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FUNDAMENTALS OF

NUMERICAL COMPUTING

L. F. Shampine

Southern Methodist University

R. C. Allen, Jr.

Sandia National Laboratories

S. Pruess

Colorado School of Mines

JOHN WILEY & SONS, INC.

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Library of Congress Cataloging-in-Publication Data:

Shampine, Lawrence

Fundamentals of numerical computing / Richard Allen, Steve

Pruess, Lawrence Shampine.

p. cm.

Includes bibliographical reference and index.

ISBN 0-471-16363-5 (cloth : alk. paper)

1. Numerical analysis-Data processing. I. Pruess, Steven.

II. Shampine, Lawrence F. III. Title.

QA297.A52 1997

519.4’0285’51—dc20

96-22074

CIP

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

PRELIMINARIES

The purpose of this book is to develop the understanding of basic numerical meth-

ods and their implementations as software that are necessary for solving fundamental

mathematical problems by numerical means. It is designed for the person who wants

to do numerical computing. Through the examples and exercises, the reader studies the

behavior of solutions of the mathematical problem along with an algorithm for solv-

ing the problem. Experience and understanding of the algorithm are gained through

hand computation and practice solving problems with a computer implementation. It

is essential that the reader understand how the codes provided work, precisely what

they do, and what their limitations are. The codes provided are powerful, yet simple

enough for pedagogical use. The reader is exposed to the art of numerical computing

as well as the science.

The book is intended for a one-semester course, requiring only calculus and a

modest acquaintance with FORTRAN, C, C++, or M ATLAB . These constraints of

background and time have important implications: the book focuses on the problems

that are most common in practice and accessible with the background assumed. By

concentrating on one effective algorithm for each basic task, it is possible to develop

the fundamental theory in a brief, elementary way. There are ample exercises, and

codes are provided to reduce the time otherwise required for programming and debug-

ging. The intended audience includes engineers, scientists, and anyone else interested

in scientific programming. The level is upper-division undergraduate to beginning

graduate and there is adequate material for a one semester to two quarter course.

Numerical analysis blends mathematics, programming, and a considerable amount

of art. We provide programs with the book that illustrate this. They are more than mere

implementations in a particular language of the algorithms presented, but they are not

production-grade software. To appreciate the subject fully, it will be necessary to study

the codes provided and gain experience solving problems first with these programs and

then with production-grade software.

Many exercises are provided in varying degrees of difficulty. Some are designed

to get the reader to think about the text material and to test understanding, while others

are purely computational in nature. Problem sets may involve hand calculation, alge-

braic derivations, straightforward computer solution, or more sophisticated computing

exercises.

The algorithms that we study and implement in the book are designed to avoid

severe roundoff errors (arising from the finite number of digits available on computers

and calculators), estimate truncation errors (arising from mathematical approxima-

tions), and give some indication of the sensitivity of the problem to errors in the data.

In Chapter 1 we give some basic definitions of errors arising in computations and

study roundoff errors through some simple but illuminating computations. Chapter 2

deals with one of the most frequently occurring problems in scientific computation,

the solution of linear systems of equations. In Chapter 3 we deal with the problem of

PRELIMINARIES

interpolation, one of the most fundamental and widely used tools in numerical com-

putation. In Chapter 4 we study methods for finding solutions to nonlinear equations.

Numerical integration is taken up in Chapter 5 and the numerical solution of ordinary

differential equations is examined in Chapter 6. Each chapter contains a case study

that illustrates how to combine analysis with computation for the topic of that chapter.

Before taking up the various mathematical problems and procedures for solving

them numerically, we need to discuss briefly programming languages and acquisition

of software.

PROGRAMMING LANGUAGES

The FORTRAN language was developed specifically for numerical computation and

has evolved continuously to adapt it better to the task. Accordingly, of the widely

used programming languages, it is the most natural for the programs of this book. The

C language was developed later for rather different purposes, but it can be used for

numerical computation.

At present FORTRAN 77 is very widely available and codes conforming to the

ANSI standard for the language are highly portable, meaning that they can be moved

to another hardware/software configuration with very little change. We have chosen

to provide codes in FORTRAN 77 mainly because the newer Fortran 90 is not in wide

use at this time. A Fortran 90 compiler will process correctly our FORTRAN 77

programs (with at most trivial changes), but if we were to write the programs so as

to exploit fully the new capabilities of the language, a number of the programs would

be structured in a fundamentally different way. The situation with C is similar, but

in our experience programs written in C have not proven to be nearly as portable as

programs written in standard FORTRAN 77. As with FORTRAN, the C language has

evolved into C++, and as with Fortran 90 compared to FORTRAN 77, exploiting fully

the additional capabilities of C++ (in particular, object oriented programming) would

lead to programs that are completely different from those in C. We have opted for a

middle ground in our C++ implementations.

In the last decade several computing environments have been developed. Popular

ones familiar to us are M ATLAB [l] and Mathematica [2]. M ATLAB is very much in

keeping with this book, for it is devoted to the solution of mathematical problems by

numerical means. It integrates the formation of a mathematical model, its numerical

solution, and graphical display of results into a very convenient package. Many of the

tasks we study are implemented as a single command in the M ATLAB language. As

M ATLAB has evolved, it has added symbolic capabilities. Mathematica is a similar

environment, but it approaches mathematical problems from the other direction. Orig-

inally it was primarily devoted to solving mathematical problems by symbolic means,

but as it has evolved, it has added significant numerical capabilities. In the book we

refer to the numerical methods implemented in these widely used packages, as well

as others, but we mention the packages here because they are programming languages

in their own right. It is quite possible to implement the algorithms of the text in these

languages. Indeed, this is attractive because the environments deal gracefully with a

number of issues that are annoying in general computing using languages like FOR-

TRAN or C.

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