Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed. (Fortran Numerical Recipes 1)

1002 Pages·1992·13.79 MB·English

by  William H. Press

Numerical Recipes
in Fortran 77
The Art of Scientific Computing
Second Edition
Volume 1 of
Fortran Numerical Recipes
WilliamH.Press
Harvard-SmithsonianCenterforAstrophysics
SaulA.Teukolsky
DepartmentofPhysics,CornellUniversity
WilliamT.Vetterling
PolaroidCorporation
BrianP.Flannery
EXXONResearchandEngineeringCompany
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Somesectionsofthisbookwereoriginallypublished,indifferentform,inComputers
inPhysicsmagazine,Copyright(cid:13)c AmericanInstituteofPhysics,1988–1992.
FirstEditionoriginallypublished1986;SecondEditionoriginallypublished1992as
NumericalRecipesinFORTRAN TheArtofScientificComputing
Reprintedwith corrections, 1993, 1994, 1995.
Reprintedwithcorrections,1996,1997,asNumericalRecipesinFortran77 TheArtof
ScientificComputing(Vol.1ofFortranNumericalRecipes)
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LibraryofCongressCataloginginPublicationData
NumericalrecipesinFortran77: theartofscientificcomputing/WilliamH.Press
::: [et al.]. – 2nd ed.
Includesbibliographicalreferences(p. ) andindex.
ISBN 0-521-43064-X
1. Numericalanalysis–Computerprograms.2. Science–Mathematics–Computerprograms.
3. FORTRAN(Computerprogramlanguage) I.Press, WilliamH.
QA297.N866 1992
519.400285053–dc20 92-8876
AcatalogrecordforthisbookisavailablefromtheBritishLibrary.
ISBN 052143064X Volume1(thisbook)
ISBN 0521574390 Volume2
ISBN 0521437210 ExamplebookinFORTRAN
ISBN 0521574404 FORTRANdiskette(IBM3.500)
ISBN 0521576083 CDROM(IBMPC/Macintosh)
ISBN 0521576075 CDROM(UNIX)
Contents
PlanoftheTwo-VolumeEdition xiii
PrefacetotheSecondEdition xv
PrefacetotheFirstEdition xviii
LicenseInformation xx
ComputerProgramsbyChapterandSection xxiv
1 Preliminaries 1
1.0Introduction 1
1.1ProgramOrganizationandControlStructures 5
1.2Error,Accuracy,andStability 18
2 SolutionofLinearAlgebraicEquations 22
2.0Introduction 22
2.1Gauss-JordanElimination 27
2.2GaussianEliminationwithBacksubstitution 33
2.3LUDecompositionandItsApplications 34
2.4TridiagonalandBandDiagonalSystemsofEquations 42
2.5IterativeImprovementofaSolutiontoLinearEquations 47
2.6SingularValueDecomposition 51
2.7SparseLinearSystems 63
2.8VandermondeMatricesandToeplitzMatrices 82
2.9CholeskyDecomposition 89
2.10QRDecomposition 91
2.11IsMatrixInversionanN3Process? 95
3 InterpolationandExtrapolation 99
3.0Introduction 99
3.1PolynomialInterpolationandExtrapolation 102
3.2RationalFunctionInterpolationandExtrapolation 104
3.3CubicSplineInterpolation 107
3.4HowtoSearchanOrderedTable 110
3.5CoefficientsoftheInterpolatingPolynomial 113
3.6InterpolationinTwoorMoreDimensions 116
v
vi Contents
4 IntegrationofFunctions 123
4.0Introduction 123
4.1ClassicalFormulasforEquallySpacedAbscissas 124
4.2ElementaryAlgorithms 130
4.3RombergIntegration 134
4.4ImproperIntegrals 135
4.5GaussianQuadraturesandOrthogonalPolynomials 140
4.6MultidimensionalIntegrals 155
5 EvaluationofFunctions 159
5.0Introduction 159
5.1SeriesandTheirConvergence 159
5.2EvaluationofContinuedFractions 163
5.3PolynomialsandRationalFunctions 167
5.4ComplexArithmetic 171
5.5RecurrenceRelationsandClenshaw’sRecurrenceFormula 172
5.6QuadraticandCubicEquations 178
5.7NumericalDerivatives 180
5.8ChebyshevApproximation 184
5.9DerivativesorIntegralsofaChebyshev-approximatedFunction 189
5.10PolynomialApproximationfromChebyshevCoefficients 191
5.11EconomizationofPowerSeries 192
5.12Pade´ Approximants 194
5.13RationalChebyshevApproximation 197
5.14EvaluationofFunctionsbyPathIntegration 201
6 SpecialFunctions 205
6.0Introduction 205
6.1GammaFunction,BetaFunction,Factorials,BinomialCoefficients 206
6.2IncompleteGammaFunction,ErrorFunction,Chi-Square
ProbabilityFunction,CumulativePoissonFunction 209
6.3ExponentialIntegrals 215
6.4IncompleteBetaFunction,Student’sDistribution,F-Distribution,
CumulativeBinomialDistribution 219
6.5BesselFunctionsofIntegerOrder 223
6.6ModifiedBesselFunctionsofIntegerOrder 229
6.7BesselFunctionsofFractionalOrder,AiryFunctions,Spherical
BesselFunctions 234
6.8SphericalHarmonics 246
6.9FresnelIntegrals,CosineandSineIntegrals 248
6.10Dawson’sIntegral 252
6.11EllipticIntegralsandJacobianEllipticFunctions 254
6.12HypergeometricFunctions 263
7 RandomNumbers 266
7.0Introduction 266
7.1UniformDeviates 267
Contents vii
7.2TransformationMethod: ExponentialandNormalDeviates 277
7.3RejectionMethod: Gamma, Poisson,BinomialDeviates 281
7.4GenerationofRandomBits 287
7.5RandomSequencesBasedonDataEncryption 290
7.6SimpleMonteCarloIntegration 295
7.7Quasi-(thatis,Sub-)RandomSequences 299
7.8AdaptiveandRecursiveMonteCarloMethods 306
8 Sorting 320
8.0Introduction 320
8.1StraightInsertionandShell’sMethod 321
8.2Quicksort 323
8.3Heapsort 327
8.4IndexingandRanking 329
8.5SelectingtheMthLargest 333
8.6DeterminationofEquivalenceClasses 337
9 RootFindingandNonlinearSetsofEquations 340
9.0Introduction 340
9.1BracketingandBisection 343
9.2SecantMethod,FalsePositionMethod,andRidders’Method 347
9.3VanWijngaarden–Dekker–BrentMethod 352
9.4Newton-RaphsonMethodUsingDerivative 355
9.5RootsofPolynomials 362
9.6Newton-RaphsonMethodforNonlinearSystemsofEquations 372
9.7GloballyConvergentMethodsforNonlinearSystemsofEquations 376
10 MinimizationorMaximizationofFunctions 387
10.0Introduction 387
10.1GoldenSectionSearchinOneDimension 390
10.2ParabolicInterpolationandBrent’sMethodinOneDimension 395
10.3One-DimensionalSearchwithFirstDerivatives 399
10.4DownhillSimplexMethodinMultidimensions 402
10.5DirectionSet(Powell’s)MethodsinMultidimensions 406
10.6ConjugateGradientMethodsinMultidimensions 413
10.7VariableMetricMethodsinMultidimensions 418
10.8LinearProgrammingandtheSimplexMethod 423
10.9SimulatedAnnealingMethods 436
11 Eigensystems 449
11.0Introduction 449
11.1JacobiTransformationsofaSymmetricMatrix 456
11.2ReductionofaSymmetricMatrixtoTridiagonalForm:
GivensandHouseholderReductions 462
11.3EigenvaluesandEigenvectorsofaTridiagonalMatrix 469
11.4HermitianMatrices 475
11.5ReductionofaGeneralMatrixtoHessenbergForm 476
viii Contents
11.6TheQRAlgorithmforRealHessenbergMatrices 480
11.7ImprovingEigenvaluesand/orFindingEigenvectorsby
InverseIteration 487
12 FastFourierTransform 490
12.0Introduction 490
12.1FourierTransformofDiscretelySampledData 494
12.2FastFourierTransform(FFT) 498
12.3FFTofRealFunctions,SineandCosineTransforms 504
12.4FFTinTwoorMoreDimensions 515
12.5FourierTransformsofRealDatainTwoandThreeDimensions 519
12.6ExternalStorageorMemory-LocalFFTs 525
13 FourierandSpectralApplications 530
13.0Introduction 530
13.1ConvolutionandDeconvolutionUsingtheFFT 531
13.2CorrelationandAutocorrelationUsingtheFFT 538
13.3Optimal(Wiener)FilteringwiththeFFT 539
13.4PowerSpectrumEstimationUsingtheFFT 542
13.5DigitalFilteringintheTimeDomain 551
13.6LinearPredictionandLinearPredictiveCoding 557
13.7PowerSpectrumEstimationbytheMaximumEntropy
(AllPoles)Method 565
13.8SpectralAnalysisofUnevenlySampledData 569
13.9ComputingFourierIntegralsUsingtheFFT 577
13.10WaveletTransforms 584
13.11NumericalUseoftheSamplingTheorem 600
14 StatisticalDescriptionofData 603
14.0Introduction 603
14.1MomentsofaDistribution:Mean,Variance,Skewness,
andSoForth 604
14.2DoTwoDistributionsHavetheSameMeansorVariances? 609
14.3AreTwoDistributionsDifferent? 614
14.4ContingencyTableAnalysisofTwoDistributions 622
14.5LinearCorrelation 630
14.6NonparametricorRankCorrelation 633
14.7DoTwo-DimensionalDistributionsDiffer? 640
14.8Savitzky-GolaySmoothingFilters 644
15 ModelingofData 650
15.0Introduction 650
15.1LeastSquaresasaMaximumLikelihoodEstimator 651
15.2FittingDatatoaStraightLine 655
15.3Straight-LineDatawithErrorsinBothCoordinates 660
15.4GeneralLinearLeastSquares 665
15.5NonlinearModels 675
Contents ix
15.6ConfidenceLimitsonEstimatedModelParameters 684
15.7RobustEstimation 694
16 IntegrationofOrdinaryDifferentialEquations 701
16.0Introduction 701
16.1Runge-KuttaMethod 704
16.2AdaptiveStepsizeControlforRunge-Kutta 708
16.3ModifiedMidpointMethod 716
16.4RichardsonExtrapolationandtheBulirsch-StoerMethod 718
16.5Second-OrderConservativeEquations 726
16.6StiffSetsofEquations 727
16.7Multistep,Multivalue,andPredictor-CorrectorMethods 740
17 TwoPointBoundaryValueProblems 745
17.0Introduction 745
17.1TheShootingMethod 749
17.2ShootingtoaFittingPoint 751
17.3RelaxationMethods 753
17.4AWorkedExample: SpheroidalHarmonics 764
17.5AutomatedAllocationofMeshPoints 774
17.6HandlingInternalBoundaryConditionsorSingularPoints 775
18 IntegralEquationsandInverseTheory 779
18.0Introduction 779
18.1FredholmEquationsoftheSecondKind 782
18.2VolterraEquations 786
18.3IntegralEquationswithSingularKernels 788
18.4InverseProblemsandtheUseofAPrioriInformation 795
18.5LinearRegularizationMethods 799
18.6Backus-GilbertMethod 806
18.7MaximumEntropyImageRestoration 809
19 PartialDifferentialEquations 818
19.0Introduction 818
19.1Flux-ConservativeInitialValueProblems 825
19.2DiffusiveInitialValueProblems 838
19.3InitialValueProblemsinMultidimensions 844
19.4FourierandCyclicReductionMethodsforBoundary
ValueProblems 848
19.5RelaxationMethodsforBoundaryValueProblems 854
19.6MultigridMethodsforBoundaryValueProblems 862
20 Less-NumericalAlgorithms 881
20.0Introduction 881
20.1DiagnosingMachineParameters 881
20.2GrayCodes 886
x Contents
20.3CyclicRedundancyandOtherChecksums 888
20.4HuffmanCodingandCompressionofData 896
20.5ArithmeticCoding 902
20.6ArithmeticatArbitraryPrecision 906
ReferencesforVolume1 916
IndexofProgramsandDependencies(Vol.1) 921
GeneralIndextoVolumes1and2
ContentsofVolume2:NumericalRecipesinFortran90
PrefacetoVolume2 viii
ForewordbyMichaelMetcalf x
LicenseInformation xvii
21 IntroductiontoFortran90LanguageFeatures 935
22 IntroductiontoParallelProgramming 962
23 NumericalRecipesUtilitiesforFortran90 987
Fortran90CodeChapters 1009
B1 Preliminaries 1010
B2 SolutionofLinearAlgebraicEquations 1014
B3 InterpolationandExtrapolation 1043
B4 IntegrationofFunctions 1052
B5 EvaluationofFunctions 1070
B6 SpecialFunctions 1083
B7 RandomNumbers 1141
B8 Sorting 1167
B9 RootFindingandNonlinearSetsofEquations 1182
B10 MinimizationorMaximizationofFunctions 1201
B11 Eigensystems 1225
B12 FastFourierTransform 1235
Contents xi
B13 FourierandSpectralApplications 1253
B14 StatisticalDescriptionofData 1269
B15 ModelingofData 1285
B16 IntegrationofOrdinaryDifferentialEquations 1297
B17 TwoPointBoundaryValueProblems 1314
B18 IntegralEquationsandInverseTheory 1325
B19 PartialDifferentialEquations 1332
B20 Less-NumericalAlgorithms 1343
ReferencesforVolume2 1359
Appendices
C1 ListingofUtilityModules(nrtypeandnrutil) 1361
C2 ListingofExplicitInterfaces 1384
C3 IndexofProgramsandDependencies(Vol.2) 1434
GeneralIndextoVolumes1and2 1447
xii