Applied Numerical Methods With MATLAB by Steven Chapra, Study notes of Numerical Methods in Engineering

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APPLIED NUMERICAL METHODS WITH MATLAB FOR ENGINEERS AND SCIENTISTS, THIRD EDITION

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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.

QA297.C4185 2012 518–dc22 2010044481

www.mhhe.com

To

My brothers,

John and Bob Chapra

v

CONTENTS

About the Author iv

• PART ONE Modeling, Computers, and Error Analysis Preface xiii
  • 1.1 Motivation
  • 1.2 Part Organization
• CHAPTER - and Problem Solving Mathematical Modeling, Numerical Methods, - 1.1 A Simple Mathematical Model - 1.2 Conservation Laws in Engineering and Science - 1.3 Numerical Methods Covered in This Book - 1.4 Case Study: It’s a Real Drag - Problems
• CHAPTER - MATLAB Fundamentals - 2.1 The MATLAB Environment - 2.2 Assignment - 2.3 Mathematical Operations - 2.4 Use of Built-In Functions - 2.5 Graphics - 2.6 Other Resources - 2.7 Case Study: Exploratory Data Analysis - Problems
• CHAPTER - Programming with MATLAB - 3.1 M-Files - 3.2 Input-Output - 3.3 Structured Programming - 3.4 Nesting and Indentation - 3.5 Passing Functions to M-Files - 3.6 Case Study: Bungee Jumper Velocity - Problems
• CHAPTER - Roundoff and Truncation Errors - 4.1 Errors - 4.2 Roundoff Errors - 4.3 Truncation Errors - 4.4 Total Numerical Error - 4.5 Blunders, Model Errors, and Data Uncertainty - Problems
  • PART TWO Roots and Optimization
    • 2.1 Overview
    • 2.2 Part Organization
• CHAPTER - Roots: Bracketing Methods - 5.1 Roots in Engineering and Science - 5.2 Graphical Methods - 5.3 Bracketing Methods and Initial Guesses - 5.4 Bisection - 5.5 False Position - 5.6 Case Study: Greenhouse Gases and Rainwater - Problems
• CHAPTER - Roots: Open Methods - 6.1 Simple Fixed-Point Iteration - 6.2 Newton-Raphson - 6.3 Secant Methods - 6.4 Brent’s Method - 6.5 MATLAB Function: fzero - 6.6 Polynomials - 6.7 Case Study: Pipe Friction - Problems - CHAPTER - Optimization - 7.1 Introduction and Background - 7.2 One-Dimensional Optimization - 7.3 Multidimensional Optimization - 7.4 Case Study: Equilibrium and Minimum Potential Energy - Problems
  • PART THREE Linear Systems - 3.1 Overview - 3.2 Part Organization
    • CHAPTER
      • Linear Algebraic Equations and Matrices
      • 8.1 Matrix Algebra Overview
      • 8.2 Solving Linear Algebraic Equations with MATLAB
      • 8.3 Case Study: Currents and Voltages in Circuits
      • Problems
    • CHAPTER
      • Gauss Elimination
      • 9.1 Solving Small Numbers of Equations
      • 9.2 Naive Gauss Elimination
      • 9.3 Pivoting
      • 9.4 Tridiagonal Systems
      • 9.5 Case Study: Model of a Heated Rod
      • Problems
• CHAPTER - LU Factorization - 10.1 Overview of LU Factorization - 10.2 Gauss Elimination as LU Factorization - 10.3 Cholesky Factorization - 10.4 MATLAB Left Division - Problems
• CHAPTER - Matrix Inverse and Condition - 11.1 The Matrix Inverse - 11.2 Error Analysis and System Condition - 11.3 Case Study: Indoor Air Pollution - Problems
• CHAPTER - Iterative Methods - 12.1 Linear Systems: Gauss-Seidel - 12.2 Nonlinear Systems - 12.3 Case Study: Chemical Reactions - Problems
• CHAPTER - Eigenvalues - 13.1 Mathematical Background - 13.2 Physical Background - 13.3 The Power Method - 13.4 MATLAB Function: eig - 13.5 Case Study: Eigenvalues and Earthquakes - Problems
  • PART FOUR Curve Fitting
    • 4.1 Overview
    • 4.2 Part Organization
• CHAPTER - Linear Regression - 14.1 Statistics Review - 14.2 Random Numbers and Simulation - 14.3 Linear Least-Squares Regression - 14.4 Linearization of Nonlinear Relationships - 14.5 Computer Applications - 14.6 Case Study: Enzyme Kinetics - Problems
• CHAPTER
  • General Linear Least-Squares and Nonlinear Regression
  • 15.1 Polynomial Regression
  • 15.2 Multiple Linear Regression
  • 15.3 General Linear Least Squares
  • 15.4 QR Factorization and the Backslash Operator
  • 15.5 Nonlinear Regression
  • 15.6 Case Study: Fitting Experimental Data
  • Problems
• CHAPTER
  • Fourier Analysis
  • 16.1 Curve Fitting with Sinusoidal Functions
  • 16.2 Continuous Fourier Series
  • 16.3 Frequency and Time Domains
  • 16.4 Fourier Integral and Transform
  • 16.5 Discrete Fourier Transform (DFT)
  • 16.6 The Power Spectrum
  • 16.7 Case Study: Sunspots
  • Problems
• CHAPTER
  • Polynomial Interpolation
  • 17.1 Introduction to Interpolation
  • 17.2 Newton Interpolating Polynomial
  • 17.3 Lagrange Interpolating Polynomial
  • 17.4 Inverse Interpolation
  • 17.5 Extrapolation and Oscillations
  • Problems
• CHAPTER
  • Splines and Piecewise Interpolation
  • 18.1 Introduction to Splines
  • 18.2 Linear Splines
  • 18.3 Quadratic Splines
  • 18.4 Cubic Splines
  • 18.5 Piecewise Interpolation in MATLAB
  • 18.6 Multidimensional Interpolation
  • 18.7 Case Study: Heat Transfer
  • Problems
  • PART FIVE Integration and Differentiation
    • 5.1 Overview
    • 5.2 Part Organization
• CHAPTER - Numerical Integration Formulas - 19.1 Introduction and Background - 19.2 Newton-Cotes Formulas - 19.3 The Trapezoidal Rule - 19.4 Simpson’s Rules - 19.5 Higher-Order Newton-Cotes Formulas - 19.6 Integration with Unequal Segments - 19.7 Open Methods - 19.8 Multiple Integrals - 19.9 Case Study: Computing Work with Numerical Integration - Problems
• CHAPTER - Numerical Integration of Functions - 20.1 Introduction - 20.2 Romberg Integration - 20.3 Gauss Quadrature - 20.4 Adaptive Quadrature - 20.5 Case Study: Root-Mean-Square Current - Problems
• CHAPTER - Numerical Differentiation - 21.1 Introduction and Background - 21.2 High-Accuracy Differentiation Formulas - 21.3 Richardson Extrapolation - 21.4 Derivatives of Unequally Spaced Data - 21.5 Derivatives and Integrals for Data with Errors - 21.6 Partial Derivatives - 21.7 Numerical Differentiation with MATLAB - 21.8 Case Study: Visualizing Fields - Problems
  • PART SIX Ordinary Differential Equations
    • 6.1 Overview
    • 6.2 Part Organization
• CHAPTER - Initial-Value Problems - 22.1 Overview - 22.2 Euler’s Method - 22.3 Improvements of Euler’s Method - 22.4 Runge-Kutta Methods - 22.5 Systems of Equations - 22.6 Case Study: Predator-Prey Models and Chaos - Problems
• CHAPTER - Adaptive Methods and Stiff Systems - 23.1 Adaptive Runge-Kutta Methods - 23.2 Multistep Methods - 23.3 Stiffness - 23.4 MATLAB Application: Bungee Jumper with Cord - 23.5 Case Study: Pliny’s Intermittent Fountain - Problems
• CHAPTER - Boundary-Value Problems - 24.1 Introduction and Background - 24.2 The Shooting Method - 24.3 Finite-Difference Methods - Problems - APPENDIX A: MATLAB BUILT-IN FUNCTIONS - APPENDIX B: MATLAB M-FILE FUNCTIONS - BIBLIOGRAPHY - INDEX

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FIGURE P.1 An outline of this edition. The shaded areas represent new material. In addition, several of the original chapters have been supplemented withnew topics. xiv

PART ONE

PART TWO

PART THREE

PART FOUR

PART FIVE

PART SIX

Modeling, Computers,

Roots and

Linear Systems

Curve Fitting

Integration and

Ordinary Differential

and Error Analysis

Optimization

Differentiation

Equations

CHAPTER 1

CHAPTER 5

CHAPTER 8

CHAPTER 14

CHAPTER 19

CHAPTER 22

Mathematical

Roots: Bracketing

Linear Algebraic

Linear Regression

Numerical Integration

Initial-Value

Modeling, Numerical

Methods

Equations

Formulas

Problems

Methods, and Problem

and Matrices

SolvingCHAPTER 2

CHAPTER 6

CHAPTER 9

CHAPTER 15

CHAPTER 20

CHAPTER 23

MATLAB

Roots: Open

Gauss Elimination

General Linear

Numerical lntegration

Adaptive Methods

Fundamentals

Methods

Least-Squares and

of Functions

and Stiff Systems

Nonlinear Regression

CHAPTER 3

CHAPTER 7

CHAPTER 10

CHAPTER 16

CHAPTER 21

CHAPTER 24

Programming

Optimization

LU

Factorization

Fourier Analysis

Numerical

Boundary-Value

with MATLAB

Differentiation

Problems

CHAPTER 4

CHAPTER 11

CHAPTER 17

Roundoff and

Matrix Inverse

Polynomial

Truncation Errors

and Condition

Interpolation

CHAPTER 12

CHAPTER 18

Iterative Methods

Splines and PiecewiseInterpolation

CHAPTER 13Eigenvalues

PREFACE xv

to move this material up to what I consider to be its more natural mathematical po-

sition at the end of the section on linear algebraic equations. By stressing applica-

tions (in particular, the use of eigenvalues to study vibrations), I have found that

students respond very positively to the subject in this position. In addition, it allows

me to return to the topic in subsequent chapters which serves to enhance the

students’ appreciation of the topic.

• Fourier Analysis. In past years, if time permitted, I also usually presented a lecture

at the end of the semester on Fourier analysis. Over the past two years, I have begun

presenting this material at its more natural position just after the topic of linear least

squares. I motivate the subject matter by using the linear least-squares approach to

fit sinusoids to data. Then, by stressing applications (again vibrations), I have found

that the students readily absorb the topic and appreciate its value in engineering and

science.

It should be noted that both chapters are written in a modular fashion and could

be skipped without detriment to the course’s pedagogical arc. Therefore, if you

choose, you can either omit them from your course or perhaps move them to the

end of the semester. In any event, I would not have included them in the current

edition if they did not represent an enhancement within my current experience in

the classroom. In particular, based on my teaching evaluations, I find that the

stronger, more motivated students actually see these topics as highlights. This is

particularly true because MATLAB greatly facilitates their application and inter-

pretation.

2. New Content. Beyond the new chapters, I have included new and enhanced sections on a

number of topics. The primary additions include sections on animation (Chap. 3), Brent’s

method for root location (Chap. 6), LU factorization with pivoting (Chap. 8), ran-

dom numbers and Monte Carlo simulation (Chap. 14), adaptive quadrature (Chap. 20),

and event termination of ODEs (Chap. 23).

3. New Homework Problems. Most of the end-of-chapter problems have been modi-

fied, 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

difficult than the problems in the previous edition.

Aside from the new material and problems, the third edition is very similar to the second.

In particular, I have endeavored to maintain most of the features contributing to its pedagog-

ical 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 straightfor-

ward 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.

PEDAGOGICAL TOOLS

Theory Presented as It Informs Key Concepts. The text is intended for Numerical Meth-

ods users, not developers. Therefore, theory is not included for “theory’s sake,” for example no

proofs. Theory is included as it informs key concepts such as the Taylor series, convergence,

condition, etc. Hence, the student is shown how the theory connects with practical issues in

problem solving.

Introductory MATLAB Material. The text includes two introductory chapters on how to

use MATLAB. Chapter 2 shows students how to perform 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 numerical 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 engi-

neering 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, and

Appendix B contains M-file functions.

Textbook Website. A text-specific website is available at www.mhhe.com/chapra. Re-

sources include the text images in PowerPoint, M-files, and additional MATLAB resources.

PREFACE xvii

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