Numerical Recipes, 3rd Ed

1262 Pages·2007·20.62 MB·English

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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
WilliamH.Press
RaymerChairinComputerSciencesandIntegrativeBiology
TheUniversityofTexasatAustin
SaulA.Teukolsky
HansA.BetheProfessorofPhysicsandAstrophysics
CornellUniversity
WilliamT.Vetterling
ResearchFellowandDirectorofImageScience
ZINKImaging,LLC
BrianP.Flannery
Science,StrategyandProgramsManager
ExxonMobilCorporation
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CAMBRIDGEUNIVERSITY PRESS
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Cambridge University Press
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Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521880688
© Cambridge University Press 1988, 1992, 2002, 2007 except for 13.10, which is placed into
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This publication is in copyright. Subject to statutory exception and to the provision of
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without the written permission of Cambridge University Press.
First published in print format 2007
ISBN-13 978-0-511-33555-6 eBook (NetLibrary)
ISBN-10 0-511-33555-5 eBook (NetLibrary)
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Recipes Software, P.O. Box 380243, Cambridge, MA 02238-0243 (USA), email info@nr.
com, or fax [phone].
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Contents
PrefacetotheThirdEdition(2007) xi
PrefacetotheSecondEdition(1992) xiv
PrefacetotheFirstEdition(1985) xvii
LicenseandLegalInformation xix
1 Preliminaries 1
1.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Error,Accuracy,andStability . . . . . . . . . . . . . . . . . . . . 8
1.2 CFamilySyntax . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Objects,Classes,andInheritance . . . . . . . . . . . . . . . . . . 17
1.4 VectorandMatrixObjects . . . . . . . . . . . . . . . . . . . . . . 24
1.5 SomeFurtherConventionsandCapabilities . . . . . . . . . . . . . 30
2 SolutionofLinearAlgebraicEquations 37
2.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.1 Gauss-JordanElimination . . . . . . . . . . . . . . . . . . . . . . 41
2.2 GaussianEliminationwithBacksubstitution . . . . . . . . . . . . 46
2.3 LU DecompositionandItsApplications . . . . . . . . . . . . . . 48
2.4 TridiagonalandBand-DiagonalSystemsofEquations . . . . . . . 56
2.5 IterativeImprovementofaSolutiontoLinearEquations . . . . . . 61
2.6 SingularValueDecomposition. . . . . . . . . . . . . . . . . . . . 65
2.7 SparseLinearSystems . . . . . . . . . . . . . . . . . . . . . . . . 75
2.8 VandermondeMatricesandToeplitzMatrices . . . . . . . . . . . . 93
2.9 CholeskyDecomposition . . . . . . . . . . . . . . . . . . . . . . 100
2.10 QRDecomposition . . . . . . . . . . . . . . . . . . . . . . . . . 102
2.11 IsMatrixInversionanN3 Process? . . . . . . . . . . . . . . . . . 106
3 InterpolationandExtrapolation 110
3.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.1 Preliminaries: SearchinganOrderedTable . . . . . . . . . . . . . 114
3.2 PolynomialInterpolationandExtrapolation . . . . . . . . . . . . . 118
3.3 CubicSplineInterpolation . . . . . . . . . . . . . . . . . . . . . . 120
3.4 RationalFunctionInterpolationandExtrapolation . . . . . . . . . 124
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3.5 CoefficientsoftheInterpolatingPolynomial . . . . . . . . . . . . 129
3.6 InterpolationonaGridinMultidimensions . . . . . . . . . . . . . 132
3.7 InterpolationonScatteredDatainMultidimensions . . . . . . . . 139
3.8 LaplaceInterpolation . . . . . . . . . . . . . . . . . . . . . . . . 150
4 IntegrationofFunctions 155
4.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.1 ClassicalFormulasforEquallySpacedAbscissas . . . . . . . . . . 156
4.2 ElementaryAlgorithms . . . . . . . . . . . . . . . . . . . . . . . 162
4.3 RombergIntegration . . . . . . . . . . . . . . . . . . . . . . . . . 166
4.4 ImproperIntegrals . . . . . . . . . . . . . . . . . . . . . . . . . . 167
4.5 QuadraturebyVariableTransformation . . . . . . . . . . . . . . . 172
4.6 GaussianQuadraturesandOrthogonalPolynomials . . . . . . . . 179
4.7 AdaptiveQuadrature . . . . . . . . . . . . . . . . . . . . . . . . . 194
4.8 MultidimensionalIntegrals . . . . . . . . . . . . . . . . . . . . . 196
5 EvaluationofFunctions 201
5.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
5.1 PolynomialsandRationalFunctions. . . . . . . . . . . . . . . . . 201
5.2 EvaluationofContinuedFractions. . . . . . . . . . . . . . . . . . 206
5.3 SeriesandTheirConvergence . . . . . . . . . . . . . . . . . . . . 209
5.4 RecurrenceRelationsandClenshaw’sRecurrenceFormula. . . . . 219
5.5 ComplexArithmetic . . . . . . . . . . . . . . . . . . . . . . . . . 225
5.6 QuadraticandCubicEquations . . . . . . . . . . . . . . . . . . . 227
5.7 NumericalDerivatives . . . . . . . . . . . . . . . . . . . . . . . . 229
5.8 ChebyshevApproximation. . . . . . . . . . . . . . . . . . . . . . 233
5.9 DerivativesorIntegralsofaChebyshev-ApproximatedFunction . . 240
5.10 PolynomialApproximationfromChebyshevCoefficients . . . . . 241
5.11 EconomizationofPowerSeries . . . . . . . . . . . . . . . . . . . 243
5.12 Pade´ Approximants . . . . . . . . . . . . . . . . . . . . . . . . . 245
5.13 RationalChebyshevApproximation . . . . . . . . . . . . . . . . . 247
5.14 EvaluationofFunctionsbyPathIntegration . . . . . . . . . . . . . 251
6 SpecialFunctions 255
6.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
6.1 GammaFunction,BetaFunction,Factorials,BinomialCoefficients 256
6.2 IncompleteGammaFunctionandErrorFunction . . . . . . . . . . 259
6.3 ExponentialIntegrals . . . . . . . . . . . . . . . . . . . . . . . . 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 InverseoftheFunctionxlog.x/ . . . . . . . . . . . . . . . . . . . 307
6.12 EllipticIntegralsandJacobianEllipticFunctions . . . . . . . . . . 309
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6.13 HypergeometricFunctions . . . . . . . . . . . . . . . . . . . . . . 318
6.14 StatisticalFunctions . . . . . . . . . . . . . . . . . . . . . . . . . 320
7 RandomNumbers 340
7.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
7.1 UniformDeviates . . . . . . . . . . . . . . . . . . . . . . . . . . 341
7.2 CompletelyHashingaLargeArray . . . . . . . . . . . . . . . . . 358
7.3 DeviatesfromOtherDistributions . . . . . . . . . . . . . . . . . . 361
7.4 MultivariateNormalDeviates . . . . . . . . . . . . . . . . . . . . 378
7.5 LinearFeedbackShiftRegisters . . . . . . . . . . . . . . . . . . . 380
7.6 HashTablesandHashMemories . . . . . . . . . . . . . . . . . . 386
7.7 SimpleMonteCarloIntegration . . . . . . . . . . . . . . . . . . . 397
7.8 Quasi-(thatis,Sub-)RandomSequences . . . . . . . . . . . . . . 403
7.9 AdaptiveandRecursiveMonteCarloMethods . . . . . . . . . . . 410
8 SortingandSelection 419
8.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
8.1 StraightInsertionandShell’sMethod . . . . . . . . . . . . . . . . 420
8.2 Quicksort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
8.3 Heapsort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
8.4 IndexingandRanking . . . . . . . . . . . . . . . . . . . . . . . . 428
8.5 SelectingtheMthLargest . . . . . . . . . . . . . . . . . . . . . . 431
8.6 DeterminationofEquivalenceClasses . . . . . . . . . . . . . . . . 439
9 RootFindingandNonlinearSetsofEquations 442
9.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
9.1 BracketingandBisection . . . . . . . . . . . . . . . . . . . . . . 445
9.2 SecantMethod,FalsePositionMethod,andRidders’Method . . . 449
9.3 VanWijngaarden-Dekker-BrentMethod . . . . . . . . . . . . . . 454
9.4 Newton-RaphsonMethodUsingDerivative . . . . . . . . . . . . . 456
9.5 RootsofPolynomials . . . . . . . . . . . . . . . . . . . . . . . . 463
9.6 Newton-RaphsonMethodforNonlinearSystemsofEquations . . . 473
9.7 GloballyConvergentMethodsforNonlinearSystemsofEquations 477
10 MinimizationorMaximizationofFunctions 487
10.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
10.1 InitiallyBracketingaMinimum . . . . . . . . . . . . . . . . . . . 490
10.2 GoldenSectionSearchinOneDimension . . . . . . . . . . . . . . 492
10.3 ParabolicInterpolationandBrent’sMethodinOneDimension . . . 496
10.4 One-DimensionalSearchwithFirstDerivatives . . . . . . . . . . . 499
10.5 DownhillSimplexMethodinMultidimensions . . . . . . . . . . . 502
10.6 LineMethodsinMultidimensions . . . . . . . . . . . . . . . . . . 507
10.7 DirectionSet(Powell’s)MethodsinMultidimensions . . . . . . . 509
10.8 ConjugateGradientMethodsinMultidimensions . . . . . . . . . . 515
10.9 Quasi-NewtonorVariableMetricMethodsinMultidimensions . . 521
10.10 LinearProgramming: TheSimplexMethod . . . . . . . . . . . . . 526
10.11 LinearProgramming: Interior-PointMethods . . . . . . . . . . . . 537
10.12 SimulatedAnnealingMethods . . . . . . . . . . . . . . . . . . . . 549
10.13 DynamicProgramming . . . . . . . . . . . . . . . . . . . . . . . 555
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11 Eigensystems 563
11.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
11.1 JacobiTransformationsofaSymmetricMatrix . . . . . . . . . . . 570
11.2 RealSymmetricMatrices . . . . . . . . . . . . . . . . . . . . . . 576
11.3 Reduction of a Symmetric Matrix to Tridiagonal Form: Givens
andHouseholderReductions . . . . . . . . . . . . . . . . . . . . 578
11.4 EigenvaluesandEigenvectorsofaTridiagonalMatrix . . . . . . . 583
11.5 HermitianMatrices . . . . . . . . . . . . . . . . . . . . . . . . . 590
11.6 RealNonsymmetricMatrices . . . . . . . . . . . . . . . . . . . . 590
11.7 TheQRAlgorithmforRealHessenbergMatrices . . . . . . . . . 596
11.8 ImprovingEigenvaluesand/orFindingEigenvectorsbyInverse
Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597
12 FastFourierTransform 600
12.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600
12.1 FourierTransformofDiscretelySampledData . . . . . . . . . . . 605
12.2 FastFourierTransform(FFT) . . . . . . . . . . . . . . . . . . . . 608
12.3 FFTofRealFunctions . . . . . . . . . . . . . . . . . . . . . . . . 617
12.4 FastSineandCosineTransforms . . . . . . . . . . . . . . . . . . 620
12.5 FFTinTwoorMoreDimensions . . . . . . . . . . . . . . . . . . 627
12.6 FourierTransformsofRealDatainTwoandThreeDimensions . . 631
12.7 ExternalStorageorMemory-LocalFFTs . . . . . . . . . . . . . . 637
13 FourierandSpectralApplications 640
13.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640
13.1 ConvolutionandDeconvolutionUsingtheFFT . . . . . . . . . . . 641
13.2 CorrelationandAutocorrelationUsingtheFFT . . . . . . . . . . . 648
13.3 Optimal(Wiener)FilteringwiththeFFT . . . . . . . . . . . . . . 649
13.4 PowerSpectrumEstimationUsingtheFFT . . . . . . . . . . . . . 652
13.5 DigitalFilteringintheTimeDomain . . . . . . . . . . . . . . . . 667
13.6 LinearPredictionandLinearPredictiveCoding . . . . . . . . . . . 673
13.7 PowerSpectrumEstimationbytheMaximumEntropy(All-Poles)
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681
13.8 SpectralAnalysisofUnevenlySampledData . . . . . . . . . . . . 685
13.9 ComputingFourierIntegralsUsingtheFFT . . . . . . . . . . . . . 692
13.10 WaveletTransforms . . . . . . . . . . . . . . . . . . . . . . . . . 699
13.11 NumericalUseoftheSamplingTheorem . . . . . . . . . . . . . . 717
14 StatisticalDescriptionofData 720
14.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720
14.1 MomentsofaDistribution:Mean,Variance,Skewness,andSoForth 721
14.2 DoTwoDistributionsHavetheSameMeansorVariances?. . . . . 726
14.3 AreTwoDistributionsDifferent? . . . . . . . . . . . . . . . . . . 730
14.4 ContingencyTableAnalysisofTwoDistributions . . . . . . . . . 741
14.5 LinearCorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . 745
14.6 NonparametricorRankCorrelation . . . . . . . . . . . . . . . . . 748
14.7 Information-TheoreticPropertiesofDistributions. . . . . . . . . . 754
14.8 DoTwo-DimensionalDistributionsDiffer? . . . . . . . . . . . . . 762
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