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1 This page intentionally left blank 12/17/10 8:58 AM Page i Applied Numerical Methods with MATLAB for Engineers and Scientists Third Edition Steven C. Chapra Berger Chair in Computing and Engineering Tufts University TM. TM. APPLIED NUMERICAL METHODS WITH MATLAB FOR ENGINEERS AND SCIENTISTS, THIRD EDITION. Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020. Copyright 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Previous editions 2008 and 2005. 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.

2 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 DOC/DOC 1 0 9 8 7 6 5 4 3 2 1. ISBN 978-0-07-340110-2. MHID 0-07-340110-2. Vice President & Editor-in-Chief: Marty Lange Design Coordinator: Brenda A. Rolwes Vice President EDP/Central Publishing Services: Cover Design: Studio Montage, St. Louis, Missouri Kimberly Meriwether David Cover Credit: Brand X/Jupiter Images RF. Publisher: Raghothaman Srinivasan Buyer: Kara Kudronowicz Sponsoring Editor: Peter E. Massar Media Project Manager: Balaji Sundararaman Marketing Manager: Curt Reynolds Compositor: MPS Limited, a Macmillan Company Development Editor: Lorraine Buczek Typeface: 10/12 Times Project Manager: Melissa M.

3 Leick Printer: Donnelley All credits appearing on page or at the end of the book are considered to be an extension of the copyright page. MATLAB and Simulink are registered trademarks of The MathWorks, Inc. See for a list of additional trademarks. The MathWorks Publisher Logo identifies books that contain MATLAB content. Used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book's use or discussion MATLAB . software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular use of the MATLAB software or related products. For MATLAB and Simulink product information, or information on other related products, please contact: The MathWorks, Inc. 3 Apple Hill Drive Natick, MA, 01760-2098 USA.

4 Tel: 508-647-7000. Fax: 508-647-7001. E-mail: Web: Library of Congress Cataloging-in-Publication Data Chapra, Steven C. Applied numerical methods with MATLAB for engineers and scientists / Steven C. Chapra. 3rd ed. p. cm. ISBN 978-0-07-340110-2 (alk. paper). 1. Numerical analysis Data processing Textbooks. 2. MATLAB Textbooks. I. Title. 2012. 518 dc22 2010044481. 12/17/10 8:58 AM Page iii To My brothers, John and Bob Chapra ABOUT THE AUTHOR. Steve Chapra teaches in the Civil and Environmental Engineering Department at Tufts University, where he holds the Louis Berger Chair in Computing and Engineering. His other books include Numerical Methods for Engineers and Surface Water-Quality Modeling. Steve received engineering degrees from Manhattan College and the University of Michigan.

5 Before joining the faculty at Tufts, he worked for the Environmental Protection Agency and the National Oceanic and Atmospheric Administration, and taught at Texas A&M University and the University of Colorado. His general research interests focus on surface water-quality modeling and advanced computer applications in environmental engineering. He has received a number of awards for his scholarly contributions, including the Rudolph Hering Medal, the Meriam/Wiley Distinguished Author Award, and the Chandler- Misener Award. He has also been recognized as the outstanding teacher among the engi- neering faculties at both Texas A&M University (1986 Tenneco Award) and the University of Colorado (1992 Hutchinson Award). Steve was originally drawn to environmental engineering and science because of his love of the outdoors.

6 He is an avid fly fisherman and hiker. An unapologetic nerd, his love affair with computing began when he was first introduced to Fortran programming as an undergraduate in 1966. Today, he feels truly blessed to be able to meld his love of mathe- matics, science, and computing with his passion for the natural environment. In addition, he gets the bonus of sharing it with others through his teaching and writing! Beyond his professional interests, he enjoys art, music (especially classical music, jazz, and bluegrass), and reading history. Despite unfounded rumors to the contrary, he never has, and never will, voluntarily bungee jump or sky dive. If you would like to contact Steve, or learn more about him, visit his home page at or e-mail him at iv 12/17/10 8:58 AM Page v CONTENTS.

7 About the Author iv Preface xiii PART ONE Modeling, Computers, and Error Analysis 1. Motivation 1. Part Organization 2. CHAPTER 1. Mathematical Modeling, Numerical Methods, and Problem Solving 4. A Simple Mathematical Model 5. Conservation Laws in Engineering and Science 12. Numerical Methods Covered in This Book 13. Case Study: It's a Real Drag 17. Problems 20. CHAPTER 2. MATLAB Fundamentals 24. The MATLAB Environment 25. Assignment 26. Mathematical Operations 32. Use of Built-In Functions 35. Graphics 38. Other Resources 40. Case Study: Exploratory Data Analysis 42. Problems 44. CHAPTER 3. Programming with MATLAB 48. M-Files 49. Input-Output 53. v vi CONTENTS. Structured Programming 57. Nesting and Indentation 71. Passing Functions to M-Files 74. Case Study: Bungee Jumper Velocity 79.

8 Problems 83. CHAPTER 4. Roundoff and Truncation Errors 88. Errors 89. Roundoff Errors 95. Truncation Errors 103. Total Numerical Error 114. Blunders, Model Errors, and Data Uncertainty 119. Problems 120. PART TWO Roots and Optimization 123. Overview 123. Part Organization 124. CHAPTER 5. Roots: Bracketing Methods 126. Roots in Engineering and Science 127. Graphical Methods 128. Bracketing Methods and Initial Guesses 129. Bisection 134. False Position 140. Case Study: Greenhouse Gases and Rainwater 144. Problems 147. CHAPTER 6. Roots: Open Methods 151. Simple Fixed-Point Iteration 152. Newton-Raphson 156. Secant Methods 161. Brent's Method 163. MATLAB Function: fzero 168. Polynomials 170. Case Study: Pipe Friction 173. Problems 178. 12/17/10 8:58 AM Page vii CONTENTS vii CHAPTER 7.

9 Optimization 182. Introduction and Background 183. One-Dimensional Optimization 186. Multidimensional Optimization 195. Case Study: Equilibrium and Minimum Potential Energy 197. Problems 199. PART THREE linear Systems 205. Overview 205. Part Organization 207. CHAPTER 8. linear Algebraic Equations and Matrices 209. Matrix Algebra Overview 211. Solving linear Algebraic Equations with MATLAB 220. Case Study: Currents and Voltages in Circuits 222. Problems 226. CHAPTER 9. Gauss Elimination 229. Solving Small Numbers of Equations 230. Naive Gauss Elimination 235. Pivoting 242. Tridiagonal Systems 245. Case Study: Model of a Heated Rod 247. Problems 251. CHAPTER 10. LU Factorization 254. Overview of LU Factorization 255. Gauss Elimination as LU Factorization 256. Cholesky Factorization 263.

10 MATLAB Left Division 266. Problems 267. viii CONTENTS. CHAPTER 11. Matrix Inverse and Condition 268. The Matrix Inverse 268. Error Analysis and System Condition 272. Case Study: Indoor Air Pollution 277. Problems 280. CHAPTER 12. Iterative Methods 284. linear Systems: Gauss-Seidel 284. nonlinear Systems 291. Case Study: Chemical Reactions 298. Problems 300. CHAPTER 13. Eigenvalues 303. Mathematical Background 305. Physical Background 308. The Power Method 310. MATLAB Function: eig 313. Case Study: Eigenvalues and Earthquakes 314. Problems 317. PART FOUR Curve Fitting 321. Overview 321. Part Organization 323. CHAPTER 14. linear Regression 324. Statistics Review 326. Random Numbers and Simulation 331. linear Least-Squares Regression 336. Linearization of nonlinear Relationships 344.