Conte, de Boor - Elementary Numerical Analysis. An Algorithmic Approach (523140)
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HomeNextELEMENTARY NUMERICAL ANALYSISAn Algorithmic ApproachInternational Series in Pure and Applied MathematicsG. SpringerConsulting EditorAhlfors: Complex AnalysisBender and Orszag: Advanced Mathematical Methods for Scientists and EngineersBuck: Advanced CalculusBusacker and Saaty: Finite Graphs and NetworksCheney: Introduction to Approximation TheoryChester: Techniques in Partial Differential EquationsCoddington and Levinson: Theory of Ordinary Differential EquationsConte and de Boor: Elementary Numerical Analysis: An Algorithmic ApproachDennemeyer: Introduction to Partial Differential Equations and Boundary ValueProblemsDettman: Mathematical Methods in Physics and EngineeringHamming: Numerical Methods for Scientists and EngineersHildebrand: Introduction to Numerical AnalysisHouseholder: The Numerical Treatment of a Single Nonlinear EquationKalman, Falb, and Arbib: Topics in Mathematical Systems TheoryMcCarty: Topology: An Introduction with Applications to Topological GroupsMoore: Elements of Linear Algebra and Matrix TheoryMoursund and Duris: Elementary Theory and Application of Numerical AnalysisPipes and Harvill: Applied Mathematics for Engineers and PhysicistsRalston and Rabinowitz: A First Course in Numerical AnalysisRitger and Rose: Differential Equations with ApplicationsRudin: Principles of Mathematical AnalysisShapiro: Introduction to Abstract AlgebraSimmons: Differential Equations with Applications and Historical NotesSimmons: Introduction to Topology and Modern AnalysisStruble: Nonlinear Differential EquationsELEMENTARYNUMERICALANALYSISAn Algorithmic ApproachThird EditionS.
D. ContePurdue UniversityCarl de BoorUniversiry of Wisconsin—MadisonMcGraw-Hill Book CompanyNew York St. Louis San Francisco Auckland Bogotá HamburgJohannesburg London Madrid Mexico Montreal New DelhiPanama Paris São Paulo Singapore Sydney Tokyo TorontoELEMENTARY NUMERICAL ANALYSISAn Algorithmic ApproachCopyright © 1980, 1972, 1965 by McGraw-Hill, inc. All rights reserved.Printed in the United States of America. No part of this publicationmay be reproduced, stored in a retrieval system, or transmitted, in anyform or by any means, electronic, mechanical, photocopying, recording orotherwise, without the prior written permission of the publisher.234567890 DODO 89876543210This book was set in Times Roman by Science Typographers, Inc. Theeditors were Carol Napier and James S. Amar; the production supervisorwas 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 DataConte, Samuel Daniel, dateElementary numerical analysis.(International series in pure and appliedmathematics)Includes index.1. Numerical analysis-Data processing.I . de Boor, Carl, joint author. II. Title.1980519.479-24641QA297.C65ISBN 0-07-012447-7CONTENTSPrefaceIntroductionChapter 11.11.21.31.41.51.61.7Chapter 22.12.22.3*2.42.52.6*2.7ixxiNumber Systems and Errors1The Representation of IntegersThe Representation of FractionsFloating-Point ArithmeticLoss of Significance and Error Propagation;Condition and InstabilityComputational Methods for Error EstimationSome Comments on Convergence of SequencesSome Mathematical Preliminaries14712181925Interpolation by Polynomial31Polynomial FormsExistence and Uniqueness of the Interpolating PolynomialThe Divided-Difference TableInterpolation at an Increasing Number ofInterpolation PointsThe Error of the Interpolating PolynomialInterpolation in a Function Table Based on EquallySpaced PointsThe Divided Difference as a Function of Its Argumentsand Osculatory Interpolation31384146515562* Sections marked with an asterisk may be omitted without loss of continuity.VviCONTETSChapter 3The Solution of Nonlinear Equations72A Survey of Iterative MethodsFortran Programs for Some Iterative MethodsFixed-Point IterationConvergence Acceleration for Fixed-Point IterationConvergence of the Newton and Secant MethodsPolynomial Equations: Real RootsComplex Roots and Müller’s Method74818895100110120Chapter 4Matrices and Systems of Linear Equations4.14.24.34.44.54.6*4.7*4.8Properties of MatricesThe Solution of Linear Systems by EliminationThe Pivoting StrategyThe Triangular FactorizationError and Residual of an Approximate Solution; NormsBackward-Error Analysis and Iterative ImprovementDeterminantsThe Eigenvalue Problem128128147157160169177185189Chapter *5 Systems of Equations and UnconstrainedOptimization2083.13.23.33.4*3.53.6*3.7Optimization and Steepest DescentNewton’s MethodFixed-Point Iteration and Relaxation Methods209216223Approximation235Uniform Approximation by PolynomialsData FittingOrthogonal PolynomialsLeast-Squares Approximation by PolynomialsApproximation by Trigonometric PolynomialsFast Fourier TransformsPiecewise-Polynomial Approximation235245251259268277284Chapter 7Differentiation and Integration2947.17.27.37.47.5l 7.6l 7.7Numerical DifferentiationNumerical Integration: Some Basic RulesNumerical Integration: Gaussian RulesNumerical Integration: Composite RulesAdaptive QuadratureExtrapolation to the LimitRomberg Integration295303311319328333340*5.1*5.2*5.3Chapter 66.16.2*6.3*6.4*6.5*6.66.7CONTENTSChapter 88.18.28.38.48.58.68.78.88.9*8.10*8.11*8.12*8.13viiThe Solution of Differential Equations346Mathematical PreliminariesSimple Difference EquationsNumerical Integration by Taylor SeriesError Estimates and Convergence of Euler’s MethodRunge-Kutta MethodsStep-Size Control with Runge-Kutta MethodsMultistep FormulasPredictor-Corrector MethodsThe Adams-Moulton MethodStability of Numerical MethodsRound-off-Error Propagation and ControlSystems of Differential EquationsStiff Differential Equations346349354359362366373379382389395398401Chapter 9 Boundary Value Problems9.1 Finite Difference Methods9.2 Shooting Methods9.3 Collocation MethodsAppendix: Subroutine LibrariesReferencesIndex406406412416421423425PREFACEThis is the third edition of a book on elementary numerical analysis whichis designed specifically for the needs of upper-division undergraduatestudents in engineering, mathematics, and science including, in particular,computer science.
On the whole, the student who has had a solid collegecalculus sequence should have no difficulty following the material.Advanced mathematical concepts, such as norms and orthogonality, whenthey are used, are introduced carefully at a level suitable for undergraduatestudents and do not assume any previous knowledge. Some familiaritywith matrices is assumed for the chapter on systems of equations and withdifferential equations for Chapters 8 and 9. This edition does contain somesections which require slightly more mathematical maturity than the previous edition. However, all such sections are marked with asterisks and allcan be omitted by the instructor with no loss in continuity.This new edition contains a great deal of new material and significantchanges to some of the older material.
The chapters have been rearrangedin what we believe is a more natural order. Polynomial interpolation(Chapter 2) now precedes even the chapter on the solution of nonlinearsystems (Chapter 3) and is used subsequently for some of the material inall chapters. The treatment of Gauss elimination (Chapter 4) has beensimplified. In addition, Chapter 4 now makes extensive use of Wilkinson’sbackward error analysis, and contains a survey of many well-knownmethods for the eigenvalue-eigenvector problem. Chapter 5 is a newchapter on systems of equations and unconstrained optimization. It contains an introduction to steepest-descent methods, Newton’s method fornonlinear systems of equations, and relaxation methods for solving largelinear systems by iteration. The chapter on approximation (Chapter 6) hasbeen enlarged.
It now treats best approximation and good approximationixxPREFACEby polynomials, also approximation by trigonometric functions, includingthe Fast Fourier Transforms, as well as least-squares data fitting, orthogonal polynomials, and curve fitting by splines. Differentiation and integration are now treated in Chapter 7, which contains a new section onadaptive quadrature. Chapter 8 on ordinary differential equations containsconsiderable new material and some new sections. There is a new sectionon step-size control in Runge-Kutta methods and a new section on stiffdifferential equations as well as an extensively revised section on numericalinstability.
Chapter 9 contains a brief introduction to collocation as amethod for solving boundary-value problems.This edition, as did the previous one, assumes that students haveaccess to a computer and that they are familiar with programming in someprocedure-oriented language. A large number of algorithms are presentedin the text, and FORTRAN programs for many of these algorithms havebeen provided. There are somewhat fewer complete programs in thisedition.
All the programs have been rewritten in the FORTRAN 77language which uses modern structured-programming concepts. All theprograms have been tested on one or more computers, and in most casesmachine results are presented. When numerical output is given, the textwill indicate which machine (IBM, CDC, UNIVAC) was used to obtainthe results.The book contains more material than can usually be covered in atypical one-semester undergraduate course for general science majors. Thisgives the instructor considerable leeway in designing the course. For this, itis important to point out that only the material on polynomial interpolation in Chapter 2, on linear systems in Chapter 4, and on differentiationand integration in Chapter 7, is required in an essential way in subsequentchapters. The material in the first seven chapters (exclusive of the starredsections) would make a reasonable first course.We take this opportunity to thank those who have communicated to usmisprints and errors in the second edition and have made suggestions forimprovement.
We are especially grateful to R. E. Barnhill, D. Chambless,A. E. Davidoff, P. G. Davis, A. G. Deacon, A. Feldstein, W. Ferguson,A. O. Garder, J. Guest, T. R. Hopkins, D. Joyce, K. Kincaid, J. T. King,N. Krikorian, and W. E. McBride.S. D. ConteCarl de BoorINTRODUCTIONThis book is concerned with the practical solution of problems on computers. In the process of problem solving, it is possible to distinguishseveral more or less distinct phases. The first phase is formulation. Informulating a mathematical model of a physical situation, scientists shouldtake into account beforehand the fact that they expect to solve a problemon a computer. They will therefore provide for specific objectives, properinput data, adequate checks, and for the type and amount of output.Once a problem has been formulated, numerical methods, togetherwith a preliminary error analysis, must be devised for solving the problem.A numerical method which can be used to solve a problem will be calledan algorithm.
An algorithm is a complete and unambiguous set of procedures leading to the solution of a mathematical problem. The selection orconstruction of appropriate algorithms properly falls within the scope ofnumerical analysis. Having decided on a specific algorithm or set ofalgorithms for solving the problem, numerical analysts should consider allthe sources of error that may affect the results. They must consider howmuch accuracy is required, estimate the magnitude of the round-off anddiscretization errors, determine an appropriate step size or the number ofiterations required, provide for adequate checks on the accuracy, and makeallowance for corrective action in cases of nonconvergence.The third phase of problem solving is programming.
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