Introduction to Numerical Methods and MATLAB Implementations and Applications

by Recktenwald, Gerald W.

• ISBN13: 9780201308600
• ISBN10: 0201308606
• Additional ISBN(s): 9780131865518
• Edition: 1st
• Format: Paperback
• Copyright: 2000-08-24
• Publisher: Pearson
• More Book Details
• Purchase Benefits

Summary

This thorough, modern exposition of classic numerical methods using MATLAB briefly develops the fundamental theory of each method. Rather than providing a detailed numerical analysis, the behavior of the methods is exposed by carefully designed numerical experiments. The methods are then exercised on several nontrivial example problems from engineering practice.This structured, concise, and efficient book contains a large number of examples of two basic typesOne type of example demonstrates a principle or numerical method in the simplest possible terms. Another type of example demonstrates how a particular method can be used to solve a more complex practical problem. The material in each chapter is organized as a progression from the simple to the complex. Contains an extensive reference to using MATLAB. This includes interactive (command line) use of MATLAB, MATLAB programming, plotting, file input and output.For a practical and rigorous introduction to the fundamentals of numerical computation.

Author Biography

GERALD RECKTENWALD is an Associate Professor of Mechanical Engineering at Portland State University, and regularly teaches courses in Numerical Methods.

Table of Contents

(NOTE: Chapters 2-12 conclude with Summary.)
1. Introduction.

Terminology. MATLAB Overview. Organization of the Book. Rating Systems for Exercises.

I. MATLAB BASICS.

2. Interactive Computing with MATLAB.

Running MATLAB. Matrices and Vectors. Additional Types of Variables. Managing the Interactive Environment. Plotting in MATLAB.

3. MATLAB Programming.

Script m-Files. Function m-Files. Input and Output. Flow Control. Vectorization. Deus ex Machina.

4. Organizing and Debugging MATLAB Programs.

Organizing and Documenting m-Files. Organizing a Numerical Solution. Debugging.

II. NUMERICAL TECHNIQUES.

5. Unavoidable Errors in Computing.

Digital Representation of Numbers. Finite Precision Arithmetic. Truncation Error of Algorithms.

6. Finding the Roots of f(x)=0.

Preliminaries. Fixed-Point Iteration. Bisection. Newton's Method. The Secant Method. Hybrid Methods. Roots of Polynomials.

7. A Review of Linear Algebra.

Vectors. Matrices. Mathematical Properties of Vectors and Matrices. Special Matrices.

8. Solving Systems of Equations.

Basic Concepts. Gaussian Elimination. Limitations on Numerical Solutions to Ax = b. Factorization Methods. Nonlinear Systems of Equations.

9. Least-Squares Fitting of a Curve to Data.

Fitting a Line to Data. Least-Squares Fit to a Linear Combination of Functions. Multivariate Linear Least-Squares Fitting.

10. Interpolation.

Basic Ideas. Interpolating Polynomials of Arbitrary Degree. Piecewise Polynomial Interpolation. MATLAB's Built in Interpolation Functions.

11. Numerical Integration.

Basic Ideas and Nomenclature. Newton-Cotes Rules. Gaussian Quadrature. Adaptive Quadrature. Improper Integrals and Other Complications.

12. Numerical Integration of Ordinary Differential Equations.

Basic Ideas and Nomenclature. Euler's Method. Higher Order One-Step Methods. Adaptive Stepsize Algorithms. Coupled ODEs. Additional Topics.

Bibliography.
Appendix A: Eigenvalues and Eigensystems.

Eigenvectors Map onto Themselves. Mathematical Preliminaries. The Power Method. Built-in Functions for Eigenvalue Computation. Singular Value Decomposition.

Appendix B: Sparse Matrices.

Storage and Flop Savings. MATLAB Sparse Matrix Format.

MATLAB Toolbox Functions.
Listings for NMM Toolbox m-Files.
Subject Index.

Excerpts

Preface This book is about using MATLAB to obtain numerical solutions to problems in engineering and science. The emphasis is on the application of standard numerical techniques, not on the analysis of those techniques. Mathematical theory is used primarily to develop the logic of the algorithms and to identify the strengths and weaknesses of different numerical methods. In most cases, the behavior of the methods is exposed by numerical experiment. The intended audience is students in engineering and science learning how to use numerical methods in their own disciplines. I also hope the book will be useful to practicing engineers and scientists who have a specific numerical task and wish to implement a solution in MATLAB. The reader needs no previous experience in numerical analysis. A solid understanding of basic calculus is assumed, as is some previous exposure to linear algebra, differential equations, and computer programming. This is not a "how to A " book. Although there are many examples that show how to use MATLAB to solve practical problems, the "how" is accompanied by a strong dose of "why." The basic theory of each method is developed before that method is implemented in MATLAB. If you are interested in a set of recipes for plugging numbers into MATLAB statements, you should put this book down and select another one from the shelf. If you are interested in developing an understanding of how certain numerical methods work and how different methods used to accomplish the same task compare, read on. Not every routine presented here is recommended for solving practical problems. Elementary algorithms are implemented and their performance is investigated. Advanced techniques are then developed as improvements on the elementary algorithms. By progressing from the simple to the sophisticated, the reader is given -an understanding of the fundamentals, as well as guidance on how to apply more reliable arid efficient algorithms to practical problems. Numerous sample problems are provided in each chapter. Where applicable, the same problem is solved with different methods, and the results are compared. Simple examples allow the reader to focus on the fundamental aspects of the numerical methods. More sophisticated examples demonstrate the solution to realistic and complicated practical problems. The first part of the book provides an introduction to MATLAB as an interactive computing system and as a programming language. It is assumed that the reader has some prior experience with computer programming. The material is presented as a first course in numerical methods, not a first course in programming. An ambitious and determined person with significant computer experience could probably use this material as a condensed introduction to programming, although, for that purpose, there are better books. The routines and data files presented in the book constitute the Numerical Methods with MATLAB (NMM) toolbox, which is available at http:// www.prenhall.com/recktenwald. Once the NMM toolbox is installed on your computer, you will be able to interactively execute all of the sample problems in this book, as well as apply the NMM toolbox routines to practical problems in your own work. To the Student Learning to use computers can be an extremely frustrating experience. It is very easy; in a state of exhaustion and despair, to conclude that numerical methods are (choose one or all of the following) (1) a waste of effort, (2) solely a means of torture, (3) only to be learned by computer geeks, (4) making me less, not more, productive. When I teach numerical methods, I warn my students that they can expect an initial drop in their productivity as they learn a new tool such as MATLAB. It is inevitable that mastering something as powerful and multifaceted as MATLAB will take patience and commitment. The rewards, however, are significant. The best way to learn a computer tool is to use it. With the knowledge that it may at times be frustrating, jump in. Try it. Let mistakes happen. You will not damage the computer by entering an incorrect MATLAB command. Have this book open when you are running MATLAB. Enter the commands that appear in the text, and explain the outcome in your own words. Try not to rush through the examples. Do not treat them as recipes that must never be altered. Experiment. See if you can break the examples by changing the input parameters. The failure of a numerical solution is a learning experience. Why does one set of inputs work when another does not? What is the smallest difference between two set of inputs that differentiates success from failure? Does the method fail because of an inherent weakness, or are the inputs not valid? Will a more powerful method work for the same inputs? Use the study guides, lecture notes, solutions to selected end-of-chapter exercises, and other supplemental learning material at http://www.prenhall.com/recktenwald to complement the information in the book. The study guides are designed to help you master the material presented in each chapter, and contain learning goals that will be most useful if consulted after reading the chapter. Your instructor may highlight specific learning goals from the study guides. In each chapter solutions to selected end-of-chapter exercises, designated by SOLUTION,are provided. To the Instructor Various subsets of the material in this book may be organized into college-level courses. To help orient the reader, the beginning of each chapter contains a brief outline of the major topics covered. Study guides for each chapter provide a series of learning objectives divided into basic, intermediate, and advanced topics. Recognizing that the classification of topics is subjective, I am making the study guides available as PDF and as plain text documents at http://www.prenhall.com/recktenwald. Instructors are encouraged to download and adapt the study guides to their own selection of topics. Other course materials, including transparency masters, are also available at http://www.prenhall.com/recktenwald. Some very important practical areas of numerical computation are not covered in the book: iterative methods for linear systems of equations, the fast Fourier transform, partial differential equations, and minimization (optimization). These topics were included in the original outline, but as the manuscript evolved, and the sabbatical allocated for writing expired, it became clear that not everything in my plan would make it into the final draft. The absence of some numerical methods will be viewed as a limitation by some instructors. This has not been a problem for the course that was the impetus for writing the book. I believe that undergraduate students are better served by providing a sound foundation in the fundamentals than by a quickly paced survey of many numerical methods. The focus of this book is on providing such a foundation. I believe that it is counterproductive to require engineering students to develop all of their own programs from pseudocode. This is not to suggest that numerical methods should be treated superficially or that the reader use the NMM toolbox routines as black boxes. The source code for the NMM toolbox is provided with the intention that it is to be read as an integral part of the text. Many of the end-of-chapter assignments require modifications to the source code presented in the chapter, or application of the codes to new problems. Starting with working codes allows you to challenge your students through the application of numerical methods to your discipline. This frees you to focus on the behavior of each numerical method without having to guide students as they reinvent standard methods. Having a library of working codes at their disposal, students are also more likely to apply numerical methods in other classes. This provides an