Numerical Methods: Algorithms and Applications / Edition 1

by Laurene V. Fausett

Overview

This book present the fundamental numerical techniques used in engineering, applied mathematics, computer science, and the physical and life sciences in a manner that is both interesting and understandable. Numerical Analysis with Applications and Algorithms includes comprehensive coverage of solving nonlinear equations of a single variable, numerical linear algebra, nonlinear functions of several variables, numerical methods for data interpolations and approximation, numerical differentiation and integration, and numerical techniques for solving differential equations. This book is useful as a reference for self study.

Product Details

ISBN-13:	9780130314000
Publisher:	Pearson
Publication date:	08/05/2002
Series:	Featured Titles for Numerical Analysis Series
Edition description:	New Edition
Pages:	649
Product dimensions:	8.10(w) x 9.00(h) x 1.50(d)

Read an Excerpt

The purpose of this text is to present the fundamental numerical techniques used in engineering, applied mathematics, computer science, and the physical and life sciences in a manner that is both interesting and understandable to undergraduate and beginning graduate students in those fields. The organization of the chapters, and of the material within each chapter, is designed with student learning as the primary objective. A detailed algorithm is given for each method presented, so that students can write simple programs implementing the technique in the computer language of their choice. Numerous examples of the use of the methods are also included.

The first chapter sets the stage for the material in the rest of the text, giving a brief introduction to the long history of numerical techniques and a "preview of coming attractions" for some of the recurring themes in the remainder of the text. It also presents a summary of the key components of a computer program for solving problems involving numerical techniques such as those given in the text. The trapezoid rule for numerical integration is used to illustrate the relationship between a numerical algorithm and a computer program implementing the algorithm. Sample programs are given in several different languages.

Each of the subsequent chapters begins with a one page overview of the subject matter, together with an indication as to how the topics presented in the chapter are related to those in previous and subsequent chapters. Introductory examples are presented to suggest a few of the types of problems for which the topics of the chapter may be used. Following the sections in which the methods are presented, each chapter concludes with a summary of the most important formulas, of suggestions for further reading, and an extensive set of exercises. The first group of problems provides fairly routine practice of the techniques; the second group includes applications adapted from a variety of fields, and the final group of problems encourages students to extend their understanding of either the theoretical or the computational aspects of the methods.

The presentation of each numerical technique is based on the successful teaching methodology of providing examples and geometric motivation for a method, and a concise statement of the steps to carry out the computation, before giving a mathematical derivation of the process or a discussion of the more theoretical issues that are relevant to the use and understanding of the topic. Each topic is illustrated by examples that range in complexity from very simple to moderate. Geometrical or graphical illustrations are included whenever they are appropriate. The last section of each chapter gives a brief discussion of more advanced methods for solving the kinds of problems covered in the chapter, including methods used in MATLAB, Mathcad, Mathematica, and various software libraries.

The chapters are arranged according to the following general areas:

• Chapter 2 deals with solving nonlinear equations of a single variable.
• Chapters 3-6 treat topics from numerical linear algebra.
• Chapter 7 considers nonlinear functions of several variables
• Chapters 8-10 cover numerical methods for data interpolation and approximation.
• Chapter 11 presents numerical differentiation and integration.
• Chapters 12-15 introduce numerical techniques for solving differential equations.

For much of the material, a calculus sequence that includes an introduction to differential equations and linear algebra provides adequate background. For more in-depth coverage of the topics from linear algebra (especially the OR method for eigenvalues), a linear algebra course would be an appropriate prerequisite. The coverage of Fourier approximation and FFT (Chapter 10) and partial differential equations (Chapter 15) assumes that the students have somewhat more mathematical maturity since the material in intrinsically more challenging. The subject matter included is suitable for a two-semester sequence of classes, or for any of several different one-term courses, depending on the desired emphasis, student background, and selection of topics.

Many people have contributed to the development of this text. My colleagues at Florida Institute of Technology, the Naval Postgraduate School, the University of South Carolina Aiken, and Georgia Southern University have provided support, encouragement, and suggestions. I especially wish to thank Sharon Barrs, Jacalyn Huband, and Gary Huband for writing the sample programs in Chapter 1, and Pierre Larochelle for the example of robot motion in Ch 13. Thanks also to Jane Lybrand and Jack Leifer, who provided data for several examples and exercises. I also appreciate the many contributions my students have made to this text, which was after all written with them in mind. The comments made by the reviewers of the text have helped greatly in the fine-tuning of the final presentation. The editorial and production staff at Prentice Hall, as well as Patty Donovan and the rest of the staff at Pine Tree Composition have my heartfelt gratitude for their efforts in insuring that the text is as accurate and well designed as possible. And, saving the most important for last, I thank my husband and colleague, Don Fausett, for his patience and support.

LAURENE FAUSETT

Table of Contents

1. Foundations.

Sample Problems and Numerical Methods. Some Basic Issues. Algorithms and Computer Programs.

2. Solving Equations of One Variable.

Bisection Method. Regular Falsi and Secant Methods. Newton's Method. Muller's Method. Methods of Modern Computing.

3. Solving Systems of Linear Equations: Direct Methods.

Gaussian Elimination. Gaussian Elimination with Row Pivoting. Gaussian Elimination for Tridiagonal Systems. Methods of Modern Computing.

4. LU and QR Factorization.

LU Factorization from Gaussian Elimination. Direct LU Factorization. Applications of LU Factorization. Householder and Givens Transformations. QR Factorization. Methods of Modern Computing.

5. Eigenvalues and Eigenvectors.

Power Method. Inverse Power Method. QR Method. Methods of Modern Computing.

6. Solving Systems of Linear Equations: Iterative Methods.

Jacobi Method. Gauss-Seidel Method. Successive Over Relaxation. Methods of Modern Computing.

7. Nonlinear Functions of Several Variables.

Newton's Method for Systems of Equations. Fixed-Point Iteration for Nonlinear Systems. Minimum of a Nonlinear Function of Several Variables. Methods of Modern Computing.

8. Interpolation.

Polynomial Interpolation. Hermite Interpolation. Rational-Function Interpolation. Spline Interpolation. Methods of Modern Computing.

9. Function Approximation.

Least-Squares Approximation. Continuous Least-Squares Approximation. Function Approximation at a Point. Methods of Modern Computing.

10. Fourier Methods.

Fourier Approximation and Interpolation. Radix-2 Fast Fourier Transforms. General Fast Fourier Transforms. Methods of Modern Computing.

11. Numerical Differentiation and Integration.

Differentiation. Basic Numerical Integration. Better Numerical Integration. Gaussian Quadrature. Methods of Modern Computing.

12. Ordinary Differential Equations: Initial-Value Problems.

Taylor Methods. Runge-Kutta Methods. Multistep Methods. Stability. Methods of Modern Computing.

13. Ordinary Differential Equations: Higher-Order Equations and First-Order Systems.

Higher-Order ODEs. Systems of Two First-Order ODE. Systems of First-Order ODE. Stiff ODE and Ill-conditioned Problems. Methods of Modern Computing.

14. Ordinary Differential Equations: Boundary-Value Problems.

Shooting Method for Linear BVP. Shooting Method Nonlinear BVP. Finite-Difference Method for Linear BVP. Finite-Difference Method for Nonlinear BVP. Methods of Modern Computing.

15. Partial Differential Equations.

Classification of PDE. Heat equation: Parabolic PDE. Wave equation: Hyperbolic PDE. Poisson Equation - Elliptic PDE. Finite Element Method for an Elliptic PDE. Methods of Modern Computing.

Preface

The purpose of this text is to present the fundamental numerical techniques used in engineering, applied mathematics, computer science, and the physical and life sciences in a manner that is both interesting and understandable to undergraduate and beginning graduate students in those fields. The organization of the chapters, and of the material within each chapter, is designed with student learning as the primary objective. A detailed algorithm is given for each method presented, so that students can write simple programs implementing the technique in the computer language of their choice. Numerous examples of the use of the methods are also included.

The first chapter sets the stage for the material in the rest of the text, giving a brief introduction to the long history of numerical techniques and a "preview of coming attractions" for some of the recurring themes in the remainder of the text. It also presents a summary of the key components of a computer program for solving problems involving numerical techniques such as those given in the text. The trapezoid rule for numerical integration is used to illustrate the relationship between a numerical algorithm and a computer program implementing the algorithm. Sample programs are given in several different languages.

Each of the subsequent chapters begins with a one page overview of the subject matter, together with an indication as to how the topics presented in the chapter are related to those in previous and subsequent chapters. Introductory examples are presented to suggest a few of the types of problems for which the topics of the chapter may be used. Following the sections in which the methods are presented, each chapter concludes with a summary of the most important formulas, of suggestions for further reading, and an extensive set of exercises. The first group of problems provides fairly routine practice of the techniques; the second group includes applications adapted from a variety of fields, and the final group of problems encourages students to extend their understanding of either the theoretical or the computational aspects of the methods.

The presentation of each numerical technique is based on the successful teaching methodology of providing examples and geometric motivation for a method, and a concise statement of the steps to carry out the computation, before giving a mathematical derivation of the process or a discussion of the more theoretical issues that are relevant to the use and understanding of the topic. Each topic is illustrated by examples that range in complexity from very simple to moderate. Geometrical or graphical illustrations are included whenever they are appropriate. The last section of each chapter gives a brief discussion of more advanced methods for solving the kinds of problems covered in the chapter, including methods used in MATLAB, Mathcad, Mathematica, and various software libraries.

The chapters are arranged according to the following general areas:

• Chapter 2 deals with solving nonlinear equations of a single variable.
• Chapters 3-6 treat topics from numerical linear algebra.
• Chapter 7 considers nonlinear functions of several variables
• Chapters 8-10 cover numerical methods for data interpolation and approximation.
• Chapter 11 presents numerical differentiation and integration.
• Chapters 12-15 introduce numerical techniques for solving differential equations.

For much of the material, a calculus sequence that includes an introduction to differential equations and linear algebra provides adequate background. For more in-depth coverage of the topics from linear algebra (especially the OR method for eigenvalues), a linear algebra course would be an appropriate prerequisite. The coverage of Fourier approximation and FFT (Chapter 10) and partial differential equations (Chapter 15) assumes that the students have somewhat more mathematical maturity since the material in intrinsically more challenging. The subject matter included is suitable for a two-semester sequence of classes, or for any of several different one-term courses, depending on the desired emphasis, student background, and selection of topics.

Many people have contributed to the development of this text. My colleagues at Florida Institute of Technology, the Naval Postgraduate School, the University of South Carolina Aiken, and Georgia Southern University have provided support, encouragement, and suggestions. I especially wish to thank Sharon Barrs, Jacalyn Huband, and Gary Huband for writing the sample programs in Chapter 1, and Pierre Larochelle for the example of robot motion in Ch 13. Thanks also to Jane Lybrand and Jack Leifer, who provided data for several examples and exercises. I also appreciate the many contributions my students have made to this text, which was after all written with them in mind. The comments made by the reviewers of the text have helped greatly in the fine-tuning of the final presentation. The editorial and production staff at Prentice Hall, as well as Patty Donovan and the rest of the staff at Pine Tree Composition have my heartfelt gratitude for their efforts in insuring that the text is as accurate and well designed as possible. And, saving the most important for last, I thank my husband and colleague, Don Fausett, for his patience and support.

LAURENE FAUSETT