpolating Polynomial
18.4 Inverse Interpolation
18.5 Additional Comments
18.6 Spline Interpolation
18.7 Multidimensional Interpolation
Problems
CHAPTER 19: Fourier Approximation
19.1 Curve Fitting with Sinusoidal Functions
19.2 Continuous Fourier Series
19.3 Frequency and Time Domains
19.4 Fourier Integral and Transform
19.5 Discrete Fourier Transform (DFT)
19.6 Fast Fourier Transform (FFT)
19.7 The Power Spectrum
19.8 Curve Fitting with Software Packages
Problems
CHAPTER 20: Case Studies: Curve Fitting
20.1 Linear Regression and Population Models (Chemical/Bio Engineering)
20.2 Use of Splines to Estimate Heat Transfer (Civil/Environmental Engineering)
20.3 Fourier Analysis (Electrical Engineering)
20.4 Analysis of Experimental Data (Mechanical/Aerospace Engineering)
Problems
EPILOGUE: PART FIVE
PT5.4 Trade-Offs
PT5.5 Important Relationships and Formulas
PT5.6 Advanced Methods and Additional References
PART SIX: NUMERICAL DIFFERENTIATION AND INTEGRATION
PT6.1 Motivation
PT6.2 Mathematical Background
PT6.3 Orientation
CHAPTER 21: Newton-Cotes Integration Formulas
21.1 The Trapezoidal Rule
21.2 Simpson’s Rules
21.3 Integration with Unequal Segments
21.4 Open Integration Formulas
21.5 Multiple Integrals
Problems
CHAPTER 22: Integration of Equations
22.1 Newton-Cotes Algorithms for Equations
22.2 Romberg Integration
22.3 Adaptive Quadrature
22.4 Gauss Quadrature
22.5 Improper Integrals
Problems
CHAPTER 23: Numerical Differentiation
23.1 High-Accuracy Differentiation Formulas
23.2 Richardson Extrapolation
23.3 Derivatives of Unequally Spaced Data
23.4 Derivatives and Integrals for Data with Errors
23.5 Partial Derivatives
23.6 Numerical Integration/Differentiation with Software Packages
Problems
CHAPTER 24: Case Studies: Numerical Integration and Differentiation
24.1 Integration to Determine the Total Quantity of Heat (Chemical/Bio Engineering)
24.2 Effective Force on the Mast of a Racing Sailboat (Civil/Environmental Engineering)
24.3 Root-Mean-Square Current by Numerical Integration (Electrical Engineering)
24.4 Numerical Integration to Compute Work (Mechanical/Aerospace Engineering)
Problems
EPILOGUE: PART SIX
PT6.4 Trade-Offs
PT6.5 Important Relationships and Formulas
PT6.6 Advanced Methods and Additional References
PART SEVEN: ORDINARY DIFFERENTIAL EQUATIONS
PT7.1 Motivation
PT7.2 Mathematical Background
PT7.3 Orientation
CHAPTER 25: Runge-Kutta Methods
25.1 Euler’s Method
25.2 Improvements of Euler’s Method
25.3 Runge-Kutta Methods
25.4 Systems of Equations
25.5 Adaptive Runge-Kutta Methods
Problems
CHAPTER 26: Stiffness and Multistep Methods
26.1 Stiffness
26.2 Multistep Methods
Problems
CHAPTER 27: Boundary-Value and Eigenvalue Problems
27.1 General Methods for Boundary-Value Problems
27.2 Eigenvalue Problems
27.3 Odes and Eigenvalues with Software Packages
Problems
CHAPTER 28: Case Studies: Ordinary Differential Equations
28.1 Using ODEs to Analyze the Transient Response of a Reactor (Chemical/Bio Engineering)
28.2 Predator-Prey Models and Chaos (Civil/Environmental Engineering)
28.3 Simulating Transient Current for an Electric Circuit (Electrical Engineering)
28.4 The Swinging Pendulum (Mechanical/Aerospace Engineering)
Problems
EPILOGUE: PART SEVEN
PT7.4 Trade-Offs
PT7.5 Important Relationships and Formulas
PT7.6 Advanced Methods and Additional References
PART EIGHT: PARTIAL DIFFERENTIAL EQUATIONS
PT8.1 Motivation
PT8.2 Orientation
CHAPTER 29: Finite Difference: Elliptic Equations
29.1 The Laplace Equation
29.2 Solution Technique
29.3 Boundary Conditions
29.4 The Control-Volume Approach
29.5 Software to Solve Elliptic Equations
Problems
CHAPTER 30: Finite Difference: Parabolic Equations
30.1 The Heat-Conduction Equation
30.2 Explicit Methods
30.3 A Simple Implicit Method
30.4 The Crank-Nicolson Method
30.5 Parabolic Equations in Two Spatial Dimensions
Problems
CHAPTER 31: Finite-Element Method
31.1 The General Approach
31.2 Finite-Element Application in One Dimension
31.3 Two-Dimensional Problems
31.4 Solving PDEs with Software Packages
Problems
CHAPTER 32: Case Studies: Partial Differential Equations
32.1 One-Dimensional Mass Balance of a Reactor (Chemical/Bio Engineering)
32.2 Deflections of a Plate (Civil/Environmental Engineering)
32.3 Two-Dimensional Electrostatic Field Problems (Electrical Engineering)
32.4 Finite-Element Solution of a Series of Springs (Mechanical/Aerospace Engineering)
Problems
EPILOGUE: PART EIGHT
PT8.3 Trade-Offs
PT8.4 Important Relationships and Formulas
PT8.5 Advanced Methods and Additional References
APPENDIX A: THE FOURIER SERIES
APPENDIX B: GETTING STARTED WITH MATLAB
APPENDIX C: GETTING STARTED WITH MATHCAD
BIBLIOGRAPHY
INDEX
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 soft-
ware packages are now regularly used for numerical analysis, this eagerly anticipated revision
maintains its strong focus on appropriate use of computational tools.
Features include:
which are based on exciting new areas such as bioengineering.
adaptive quadrature.
and differential equations.
students using this text will be able to apply their new skills to their chosen fi eld.
For more information, please visit www.mhhe.com/chapra
Electronic Textbook Options
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Sixth
Edition
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cha01064_fm.qxd 3/25/09 10:51 AM Page i
Numerical Methods
for Engineers
SIXTH EDITION
Steven C. Chapra
Berger Chair in Computing and Engineering
Tufts University
Raymond P. Canale
Professor Emeritus of Civil Engineering
University of Michigan
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 10020.
Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. Previous editions © 2006, 2002, and 1998. 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.
1 2 3 4 5 6 7 8 9 0 VNH/VNH 0 9
ISBN 978–0–07–340106–5
MHID 0–07–340106–4
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 978–0–07–340106–5 — ISBN 0–07–340106–4 (hard copy : alk. paper)
1. Engineering mathematics—Data processing. 2. Numerical calculations—Data processing 3. Microcomputers—
Programming.
I. Canale, Raymond P.
II. Title.
TA345.C47 2010
518.02462—dc22
www.mhhe.com
2008054296
cha01064_fm.qxd 3/25/09 10:51 AM Page iii
To
Margaret and Gabriel Chapra
Helen and Chester Canale
cha01064_fm.qxd 3/26/09 5:25 PM Page iv
CONTENTS
PREFACE xiv
GUIDED TOUR xvi
ABOUT THE AUTHORS xviii
PART ONE
MODELING,
COMPUTERS, AND
ERROR ANALYSIS 3
PT1.1 Motivation 3
PT1.2 Mathematical Background 5
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 25
2.1 Packages and Programming 25
2.2 Structured Programming 26
2.3 Modular Programming 35
2.4 Excel 37
2.5 MATLAB 41
2.6 Mathcad 45
2.7 Other Languages and Libraries 46
Problems 47
CHAPTER 3
Approximations and Round-Off Errors 52
3.1 Significant Figures 53
3.2 Accuracy and Precision 55
3.3 Error Definitions 56
3.4 Round-Off Errors 62
Problems 76
iv
cha01064_fm.qxd 3/25/09 10:51 AM Page v
CONTENTS
v
CHAPTER 4
Truncation Errors and the Taylor Series 78
4.1 The Taylor Series 78
4.2 Error Propagation 94
4.3 Total Numerical Error 98
4.4 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
EQUATIONS 113
PT2.1 Motivation 113
PT2.2 Mathematical Background 115
PT2.3 Orientation 116
CHAPTER 5
Bracketing Methods 120
5.1 Graphical Methods 120
5.2 The Bisection Method 124
5.3 The False-Position Method 132
5.4 Incremental Searches and Determining Initial Guesses 138
Problems 139
CHAPTER 6
Open Methods 142
6.1 Simple Fixed-Point Iteration 143
6.2 The Newton-Raphson Method 148
6.3 The Secant Method 154
6.4 Brent’s Method 159
6.5 Multiple Roots 164
6.6 Systems of Nonlinear Equations 167
Problems 171
CHAPTER 7
Roots of Polynomials 174
7.1 Polynomials in Engineering and Science 174
7.2 Computing with Polynomials 177
7.3 Conventional Methods 180
cha01064_fm.qxd 3/25/09 10:51 AM Page vi
vi
CONTENTS
7.4 Müller’s Method 181
7.5 Bairstow’s Method 185
7.6 Other Methods 190
7.7 Root Location with Software Packages 190
Problems 200
CHAPTER 8
Case Studies: Roots of Equations 202
8.1 Ideal and Nonideal Gas Laws (Chemical/Bio Engineering) 202
8.2 Greenhouse Gases and Rainwater (Civil/Environmental Engineering) 205
8.3 Design of an Electric Circuit (Electrical Engineering) 207
8.4 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
EQUATIONS 227
PT3.1 Motivation 227
PT3.2 Mathematical Background 229
PT3.3 Orientation 237
CHAPTER 9
Gauss Elimination 241
9.1 Solving Small Numbers of Equations 241
9.2 Naive Gauss Elimination 248
9.3 Pitfalls of Elimination Methods 254
9.4 Techniques for Improving Solutions 260
9.5 Complex Systems 267
9.6 Nonlinear Systems of Equations 267
9.7 Gauss-Jordan 269
9.8 Summary 271
Problems 271
CHAPTER 10
LU Decomposition and Matrix Inversion 274
10.1 LU Decomposition 274
10.2 The Matrix Inverse 283
10.3 Error Analysis and System Condition 287
Problems 293
cha01064_fm.qxd 3/25/09 10:51 AM Page vii
CONTENTS
vii
CHAPTER 11
Special Matrices and Gauss-Seidel 296
11.1 Special Matrices 296
11.2 Gauss-Seidel 300
11.3 Linear Algebraic Equations with Software Packages 307
Problems 312
CHAPTER 12
Case Studies: Linear Algebraic Equations 315
12.1 Steady-State Analysis of a System of Reactors (Chemical/Bio
Engineering) 315
12.2 Analysis of a Statically Determinate Truss (Civil/Environmental
Engineering) 318
12.3 Currents and Voltages in Resistor Circuits (Electrical Engineering) 322
12.4 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 351
13.1 Golden-Section Search 352
13.2 Parabolic Interpolation 359
13.3 Newton’s Method 361
13.4 Brent’s Method 364
Problems 364
CHAPTER 14
Multidimensional Unconstrained Optimization 367
14.1 Direct Methods 368
14.2 Gradient Methods 372
Problems 385
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