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INTRODUCTION TO NUMERICAL ANALYSIS - - - - - - - - - SECOND EDITION F. B. HILDEBRAND Professor of Mathematics, Emeritus Massachusetts Institute of Technology DOVER PUBLICATraNS, INC. NEW YORK

Copyright © 1956, 1974 by Francis B Hildebrand All rights reserved under Pan American and International Copyright Conventions Published in Canada by General Publishing Company, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario This Dover edition, first published in 1987, is an unabridged, slightly corrected republication of the second edition (1974) of the work first published by McGlaw-Hill, Inc, in 1956 ManufactUied in the United States of America Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501 Library of Congress Cataloging-in-Publication Data Hildebrand, Francis Begnaud. Introduction to numerical analysis, second edition. "Unabridged, slightly corrected republication"-T.p. verso. Bibliography. p. Includes index. 1. Numerical analysis. I. Title. QA297.H54 515 ISBN 0-486-65363-3 (pbk.) 1987 87-5370

CONTENTS Preface 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 2 2.1 2.2 2.3 2.4 Introduction Numerical Analysis Approximation Errors Significant Figures Determinacy of Functions. Error Control Machine Errors Random Errors Recursive Computation Mathematical Preliminaries Supplementary References Problems Interpolation with Divided Differences Introduction Linear Interpolation Divided Differences Second-Degree Interpolation xi 1 1 3 5 10 16 19 23 28 31 38 39 51 51 52 54 58

vi CONTENTS 2.5 2.6 2.7 2.8 2.9 Newton's Fundamental Formula Error Formulas Iterated Interpolation Inverse Interpolation Supplementary References Problems Lagrangian Methods Introduction Lagrange's Interpolation Formula Numerical Differentiation and Integration Uniform-spacing Interpolation Newton-Cotes Integration Formulas Composite Integration Formulas Use of Integration Formulas Richardson Extrapolation. Romberg Integration Asymptotic Behavior of Newton-Cotes Formulas 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 Weighting Functions. Filon Integration 3.11 3.12 Differentiation Formulas Supplementary References Problems 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 5 5.1 Finite-Difference Interpolation Introduction Difference Notations Newton Forward- and Backward-difference Formulas Gaussian Formulas Stirling's Formula Bessel's Formula Everett's Formulas Use of Interpolation Formulas Propagation of Inherent Errors Throwback Techniques Interpolation Series Tables of Interpolation Coefficients Supplementary References Problems Operations with Finite Differences Introduction 60 62 66 68 70 71 80 80 81 85 89 91 95 97 99 103 107 llO ll4 ll6 129 129 130 133 136 138 140 143 144 ISO 153 154 158 161 162 174 174

5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 6 6.1 6.2 6.3 6.4 6.5 6.6 CONTENTS vii 175 Difference Operators 181 Differentiation Formulas 186 Newtonian Integration Formulas 189 Newtonian Formulas for Repeated Integration 192 Central-Difference Integration Formulas 195 Subtabulation Summation and Integration. The Euler-Maclaurin Sum Formula 197 203 Approximate Summation 208 Error Terms in Integration Formulas 217 Other Representations of Error Terms 222 Supplementary References 222 Problems Numerical Solution of Differential Equations Introduction Formulas of Open Type Formulas of Closed Type Start of Solution Methods Based on Open-Type Formulas Methods Based on Closed-Type Formulas. Prediction-Correction Methods The Special Case F = Ay Propagated-Error Bounds Application to Equations of Higher Order. Sets of Equations Special Second-order Equations Change of Interval Use of Higher Derivatives 240 240 241 244 245 250 252 257 265 269 275 280 282 285 290 293 297 301 303 303 314 314 314 315 318 6.7 6.8 6.9 6.10 6.11 6.12 6.13 A Simple Runge-Kutta Method 6.14 6.15 6.16 6.17 6.18 Runge-Kutta Methods of Higher Order Boundary-Value Problems Linear Characteristic-value Problems Selection of a Method Supplementary References Problems 7 7.1 7.2 7.3 7.4 Least-Squares Polynomial Approximation Introduction The Principle of Least Squares Least-Squares Approximation over Discrete Sets of Points Error Estimation

viii CONTENTS 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15 7.16 7.17 Orthogonal Polynomials Legendre Approximation Laguerre Approximation Hermite Approximation Chebyshev Approximation Properties of Orthogonal Polynomials. Recursive Computation Factorial Power Functions and Summation Formulas Polynomials Orthogonal over Discrete Sets of Points Gram Approximation Example: Five-Point Least-Squares Approximation Smoothing Formulas Recursive Computation of Orthogonal Polynomials on Discrete Sets of Points Supplementary References Problems Gaussian Quadrature and Related Topics Introduction Hermite Interpolation Hermite Quadrature Gaussian Quadrature Legendre-Gauss Quadrature Laguerre-Gauss Quadrature Hermite-Gauss Quadrature Chebyshev-Gauss Quadrature Jacobi-Gauss Quadrature Formulas with Assigned Abscissas Radau Quadrature Lobatto Quadrature Convergence of Gaussian-quadrature Sequences Chebyshev Quadrature Algebraic Derivations 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 8.14 8.15 8.16 Application to Trigonometric Integrals 8.17 Supplementary References Problems 9 9.1 9.2 9.3 Approximations of Various Types Introduction Fourier Approximation: Continuous Domain Fourier Approximation: Discrete Domain 327 329 332 334 336 340 344 348 350 353 357 363 365 365 379 379 382 385 387 390 392 395 398 399 402 406 409 412 414 421 427 432 433 446 446 447 452

CONTENTS ix 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12 9.13 9.14 9.15 9.16 9.17 9.18 9.19 10 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12 10.13 10.14 10.15 10.16 10.17 10.18 10.19 10.20 Exponential Approximation Determination of Constituent Periodicities Optimum Polynomial Interpolation with Selected Abscissas Chebyshev Interpolation Economization of Polynomial Approximations Uniform (Minimax) Polynomial Approximation Spline Approximation Splines with Uniform Spacing Spline Error Estimates A Special Class of Splines Approximation by Continued Fractions Rational Approximations and Continued Fractions Determination of Convergents of Continued Fractions Thiele's Continued-Fraction Approximations Uniformization of Rational Approximations Supplementary References Problems Numerical Solution of Equations Introduction Sets of Linear Equations The Gauss Reduction The Crout Reduction Intermediate Roundoff Errors Determination of the Inverse Matrix Inherent Errors Tridiagonal Sets of Equations Iterative Methods and Relaxation Iterative Methods for Nonlinear Equations The Newton-Raphson Method Iterative Methods of Higher Order Sets of Nonlinear Equations Iterated Synthetic Division of Polynomials. Lin's Method Determinacy of Zeros of Polynomials Bernoulli's Iteration Graeffe's Root-squaring Technique Quadratic Factors. Lin's Quadratic Method Bairstow Iteration Supplementary References Problems 457 462 466 469 471 475 478 482 485 488 494 498 502 506 514 518 519 539 539 539 543 545 549 553 555 559 561 567 575 578 583 588 595 598 602 609 613 618 621

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