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C++数值算法_第三版(英文)_William.H.Press
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C++数值算法_第三版(英文)_William.H.Press C++数值算法_第三版(英文)_William.H.Press
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NUMERICAL
RECIPES
The Art of Scientific Computing
Third Edition
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NUMERICAL
RECIPES
The Art of Scientific Computing
Third Edition
William H. Press
Raymer Chair in Computer Sciences and Integrative Biology
The University of Texas at A ustin
Saul A. Teukolsky
Hans A. Bethe Professor of Physics and Astrophysics
Cornell University
William T. Vetterling
Research Fellow and Director of Image Science
ZINK Imaging, LLC
Brian P. Flannery
Science, Strategy and Programs Manager
Exxon Mobil Corporation
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
First published in print format
ISBN-13 978-0-521-88068-8
ISBN-13 978-0-511-33555-6
© Cambridge University Press 1988, 1992, 2002, 2007 except for 13.10, which is placed into
the
p
ublic domain, and exce
p
t for all other com
p
uter
p
ro
g
rams and
p
rocedures, which are
W
ithout an additional license to use the contained software, this book is intended as a text
and reference book, for reading and study purposes only. However, a restricted, limited
free license for use of the software by the individual owner of a copy of this book who
personally keyboards one or more routines into a single computer is granted under terms
described on p. xix. See the section “License and Legal Information” (pp. xix–xxi) for
information on obtaining more general licenses. Machine-readable media containing the
software in this book, with included license for use by a single individual, are available
from Cambridge University Press. The software may also be downloaded, with immediate
purchase of a license also possible, from the Numerical Recipes Software Web site (http:
//www.nr.com). Unlicensed transfer of Numerical Recipes programs to any other format,
or to any computer except one that is specifically licensed, is strictly prohibited. Technical
questions, corrections, and requests for information should be addressed to Numerical
Recipes Software, P.O. Box 380243, Cambridge, MA 02238-0243 (USA), email info@nr.
com, or fax 781-863-1739.
2007
Information on this title: www.cambridge.org/9780521880688
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written
p
ermission of Cambrid
g
e University Press.
ISBN-10 0-511-33555-5
ISBN-10 0-521-88068-8
Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
g
uarantee that any content on such websites is, or will remain, accurate or a
pp
ro
p
riate.
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
hardback
eBook (NetLibrary)
eBook (NetLibrary)
hardback
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Contents
Preface to the Third Edition (2007) xi
Preface to the Second Edition (1992) xiv
Preface to the First Edition (1985) xvii
License and Legal Information xix
1 Preliminaries 1
1.0 Introduction . . . . . . ....................... 1
1.1 Error, Accuracy, and Stability . ................... 8
1.2 CFamilySyntax........................... 12
1.3 Objects,Classes,andInheritance .................. 17
1.4 VectorandMatrixObjects...................... 24
1.5 Some Further Conventions and Capabilities . . . . . ........ 30
2 Solution of Linear Algebraic Equations 37
2.0 Introduction . . . . . . ....................... 37
2.1 Gauss-JordanElimination ...................... 41
2.2 Gaussian Elimination with Backsubstitution . . . . ........ 46
2.3 LU Decomposition and Its Applications . . . . . . ........ 48
2.4 Tridiagonal and Band-Diagonal Systems of Equations . . . .... 56
2.5 IterativeImprovementofaSolutiontoLinearEquations...... 61
2.6 Singular Value Decomposition . ................... 65
2.7 SparseLinearSystems........................ 75
2.8 Vandermonde Matrices and Toeplitz Matrices . . . . ........ 93
2.9 Cholesky Decomposition . . . ................... 100
2.10 QR Decomposition . . ....................... 102
2.11 Is Matrix Inversion an N
3
Process?................. 106
3 Interpolation and Extrapolation 110
3.0 Introduction . . . . . . ....................... 110
3.1 Preliminaries:SearchinganOrderedTable ............. 114
3.2 Polynomial Interpolation and Extrapolation . . . . . ........ 118
3.3 CubicSplineInterpolation...................... 120
3.4 RationalFunctionInterpolationandExtrapolation ......... 124
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vi
Contents
3.5 Coefficients of the Interpolating Polynomial ............ 129
3.6 Interpolation on a Grid in Multidimensions . ............ 132
3.7 Interpolation on Scattered Data in Multidimensions ........ 139
3.8 LaplaceInterpolation ........................ 150
4 Integration of Functions 155
4.0 Introduction . . ........................... 155
4.1 Classical Formulas for Equally Spaced Abscissas . . ........ 156
4.2 ElementaryAlgorithms ....................... 162
4.3 RombergIntegration......................... 166
4.4 ImproperIntegrals.......................... 167
4.5 QuadraturebyVariableTransformation............... 172
4.6 Gaussian Quadratures and Orthogonal Polynomials ........ 179
4.7 AdaptiveQuadrature......................... 194
4.8 Multidimensional Integrals . . . . . ................ 196
5 Evaluation of Functions 201
5.0 Introduction . . ........................... 201
5.1 Polynomials and Rational Functions . ................ 201
5.2 EvaluationofContinuedFractions.................. 206
5.3 SeriesandTheirConvergence.................... 209
5.4 Recurrence Relations and Clenshaw’s Recurrence Formula . . . . . 219
5.5 ComplexArithmetic......................... 225
5.6 QuadraticandCubicEquations ................... 227
5.7 NumericalDerivatives........................ 229
5.8 ChebyshevApproximation...................... 233
5.9 Derivatives or Integrals of a Chebyshev-Approximated Function . . 240
5.10 Polynomial Approximation from Chebyshev Coefficients . . . . . 241
5.11 Economization of Power Series . . . ................ 243
5.12 Pad
´
eApproximants ......................... 245
5.13 RationalChebyshevApproximation................. 247
5.14 EvaluationofFunctionsbyPathIntegration............. 251
6 Special Functions 255
6.0 Introduction . . ........................... 255
6.1 Gamma Function, Beta Function, Factorials, Binomial Coefficients 256
6.2 IncompleteGammaFunctionandErrorFunction.......... 259
6.3 Exponential Integrals . ....................... 266
6.4 IncompleteBetaFunction ...................... 270
6.5 BesselFunctionsofIntegerOrder.................. 274
6.6 Bessel Functions of Fractional Order, Airy Functions, Spherical
BesselFunctions........................... 283
6.7 SphericalHarmonics......................... 292
6.8 FresnelIntegrals,CosineandSineIntegrals............. 297
6.9 Dawson’sIntegral .......................... 302
6.10 GeneralizedFermi-DiracIntegrals.................. 304
6.11 Inverse of the Function x log.x/ ................... 307
6.12 Elliptic Integrals and Jacobian Elliptic Functions . . ........ 309
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