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Applied Numerical Methods with MATLAB® for Engineers and Scientists

Fourth Edition

Steven C. Chapra Berger Chair in Computing and Engineering

Tufts University

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Published by McGraw-Hill Education, 2 Penn Plaza, New York, NY 10121. Copyright © 2018 by McGraw-Hill Education. All rights reserved. Printed in the United States of America. Previous editions © 2012, 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 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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Library of Congress Cataloging-in-Publication Data Chapra, Steven C., author. Applied numerical methods with MATLAB for engineers and scientists / Steven C. Chapra, Berger Chair in Computing and Engineering, Tufts University. Fourth edition. | New York, NY : McGraw-Hill Education, [2018] | Includes bibliographical references and index. LCCN 2016038044 | ISBN 9780073397962 (alk. paper) | ISBN 0073397962 (alk. paper) LCSH: Numerical analysis—Data processing—Textbooks. | Engineering mathematics—Textbooks. | MATLAB—Textbooks. LCC QA297 .C4185 2018 | DDC 518—dc23 LC record available at https://lccn.loc.gov/2016038044 The Internet addresses listed in the text were accurate at the time of publication. The inclusion of a website does not indicate an endorsement by the authors or McGraw-Hill Education, and McGraw-Hill Education does not guarantee the accuracy of the information presented at these sites.


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My brothers,

John and Bob Chapra


Fred Berger (1947–2015)

who I miss as a good friend, a good man.

and a comrade in bringing the light of engineering

to some of world’s darker corners.

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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. 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 at Texas A&M University (1986 Tenneco Award), the University of Colorado (1992 Hutchinson Award), and Tufts University (2011 Professor of the Year Award).

Steve was originally drawn to environmental engineering and science because of his love of the outdoors. 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 math- ematics, 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 http://engineering.tufts.edu/cee/people/chapra/ or e-mail him at [email protected]

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About the Author iv

Preface xiv

Part One Modeling, Computers, and Error Analysis 1

1.1 Motivation 1 1.2 Part Organization 2


Mathematical Modeling, Numerical Methods, and Problem Solving 4

1.1 A Simple Mathematical Model 5 1.2 Conservation Laws in Engineering and Science 12 1.3 Numerical Methods Covered in This Book 13 1.4 Case Study: It’s a Real Drag 17 Problems 20


MATLAB Fundamentals 27

2.1 The MATLAB Environment 28 2.2 Assignment 29 2.3 Mathematical Operations 36 2.4 Use of Built-In Functions 39 2.5 Graphics 42 2.6 Other Resources 46 2.7 Case Study: Exploratory Data Analysis 46 Problems 49


Programming with MATLAB 53

3.1 M-Files 54 3.2 Input-Output 61

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3.3 Structured Programming 65 3.4 Nesting and Indentation 79 3.5 Passing Functions to M-Files 81 3.6 Case Study: Bungee Jumper Velocity 87 Problems 91


Roundoff and Truncation Errors 99

4.1 Errors 100 4.2 Roundoff Errors 106 4.3 Truncation Errors 114 4.4 Total Numerical Error 125 4.5 Blunders, Model Errors, and Data Uncertainty 130 Problems 131

Part Two Roots and Optimization 135

2.1 Overview 135 2.2 Part Organization 136


Roots: Bracketing Methods 138

5.1 Roots in Engineering and Science 139 5.2 Graphical Methods 140 5.3 Bracketing Methods and Initial Guesses 141 5.4 Bisection 146 5.5 False Position 152 5.6 Case Study: Greenhouse Gases and Rainwater 156 Problems 159


Roots: Open Methods 164

6.1 Simple Fixed-Point Iteration 165 6.2 Newton-Raphson 169 6.3 Secant Methods 174 6.4 Brent’s Method 176 6.5 MATLAB Function: fzero 181 6.6 Polynomials 183 6.7 Case Study: Pipe Friction 186 Problems 191

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Optimization 198

7.1 Introduction and Background 199 7.2 One-Dimensional Optimization 202 7.3 Multidimensional Optimization 211 7.4 Case Study: Equilibrium and Minimum Potential Energy 213 Problems 215

Part Three Linear Systems 223

3.1 Overview 223 3.2 Part Organization 225


Linear Algebraic Equations and Matrices 227

8.1 Matrix Algebra Overview 229 8.2 Solving Linear Algebraic Equations with MATLAB 238 8.3 Case Study: Currents and Voltages in Circuits 240 Problems 244


Gauss Elimination 248

9.1 Solving Small Numbers of Equations 249 9.2 Naive Gauss Elimination 254 9.3 Pivoting 261 9.4 Tridiagonal Systems 264 9.5 Case Study: Model of a Heated Rod 266 Problems 270


LU Factorization 274

10.1 Overview of LU Factorization 275 10.2 Gauss Elimination as LU Factorization 276 10.3 Cholesky Factorization 283 10.4 MATLAB Left Division 286 Problems 287

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Matrix Inverse and Condition 288

11.1 The Matrix Inverse 288 11.2 Error Analysis and System Condition 292 11.3 Case Study: Indoor Air Pollution 297 Problems 300


Iterative Methods 305

12.1 Linear Systems: Gauss-Seidel 305 12.2 Nonlinear Systems 312 12.3 Case Study: Chemical Reactions 320 Problems 323


Eigenvalues 326

13.1 Mathematical Background 328 13.2 Physical Background 331 13.3 The Power Method 333 13.4 MATLAB Function: eig 336 13.5 Case Study: Eigenvalues and Earthquakes 337 Problems 340

Part Four Curve Fitting 343

4.1 Overview 343 4.2 Part Organization 345


Linear Regression 346

14.1 Statistics Review 348 14.2 Random Numbers and Simulation 353 14.3 Linear Least-Squares Regression 358 14.4 Linearization of Nonlinear Relationships 366 14.5 Computer Applications 370 14.6 Case Study: Enzyme Kinetics 373 Problems 378

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General Linear Least-Squares and Nonlinear Regression 385

15.1 Polynomial Regression 385 15.2 Multiple Linear Regression 389 15.3 General Linear Least Squares 391 15.4 QR Factorization and the Backslash Operator 394 15.5 Nonlinear Regression 395 15.6 Case Study: Fitting Experimental Data 397 Problems 399


Fourier Analysis 404

16.1 Curve Fitting with Sinusoidal Functions 405 16.2 Continuous Fourier Series 411 16.3 Frequency and Time Domains 414 16.4 Fourier Integral and Transform 415 16.5 Discrete Fourier Transform (DFT) 418 16.6 The Power Spectrum 423 16.7 Case Study: Sunspots 425 Problems 426


Polynomial Interpolation 429

17.1 Introduction to Interpolation 430 17.2 Newton Interpolating Polynomial 433 17.3 Lagrange Interpolating Polynomial 441 17.4 Inverse Interpolation 444 17.5 Extrapolation and Oscillations 445 Problems 449


Splines and Piecewise Interpolation 453

18.1 Introduction to Splines 453 18.2 Linear Splines 455 18.3 Quadratic Splines 459 18.4 Cubic Splines 462 18.5 Piecewise Interpolation in MATLAB 468 18.6 Multidimensional Interpolation 473 18.7 Case Study: Heat Transfer 476 Problems 480

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Part Five Integration and Differentiation 485

5.1 Overview 485 5.2 Part Organization 486


Numerical Integration Formulas 488

19.1 Introduction and Background 489 19.2 Newton-Cotes Formulas 492 19.3 The Trapezoidal Rule 494 19.4 Simpson’s Rules 501 19.5 Higher-Order Newton-Cotes Formulas 507 19.6 Integration with Unequal Segments 508 19.7 Open Methods 512 19.8 Multiple Integrals 512 19.9 Case Study: Computing Work with Numerical Integration 515 Problems 518


Numerical Integration of Functions 524

20.1 Introduction 524 20.2 Romberg Integration 525 20.3 Gauss Quadrature 530 20.4 Adaptive Quadrature 537 20.5 Case Study: Root-Mean-Square Current 540 Problems 544


Numerical Differentiation 548

21.1 Introduction and Background 549 21.2 High-Accuracy Differentiation Formulas 552 21.3 Richardson Extrapolation 555 21.4 Derivatives of Unequally Spaced Data 557 21.5 Derivatives and Integrals for Data with Errors 558 21.6 Partial Derivatives 559 21.7 Numerical Differentiation with MATLAB 560 21.8 Case Study: Visualizing Fields 565 Problems 567

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Part six Ordinary Differential Equations 573

6.1 Overview 573 6.2 Part Organization 577


Initial-Value Problems 579

22.1 Overview 581 22.2 Euler’s Method 581 22.3 Improvements of Euler’s Method 587 22.4 Runge-Kutta Methods 593 22.5 Systems of Equations 598 22.6 Case Study: Predator-Prey Models and Chaos 604 Problems 609


Adaptive Methods and Stiff Systems 615

23.1 Adaptive Runge-Kutta Methods 615 23.2 Multistep Methods 624 23.3 Stiffness 628 23.4 MATLAB Application: Bungee Jumper with Cord 634 23.5 Case Study: Pliny’s Intermittent Fountain 635 Problems 640


Boundary-Value Problems 646

24.1 Introduction and Background 647 24.2 The Shooting Method 651 24.3 Finite-Difference Methods 658 24.4 MATLAB Function: bvp4c 665 Problems 668






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This book is designed to support a one-semester course in numerical methods. It has been written for students who want to learn and apply numerical methods in order to solve prob- lems in engineering and science. As such, the methods are motivated by problems rather than by mathematics. That said, sufficient theory is provided so that students come away with insight into the techniques and their shortcomings.

MATLAB® provides a great environment for such a course. Although other en- vironments (e.g., Excel/VBA, Mathcad) or languages (e.g., Fortran 90, C++) could have been chosen, MATLAB presently offers a nice combination of handy program- ming features with powerful built-in numerical capabilities. On the one hand, its M-file programming environment allows students to implement moderately compli- cated algorithms in a structured and coherent fashion. On the other hand, its built-in, numerical capabilities empower students to solve more difficult problems without try- ing to “reinvent the wheel.”

The basic content, organization, and pedagogy of the third edition are essentially pre- served in the fourth edition. In particular, the conversational writing style is intentionally maintained in order to make the book easier to read. This book tries to speak directly to the reader and is designed in part to be a tool for self-teaching.

That said, this edition differs from the past edition in three major ways: (1) new material, (2) new and revised homework problems, and (3) an appendix introducing Simulink.

1. New Content. I have included new and enhanced sections on a number of topics. The primary additions include material on some MATLAB functions not included in previ- ous editions (e.g., fsolve, integrate, bvp4c), some new applications of Monte Carlo for problems such as integration and optimization, and MATLAB’s new way to pass parameters to function functions.

2. New Homework Problems. Most of the end-of-chapter problems have been modified, and a variety of new problems have been added. In particular, an effort has been made to include several new problems for each chapter that are more challenging and dif- ficult than the problems in the previous edition.

3. I have developed a short primer on Simulink which I have my students read prior to covering that topic. Although I recognize that some professors may not choose to cover Simulink, I included it as a teaching aid for those that do.

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Aside from the new material and problems, the fourth edition is very similar to the third. In particular, I have endeavored to maintain most of the features contributing to its pedagogical effectiveness including extensive use of worked examples and engineering and scien tific applications. As with the previous edition, I have made a concerted effort to make this book as “student-friendly” as possible. Thus, I’ve tried to keep my explanations straight- forward and practical.

Although my primary intent is to empower students by providing them with a sound introduction to numerical problem solving, I have the ancillary objective of making this introduction exciting and pleasurable. I believe that motivated students who enjoy engi- neering and science, problem solving, mathematics—and yes—programming, will ulti- mately make better professionals. If my book fosters enthusiasm and appreciation for these subjects, I will consider the effort a success.

Acknowledgments. Several members of the McGraw-Hill team have contributed to this  project. Special thanks are due to Jolynn Kilburg, Thomas Scaife, Ph.D., Chelsea Haupt, Ph.D., and Jeni McAtee for their encouragement, support, and direction.

During the course of this project, the folks at The MathWorks, Inc., have truly dem- onstrated their overall excellence as well as their strong commitment to engineering and science education. In particular, Naomi Fernandes of The MathWorks, Inc., Book Program has been especially helpful and Jared Wasserman of the MathWorks Technical Support Department was of great help with technical questions.

The generosity of the Berger family has provided me with the opportunity to work on creative projects such as this book dealing with computing and engineering. In addition, my colleagues in the School of Engineering at Tufts, notably Masoud Sanayei, Babak Moaveni, Luis Dorfmann, Rob White, Linda Abriola, and Laurie Baise, have been very supportive and helpful.

Significant suggestions were also given by a number of colleagues. In particular, Dave Clough (University of Colorado–Boulder), and Mike Gustafson (Duke University) pro- vided valuable ideas and suggestions. In addition, a number of reviewers provided use- ful feedback and advice including Karen Dow Ambtman (University of Alberta), Jalal Behzadi (Shahid Chamran University), Eric Cochran (Iowa State University), Frederic Gibou (University of California at Santa Barbara), Jane Grande-Allen (Rice University), Raphael Haftka (University of Florida), Scott Hendricks (Virginia Tech University), Ming Huang (University of San Diego), Oleg Igoshin (Rice University), David Jack (Baylor Uni- versity), Se Won Lee (Sungkyunkwan University), Clare McCabe (Vanderbilt University), Eckart Meiburg (University of California at Santa Barbara), Luis Ricardez (University of Waterloo), James Rottman (University of California, San Diego), Bingjing Su (University of Cincinnati), Chin-An Tan (Wayne State University), Joseph Tipton (The University of Evansville), Marion W. Vance (Arizona State University), Jonathan Vande Geest (University of Arizona), Leah J. Walker (Arkansas State University), Qiang Hu (University of Alabama, Huntsville), Yukinobu Tanimoto (Tufts University), Henning T. Søgaard (Aarhus University), and Jimmy Feng (University of British Columbia).

It should be stressed that although I received useful advice from the aforementioned individuals, I am responsible for any inaccuracies or mistakes you may find in this book. Please contact me via e-mail if you should detect any errors.


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Finally, I want to thank my family, and in particular my wife, Cynthia, for the love, patience, and support they have provided through the time I’ve spent on this project.

Steven C. Chapra Tufts University

Medford, Massachusetts [email protected]


Theory Presented as It Informs Key Concepts. The text is intended for Numerical Methods users, not developers. Therefore, theory is not included for “theory’s sake,” for ex- ample no proofs. Theory is included as it informs key concepts such as the Taylor series, con- vergence, condition, etc. Hence, the student is shown how the theory connects with practical issues in problem solving.

Introductory MATLAB Material. The text in cludes two introductory chapters on how to use MATLAB. Chapter 2 shows students how to per form computations and create graphs in MATLAB’s standard command mode. Chapter 3 provides a primer on developing numerical programs via MATLAB M-file functions. Thus, the text provides students with the means to develop their own nu merical algorithms as well as to tap into MATLAB’s powerful built-in routines.

Algorithms Presented Using MATLAB M-files. Instead of using pseudocode, this book presents algorithms as well-structured MATLAB M-files. Aside from being useful com- puter programs, these provide students with models for their own M-files that they will develop as homework exercises.

Worked Examples and Case Studies. Extensive worked examples are laid out in detail so that students can clearly follow the steps in each numerical computation. The case stud- ies consist of engineering and science applications which are more complex and richer than the worked examples. They are placed at the ends of selected chapters with the intention of (1) illustrating the nuances of the methods and (2) showing more realistically how the methods along with MATLAB are applied for problem solving.

Problem Sets. The text includes a wide variety of problems. Many are drawn from en- gineering and scientific disciplines. Others are used to illustrate numerical techniques and theoretical concepts. Problems include those that can be solved with a pocket calculator as well as others that require computer solution with MATLAB.

Useful Appendices and Indexes. Appendix A contains MATLAB commands, Appendix B contains M-file functions, and new Appendix C contains a brief Simulink primer.

Instructor Resources. Solutions Manual, Lecture PowerPoints, Text images in Power- Point, M-files and additional MATLAB resources are available through Connect®.

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Part One

Modeling, Computers, and Error Analysis


What are numerical methods and why should you study them? Numerical methods are techniques by which mathematical problems are formulated

so that they can be solved with arithmetic and logical operations. Because digital comput- ers excel at performing such operations, numerical methods are sometimes referred to as computer mathematics.

In the pre–computer era, the time and drudgery of implementing such calculations seriously limited their practical use. However, with the advent of fast, inexpensive digital computers, the role of numerical methods in engineering and scientific problem solving has exploded. Because they figure so prominently in much of our work, I believe that numerical methods should be a part of every engineer’s and scientist’s basic education. Just as we all must have solid foundations in the other areas of mathematics and science, we should also have a fundamental understanding of numerical methods. In particular, we

should have a solid appreciation of both their capabilities and their limitations.

Beyond contributing to your overall education, there are several additional reasons why you should study numerical methods:

1. Numerical methods greatly expand the types of problems you can address. They are capable of handling large sys- tems of equations, nonlinearities, and complicated geometries that are not uncommon in engineering and science and that are often impossible to solve analytically with standard calculus. As such, they greatly enhance your prob- lem-solving skills.

2. Numerical methods allow you to use “canned” software with insight. During

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2 PART 1 ModEling, CoMPuTERs, And ERRoR AnAlysis

your career, you will invariably have occasion to use commercially available prepack- aged computer programs that involve numerical methods. The intelligent use of these programs is greatly enhanced by an understanding of the basic theory underlying the methods. In the absence of such understanding, you will be left to treat such packages as “black boxes” with little critical insight into their inner workings or the validity of the results they produce.

3. Many problems cannot be approached using canned programs. If you are conversant with numerical methods, and are adept at computer programming, you can design your own programs to solve problems without having to buy or commission expensive software.

4. Numerical methods are an efficient vehicle for learning to use computers. Because nu- merical methods are expressly designed for computer implementation, they are ideal for illustrating the computer’s powers and limitations. When you successfully implement numerical methods on a computer, and then apply them to solve otherwise intractable problems, you will be provided with a dramatic demonstration of how computers can serve your professional development. At the same time, you will also learn to acknowl- edge and control the errors of approximation that are part and parcel of large-scale numerical calculations.

5. Numerical methods provide a vehicle for you to reinforce your understanding of math- ematics. Because one function of numerical methods is to reduce higher mathematics to basic arithmetic operations, they get at the “nuts and bolts” of some otherwise obscure topics. Enhanced understanding and insight can result from this alternative perspective.

With these reasons as motivation, we can now set out to understand how numerical methods and digital computers work in tandem to generate reliable solutions to mathemati- cal problems. The remainder of this book is devoted to this task.


This book is divided into six parts. The latter five parts focus on the major areas of nu- merical methods. Although it might be tempting to jump right into this material, Part One consists of four chapters dealing with essential background material.

Chapter 1 provides a concrete example of how a numerical method can be employed to solve a real problem. To do this, we develop a mathematical model of a free-falling bungee jumper. The model, which is based on Newton’s second law, results in an ordinary differential equation. After first using calculus to develop a closed-form solution, we then show how a comparable solution can be generated with a simple numerical method. We end the chapter with an overview of the major areas of numerical methods that we cover in Parts Two through Six.

Chapters 2 and 3 provide an introduction to the MATLAB® software environment. Chapter 2 deals with the standard way of operating MATLAB by entering commands one at a time in the so-called calculator, or command, mode. This interactive mode provides a straightforward means to orient you to the environment and illustrates how it is used for common operations such as performing calculations and creating plots.

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1.2 PART oRgAniZATion 3

Chapter 3 shows how MATLAB’s programming mode provides a vehicle for assem- bling individual commands into algorithms. Thus, our intent is to illustrate how MATLAB serves as a convenient programming environment to develop your own software.

Chapter 4 deals with the important topic of error analysis, which must be understood for the effective use of numerical methods. The first part of the chapter focuses on the roundoff errors that result because digital computers cannot represent some quantities …