elementary numerical analysis an algorithmic approach, 3rd ed - de boor

Next Home ELEMENTARY NUMERICAL ANALYSIS An Algorithmic Approach International Series in Pure and Applied Mathematics G. Springer Consulting Editor Ahlfors: Complex Analysis Bender and Orszag: Advanced Mathematical Methods for Scientists and Engineers Buck: Advanced Calculus Busacker and Saaty: Finite Graphs and Networks Cheney: Introduction to Approximation Theory Chester: Techniques in Partial Differential Equations Coddington and Levinson: Theory of Ordinary Differential Equations Conte and de Boor: Elementary Numerical Analysis: An Algorithmic Approach Dennemeyer: Introduction to Partial Differential Equations and Boundary Value Problems Dettman: Mathematical Methods in Physics and Engineering Hamming: Numerical Methods for Scientists and Engineers Hildebrand: Introduction to Numerical Analysis Householder: The Numerical Treatment of a Single Nonlinear Equation Kalman, Falb, and Arbib: Topics in Mathematical Systems Theory McCarty: Topology: An Introduction with Applications to Topological Groups Moore: Elements of Linear Algebra and Matrix Theory Moursund and Duris: Elementary Theory and Application of Numerical Analysis Pipes and Harvill: Applied Mathematics for Engineers and Physicists Ralston and Rabinowitz: A First Course in Numerical Analysis Ritger and Rose: Differential Equations with Applications Rudin: Principles of Mathematical Analysis Shapiro: Introduction to Abstract Algebra Simmons: Differential Equations with Applications and Historical Notes Simmons: Introduction to Topology and Modern Analysis Struble: Nonlinear Differential Equations ELEMENTARY NUMERICAL ANALYSIS An Algorithmic Approach Third Edition S. D. Conte Purdue University Carl de Boor Universiry of Wisconsin—Madison McGraw-Hill Book Company New York St. Louis San Francisco Auckland Bogotá Hamburg Johannesburg London Madrid Mexico Montreal New Delhi Panama Paris São Paulo Singapore Sydney Tokyo Toronto ELEMENTARY NUMERICAL ANALYSIS An Algorithmic Approach Copyright © 1980, 1972, 1965 by McGraw-Hill, inc. All rights reserved. Printed in the United States of America. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher. 234567890 DODO 89876543210 This book was set in Times Roman by Science Typographers, Inc. The editors were Carol Napier and James S. Amar; the production supervisor was Phil Galea. The drawings were done by Fine Line Illustrations, Inc. R. R. Donnelley & Sons Company was printer and binder. Library of Congress Cataloging in Publication Data Conte, Samuel Daniel, date Elementary numerical analysis. (International series in pure and applied mathematics) Includes index. 1. Numerical analysis-Data processing. I.de Boor, Carl, joint author. II. Title. QA297.C65 1980 519.4 79-24641 ISBN 0-07-012447-7 CONTENTS Chapter 1 1.1 1.2 1.3 1.4 1 1 4 7 1.5 1.6 1.7 Chapter 2 2.1 2.2 2.3 *2.4 2.5 2.6 *2.7 Preface Introduction Number Systems and Errors The Representation of Integers The Representation of Fractions Floating-Point Arithmetic Loss of Significance and Error Propagation; Condition and Instability Computational Methods for Error Estimation Some Comments on Convergence of Sequences Some Mathematical Preliminaries Interpolation by Polynomial Polynomial Forms Existence and Uniqueness of the Interpolating Polynomial The Divided-Difference Table Interpolation at an Increasing Number of Interpolation Points The Error of the Interpolating Polynomial Interpolation in a Function Table Based on Equally Spaced Points The Divided Difference as a Function of Its Arguments and Osculatory Interpolation * Sections marked with an asterisk may be omitted without loss of continuity. ix xi 12 18 19 25 31 31 38 41 46 51 55 62 V vi CONTETS Chapter 3 The Solution of Nonlinear Equations 72 3.1 A Survey of Iterative Methods 74 3.2 Fortran Programs for Some Iterative Methods 81 3.3 Fixed-Point Iteration 88 3.4 Convergence Acceleration for Fixed-Point Iteration 95 *3.5 Convergence of the Newton and Secant Methods 100 3.6 Polynomial Equations: Real Roots 110 *3.7 Complex Roots and Müller’s Method 120 Chapter 4 Matrices and Systems of Linear Equations 4.1 Properties of Matrices 4.2 The Solution of Linear Systems by Elimination 4.3 The Pivoting Strategy 4.4 The Triangular Factorization 4.5 Error and Residual of an Approximate Solution; Norms 4.6 Backward-Error Analysis and Iterative Improvement *4.7 Determinants *4.8 The Eigenvalue Problem Chapter *5 Systems of Equations and Unconstrained Optimization *5.1 Optimization and Steepest Descent *5.2 Newton’s Method *5.3 Fixed-Point Iteration and Relaxation Methods Chapter 6 Approximation 6.1 Uniform Approximation by Polynomials 6.2 Data Fitting *6.3 Orthogonal Polynomials *6.4 Least-Squares Approximation by Polynomials *6.5 Approximation by Trigonometric Polynomials *6.6 Fast Fourier Transforms 6.7 Piecewise-Polynomial Approximation Chapter 7 7.1 7.2 7.3 7.4 7.5 l 7.6 l 7.7 Differentiation and Integration Numerical Differentiation Numerical Integration: Some Basic Rules Numerical Integration: Gaussian Rules Numerical Integration: Composite Rules Adaptive Quadrature Extrapolation to the Limit Romberg Integration 128 128 147 157 160 169 177 185 189 208 209 216 223 235 235 245 251 259 268 277 284 294 295 303 311 319 328 333 340 CONTENTS vii Chapter 8 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 *8.10 *8.11 *8.12 *8.13 Mathematical Preliminaries Simple Difference Equations The Solution of Differential Equations Numerical Integration by Taylor Series Error Estimates and Convergence of Euler’s Method Runge-Kutta Methods Step-Size Control with Runge-Kutta Methods Multistep Formulas Predictor-Corrector Methods The Adams-Moulton Method Stability of Numerical Methods Round-off-Error Propagation and Control Systems of Differential Equations Stiff Differential Equations Chapter 9 Boundary Value Problems 9.1 Finite Difference Methods 9.2 Shooting Methods 9.3 Collocation Methods Appendix: Subroutine Libraries 421 References 423 Index 425 346 346 349 354 359 362 366 373 379 382 389 395 398 401 406 406 412 416 PREFACE This is the third edition of a book on elementary numerical analysis which is designed specifically for the needs of upper-division undergraduate students in engineering, mathematics, and science including, in particular, computer science. On the whole, the student who has had a solid college calculus sequence should have no difficulty following the material. Advanced mathematical concepts, such as norms and orthogonality, when they are used, are introduced carefully at a level suitable for undergraduate students and do not assume any previous knowledge. Some familiarity with matrices is assumed for the chapter on systems of equations and with differential equations for Chapters 8 and 9. This edition does contain some sections which require slightly more mathematical maturity than the previ- ous edition. However, all such sections are marked with asterisks and all can be omitted by the instructor with no loss in continuity. This new edition contains a great deal of new material and significant changes to some of the older material. The chapters have been rearranged in what we believe is a more natural order. Polynomial interpolation (Chapter 2) now precedes even the chapter on the solution of nonlinear systems (Chapter 3) and is used subsequently for some of the material in all chapters. The treatment of Gauss elimination (Chapter 4) has been simplified. In addition, Chapter 4 now makes extensive use of Wilkinson’s backward error analysis, and contains a survey of many well-known methods for the eigenvalue-eigenvector problem. Chapter 5 is a new chapter on systems of equations and unconstrained optimization. It con- tains an introduction to steepest-descent methods, Newton’s method for nonlinear systems of equations, and relaxation methods for solving large linear systems by iteration. The chapter on approximation (Chapter 6) has been enlarged. It now treats best approximation and good approximation ix [...]... used, standard conventions are followed, whereas all procedural statement charts use a self-explanatory ALGOL-like statement language Having produced a flow chart, the programmer must transform the indicated procedures into a set of machine instructions This may be done directly in machine language, in an assembly language, or in a procedure-oriented language In this book a dialect of FORTRAN called... 1.1 and decimal arithmetic 1.3 FLOATING-POINT ARITHMETIC Scientific calculations are usually carried out in floating-point arithmetic An n-digit floating-point number in base has the form (1.5) is a called the mantissa, and e is an where integer called the exponent Such a floating-point number is said to be normalized in case or else For most computers, although on some, and in hand calculations and... same sign 1. 3-7 Carry out a backward error analysis for the calculation of the scalar product Redo the analysis under the assumption that double-precision cumulation is used This means that the double-precision results of each multiplicatioin retained and added to the sum in double precision, with the resulting sum rounded only at end to single precision o for acare the 12 NUMBER SYSTEMS AND ERRORS 1.4... Basic concepts such as derivative, integral, and continuity are defined in terms of convergent sequences, and elementary functions such as ln x or sin x are defined by convergent series, At the same time, numerical answers to engineering and scientific problems are never needed exactly Rather, an approximation to the answer is required which is accurate “to a certain number of decimal places,” or accurate... chopping (1.8) (1.9) 1.3 FLOATING-POINT ARITHMETIC 9 See Exercise 1. 3-3 The maximum possible value for is often called the unit roundoff and is denoted by u When an arithmetic operation is applied to two floating-point numbers, the result usually fails to be a floating-point number of the same length If, for example, we deal with two-decimal-digit numbers and then Hence, if denotes one of the arithmetic... hold for floating-point arithmetic (Hint: Try laws involving three operands.) 1. 3-5 Write a FORTRAN FUNCTION FL(X) which returns the value of the n-decimal-digit floating-point number derived from X by rounding Take n to be 4 and check your calculations in Exercise 1.3-l [Use ALOG10(ABS(X)) to determine e such that 1. 3-6 Let Show that for all there exists that Show also that some provided all have the... called FORTRAN 77 is used exclusively FORTRAN 77 is a new dialect of FORTRAN which incorporates new control statements and which emphasizes modern structured-programming concepts While FORTRAN IV compilers are available on almost all computers, FORTRAN 77 may not be as readily available However, conversion from FORTRAN 77 to FORTRAN IV should be relatively straightforward A procedure-oriented language... values Using statistical methods, we can then obtain the standard deviation, the variance of distribution, and estimates of the accumulated roundoff error The statistical approach is considered in some detail by Hamming [1] and Henrici [2] The method does involve substantial analysis and additional computer time, but in the experiments conducted to date it has obtained error estimates which are in remarkable... beforehand the fact that they expect to solve a problem on a computer They will therefore provide for specific objectives, proper input data, adequate checks, and for the type and amount of output Once a problem has been formulated, numerical methods, together with a preliminary error analysis, must be devised for solving the problem A numerical method which can be used to solve a problem will be called an. .. Every floating-point operation in a computational process may give rise to an error which, once generated, may then be amplified or reduced in subsequent operations One of the most common (and often avoidable) ways of increasing the importance of an error is commonly called loss of significant digits If x* is an approximation to x, then we say that x* approximates x to r significant provided the absolute . Next Home ELEMENTARY NUMERICAL ANALYSIS An Algorithmic Approach International Series in Pure and Applied Mathematics G. Springer Consulting Editor Ahlfors: Complex Analysis Bender and Orszag: Advanced. Theory Moursund and Duris: Elementary Theory and Application of Numerical Analysis Pipes and Harvill: Applied Mathematics for Engineers and Physicists Ralston and Rabinowitz: A First Course in Numerical Analysis Ritger. to Topology and Modern Analysis Struble: Nonlinear Differential Equations ELEMENTARY NUMERICAL ANALYSIS An Algorithmic Approach Third Edition S. D. Conte Purdue University Carl de Boor Universiry