C.T.Kelley_Solving Nonlinear Equations with Newtons Method.pdf

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Solving Nonlinear Equations Newton's M thod th

fundamentals of Algorithms Editor-in-Chief: Nicholas J. Higham, University of Manchester The SIAM series on Fundamentals of Algorithms publishes monographs on state-of-the-art numerical methods to provide the reader with sufficient knowledge to choose the appropriate method for a given application and to aid the reader in understanding the limitations of each method. The monographs focus on numerical methods and algorithms to solve specific classes of problems and are written for researchers, practitioners, and students. The goal of the series is to produce a collection of short books written by experts on numerical methods that include an explanation of each method and a summary of theoretical background. What distinguishes a book in this series is its emphasis on explaining how to best choose a method, algorithm, or software program to solve a specific type of problem and its descriptions of when a given algorithm or method succeeds or fails. Kelley, C. T. Solving Nonlinear Equations with Newton's Method

C T. Kelleg North Carolina State University Raleigh, North Carolina Solving Nonlinear Equations with Newton's Method siamm Society for Industrial and Applied Mathematics Philadelphia

Copyright © 2003 by the Society for Industrial and Applied Mathematics. 10 9 8 7 6 5 4 3 21 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688. Library of Congress Cataloging-in-Publication Data Kelley, C. T. Solving nonlinear equations with Newton's method / C.T. Kelley. p. cm. — (Fundamentals of algorithms) Includes bibliographical references and index. ISBN 0-89871-546-6 (pbk.) 1. Newton-Raphson method. 2. Iterative methods (Mathematics) 3. Nonlinear theories. I. Title. II. Series. QA297.8.K455 2003 511'.4— dc21 2003050663 Apple and Macintosh are trademarks of Apple Computer, Inc., registered in the U.S. and other countries. VAIO is a registered trademark of Sony Corporation. No warranties, express or implied, are made by the publisher, author, and their employers that the programs contained in this volume are free of error. They should not be relied on as the sole basis to solve a problem whose incorrect solution could result in injury to person or property. If the programs are employed in such a manner, it is at the user's own risk and the publisher, author, and their employers disclaim all liability for such misuse. is a registered trademark.

To my students

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Contents Preface How to Get the Software 1 Introduction 1.1 What Is the Problem? Notation 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Local Convergence Theory 1.1.1 Newton's Method 1.2.1 Approximating the Jacobian Inexact Newton Methods Termination of the Iteration Global Convergence and the Armijo Rule A Basic Algorithm 1.7.1 Things to Consider 1.8.1 1.8.2 1.8.3 1.8.4 Human Time and Public Domain Codes The Initial Iterate Computing the Newton Step Choosing a Solver Warning! 1.9 What Can Go Wrong? . . Nonsmooth Functions Failure to Converge Failure of the Line Search Slow Convergence Multiple Solutions Storage Problems 1.9.1 1.9.2 1.9.3 1.9.4 1.9.5 1.9.6 Three Codes for Scalar Equations 1.10.1 1.10.2 1.10.3 1.10.4 Common Features newtsol.m chordsol.m secant.m 1.10 1.11 Projects vii xi xiii 1 1 1 2 3 5 7 9 11 12 14 15 15 15 16 16 17 17 18 19 19 20 20 20 21 21 22 23 24

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