Computing algorithms for solutions of problems in applied mathematics and their standard program realization. Part 2, Stochastic mathematics

389 Pages·2015·3.05 MB·English

by  K. J. KACHIASHVILI

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A M T
PPLIED ATHEMATICS AND HEIR
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TANDARD ROGRAM EALIZATION

P 1
ART

D M
ETERMINISTIC ATHEMATICS

K. J. KACHIASHVILI,
D. YU. MELIKDZHANIAN
AND
A. I. PRANGISHVILI

New York
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Contents
List of Figures
xi
List of Tables
xiii
Introduction xv
1 NumericalMethodsofLinearAlgebra 1
1.1 GeneralPropertiesofLinearEquations. . . . . . . . . . . . . . . . . . . . 1
1.2 SolvingSystemsofLinearEquationsUsingtheCramer
andGaussianMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 GaussianAlgorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 SolutionofLinearEquationsContainingTridiagonal
Matrixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 IterativeMethodsofSolutionofLinearEquations . . . . . . . . . . . . . . 9
1.6 PseudoinverseMatrixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7 EigenvaluesandEigenvectorsofLinearOperators . . . . . . . . . . . . . . 15
1.8 CharacteristicPolynomialsofMatrixes . . . . . . . . . . . . . . . . . . . . 21
1.9 NumericalMethodsofDeterminationofEigenvalues
andEigenvectorsofMatrixes . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.9.1 IterativeMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.9.2 RotationMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.10 Clebsch–GordanCoefficients . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.10.1 AngularMomentumOperator . . . . . . . . . . . . . . . . . . . . 31
1.10.2 AdditionofAngularMomentumOperators . . . . . . . . . . . . . 34
1.10.3 PropertiesofClebsch–GordanCoefficients . . . . . . . . . . . . . 35
2 NumericalAnalysisofPowerSeriesandPolynomials 43
2.1 ActionswithPowerSeries . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2 SomePropertiesofPolynomialsandtheirZeros . . . . . . . . . . . . . . . 52
2.2.1 SomePropertiesofPolynomials . . . . . . . . . . . . . . . . . . . 52
2.2.2 ZerosofPolynomials . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.3 DivisionofPolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.4 ExpansionofFractionalRationalFunctionsintoPartialFractions . . . . . . 60
2.5 PolynomialswithRealCoefficients . . . . . . . . . . . . . . . . . . . . . . 64
2.5.1 ElementaryPropertiesofPolynomialswithRealCoefficients . . . . 64
2.5.2 PropertiesofZerosofPolynomialsInfluencingonStability
ofDynamicalSystems . . . . . . . . . . . . . . . . . . . . . . . . 65
2.5.3 BoundariesofRealZerosofPolynomialswithRealCoefficients . . 66
2.5.4 TheNumberRealZerosofPolynomialswithRealCoefficients . . . 67
vi Contents
2.5.5 Algorithm of Determination of Real Zeros of Polynomials with
RealCoefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.6 RestorationofPolynomialbyitsZeros . . . . . . . . . . . . . . . . . . . . 70
2.6.1 ExpressionsforthePolynomialanditsCoefficients . . . . . . . . . 70
2.6.2 PropertiesofElementarySymmetricFunctions . . . . . . . . . . . 71
2.7 RestorationofPolynomialbyitsValuesinGivenPoints . . . . . . . . . . . 73
2.7.1 ExpressionsforthePolynomialanditsCoefficientsandSomeProp-
ertiesoftheAuxiliaryFunctions . . . . . . . . . . . . . . . . . . . 74
(m)
2.7.2 MainPropertiesoftheFunctionsλ (...). . . . . . . . . . . . . . 78
jk
2.8 DeterminationofZerosofPolynomialsbyMeansofExplicitExpressions . 80
2.9 Approximate Solution of Algebraic Equations by Gra¨ffe–Lobatchevsky
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
2.10 CalculationofSomeSpecialPolynomials
andTheirCoefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.10.1 BinomialCoefficients . . . . . . . . . . . . . . . . . . . . . . . . 88
2.10.2 PolynomialsofType(ξ+z)n andAnalogousPolynomialsofSev-
eralVariables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2.10.3 PolynomialsofTypeξ zn . . . . . . . . . . . . . . . . . . . . . 90
±
2.10.4 PochhammerSymbol . . . . . . . . . . . . . . . . . . . . . . . . . 91
2.10.5 MainPropertiesofStirlingNumbers . . . . . . . . . . . . . . . . . 92
2.10.6 BernoulliPolynomialsandEulerPolynomials . . . . . . . . . . . . 94
2.10.7 MainPropertiesofBernoulliNumbersandEulerNumbers . . . . . 95
2.11 CalculationofValuesofClassicalOrthogonalPolynomials . . . . . . . . . 96
2.11.1 GeneralPropertiesofOrthogonalPolynomials . . . . . . . . . . . 96
2.11.2 JacobiPolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 98
2.11.3 LaguerrePolynomials . . . . . . . . . . . . . . . . . . . . . . . . 99
2.11.4 HermitePolynomials . . . . . . . . . . . . . . . . . . . . . . . . . 100
2.11.5 LegendrePolynomials . . . . . . . . . . . . . . . . . . . . . . . . 102
2.11.6 TchebyshevPolynomials . . . . . . . . . . . . . . . . . . . . . . . 104
2.11.7 SomeFunctionsConnectedwithOrthogonalPolynomials . . . . . 106
2.11.8 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
2.12 SumsContainingPolynomialsandFractionalRationalFunctions . . . . . . 113
3 SolutionofNonlinearEquations
andDeterminationofExtremums 117
3.1 AuxiliaryTheoremsforNumericalSolutionofEquations . . . . . . . . . . 117
3.2 Numericalequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
3.2.1 GeneralPropertiesofNumericalEquations . . . . . . . . . . . . . 120
3.2.2 NumericalSolutionofEquationsContainingRealVariables . . . . 121
3.2.3 NumericalSolutionofEquationsContainingComplexVariables . . 124
3.2.4 NumericalSolutionofSystemsofEquations . . . . . . . . . . . . 125
3.3 Maximumsandminimums . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.3.1 Conditions of Existence of Maximums and Minimums for Func-
tionsofOneRealVariable . . . . . . . . . . . . . . . . . . . . . . 128
Contents
vii
3.3.2 Conditions of Existence of Maximums and Minimums for Func-
tionsofSeveralRealVariables . . . . . . . . . . . . . . . . . . . . 129
3.3.3 NumericalMethods. . . . . . . . . . . . . . . . . . . . . . . . . . 131
4 InterpolationandApproximation
ofFunctions 135
4.1 AuxiliaryTheoremforApproximationofFunctionsandtheQuestionsCon-
nectedwithIt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.2 Interpolation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.2.1 DifferentTypesofRestoredFunction . . . . . . . . . . . . . . . . 138
4.2.2 InterpolationError . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.3 InterpolationbyMeansofPolynomialsandRelatedFunctions . . . . . . . 140
4.4 InterpolationofNonlinearFunctionsoftheCertainClass . . . . . . . . . . 144
4.4.1 Functiona+becx . . . . . . . . . . . . . . . . . . . . . . . . . . 144
4.4.2 Function(a+bx) ecx . . . . . . . . . . . . . . . . . . . . . . . 145
·
4.4.3 Functionh+(a+bx) ecx . . . . . . . . . . . . . . . . . . . . . 146
·
4.4.4 Functionaxc (1 bx)d . . . . . . . . . . . . . . . . . . . . . . 148
· −
4.4.5 Functionaecx+bedx . . . . . . . . . . . . . . . . . . . . . . . . 149
4.4.6 Functionh+aecx+bedx . . . . . . . . . . . . . . . . . . . . . . 150
4.4.7 Functionesx A cos(ωx)+B sin(ωx) . . . . . . . . . . . . . . 151
·
4.4.8 Functionh+esx A cos(ωx)+B sin(ωx) . . . . . . . . . . . 153
(cid:0) · (cid:1)
4.4.9 RealizationoftheAlgorithms . . . . . . . . . . . . . . . . . . . . 154
(cid:0) (cid:1)
4.5 Spline-Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.5.1 InterpolationofFunctionofOneVariable . . . . . . . . . . . . . . 155
4.5.2 InterpolationofFunctionsofTwoandThreeVariables . . . . . . . 158
4.6 Approximationof FunctionsbytheGeneralized LeastSquares Method on
DiscreteSetofPoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
4.6.1 LinearDependenceofRestoredFunctiononParameters . . . . . . 160
4.6.2 NonlinearDependenceoftheRestoredFunctiononParameters . . 162
4.7 DeterminationofBoundariesofSearchof
ApproximationParameters . . . . . . . . . . . . . . . . . . . . . . . . . . 164
4.8 OtherMethodsofDeterminationofBoundaries . . . . . . . . . . . . . . . 167
4.9 ApproximationofFunctionsbytheGeneralized
LeastSquaresMethodinIntervalorRegion . . . . . . . . . . . . . . . . . 169
4.10 OtherMethodsofApproximationofFunctions . . . . . . . . . . . . . . . 172
5 NumericalDifferentiationandIntegration 175
5.1 NumericalDifferentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.1.1 SomeFormulasforDerivatives. . . . . . . . . . . . . . . . . . . . 175
5.1.2 ApproximationofDifferentialOperatorswithDifferenceOperators 176
5.2 NumericalIntegrationofFunctionsbyMeans
ofNewton–CotesFormulas . . . . . . . . . . . . . . . . . . . . . . . . . . 179
5.3 NumericalIntegrationofFunctionsbyMeansofthe
FormulasofGaussianType . . . . . . . . . . . . . . . . . . . . . . . . . . 184
5.3.1 MostImportantQuadratureFormulas . . . . . . . . . . . . . . . . 184
viii Contents
5.3.2 AdditionalQuadratureFormulas . . . . . . . . . . . . . . . . . . . 188
6 CalculationofValuesofSomeFunctions 191
6.1 MainTranscendentalMathematicalConstants . . . . . . . . . . . . . . . . 192
6.2 SolutionofTranscendentalEquationsofSpecialTypes . . . . . . . . . . . 193
6.2.1 Equations Containing Linear-Exponential or Geometric-
ExponentialDependence . . . . . . . . . . . . . . . . . . . . . . . 193
6.2.2 EquationsContainingProductofGeometricalDependences . . . . 197
6.2.3 EquationsContainingSumofExponents . . . . . . . . . . . . . . 202
6.3 CalculationofValuesofGamma-FunctionandConnectedwithitFunctions 206
6.3.1 MainPropertiesofConsideredFunctions . . . . . . . . . . . . . . 206
6.3.2 RepresentationoftheFunctionsintheForms ofConvergentSeries
andIntegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
6.3.3 AsymptoticExpansions . . . . . . . . . . . . . . . . . . . . . . . 213
6.3.4 RiemannZetaFunctionandFunctionsConnectedwithit . . . . . . 216
6.3.5 CalculationofValuesoftheFunctions . . . . . . . . . . . . . . . . 218
6.4 HypergeometricFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 218
6.4.1 ElementaryPropertiesofHypergeometricFunction . . . . . . . . . 219
6.4.2 DifferentialEquations . . . . . . . . . . . . . . . . . . . . . . . . 220
6.4.3 PowerSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
6.4.4 FunctionalEquationsandLimits . . . . . . . . . . . . . . . . . . . 223
6.4.5 FunctionalEquationsforHypergeometricFunctionsSatisfying
Second-OrderDifferentialEquations. . . . . . . . . . . . . . . . . 225
6.4.6 DifferentiationandIntegrationFormulas. . . . . . . . . . . . . . . 229
6.4.7 IntegralRepresentations . . . . . . . . . . . . . . . . . . . . . . . 230
6.4.8 InequalitiesforHypergeometricFunctions . . . . . . . . . . . . . 232
6.5 CylindricalFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
6.6 UseofHypergeometricFunctionsforSolvingtheLinearDifferentialEqua-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
6.6.1 DifferentialEquationsofanyOrderN 2 . . . . . . . . . . . . . 242
≥
6.6.2 Second-OrderDifferentialEquations. . . . . . . . . . . . . . . . . 244
6.7 ReductionFormulasforHypergeometricFunction . . . . . . . . . . . . . . 247
6.7.1 ReductionFormulasGenerally . . . . . . . . . . . . . . . . . . . . 248
6.7.2 ReductionFormulasfortheFunction F (γ,z) . . . . . . . . . . . 253
0 1
6.7.3 ReductionFormulasfortheFunction F (α ,α ,z) . . . . . . . . 255
2 0 1 2
6.7.4 ReductionFormulasfortheKummerHypergeometricFunction . . 256
6.7.5 ReductionFormulasfortheGaussHypergeometricFunction . . . . 259
6.8 AsymtoticExpansionofHypergeometricFunctions
inTermsofParameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
6.8.1 AsymptoticExpansions . . . . . . . . . . . . . . . . . . . . . . . 265
6.8.2 MainPropertiesofFunctionsh (z). . . . . . . . . . . . . . . . . . 270
r
6.8.3 MainPropertiesofFunctionsχ (λ,z) . . . . . . . . . . . . . . . . 271
r
6.8.4 MainPropertiesofFunctionsU (λ),V (λ),W (λ)andΥ (...) 272
jk jk jk jk
6.8.5 AsymptoticExpansionofHypergeometricFunctionsSatisfyingthe
Second-OrderDifferentialEquations. . . . . . . . . . . . . . . . . 275
Contents ix
6.8.6 ControlExamples . . . . . . . . . . . . . . . . . . . . . . . . . . 277
6.8.7 FinalRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
6.9 ElementaryMethodsofCalculationofValuesof
HypergeometricandCylindricalFunctions . . . . . . . . . . . . . . . . . . 278
6.10 Calculation of Values of Hypergeometric Functions by Means of Power
SeriesandRecurrenceRelations . . . . . . . . . . . . . . . . . . . . . . . 284
6.10.1 DescriptionoftheMethod . . . . . . . . . . . . . . . . . . . . . . 284
6.10.2 DeterminationofParameter ξ . . . . . . . . . . . . . . . . . . . . 290
6.10.3 FinalRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
7 NumericalMethodsforSolvingDifferentialEquations 295
7.1 Numerical Solution of Ordinary Differential Equations by Runge–Kutta
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
7.2 NumericalSolvingofOrdinaryDifferentialEquationsbyMultistepDiffer-
enceMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
7.3 One-dimensionalBoundaryProblems . . . . . . . . . . . . . . . . . . . . 307
7.4 BoundaryProblemsofGeneralForm . . . . . . . . . . . . . . . . . . . . . 310
7.5 MultidimensionalBoundaryProblemsofSpecialType . . . . . . . . . . . 315
7.6 DiffusionEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
7.6.1 ExplicitScheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
7.6.2 ClassicalDifferenceSchemeintheGeneralForm . . . . . . . . . . 320
7.6.3 MethodofDecompositionoftheOperator . . . . . . . . . . . . . . 324
7.7 WaveEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
7.8 EstimationofDerivativesofUnknownFunction . . . . . . . . . . . . . . . 328
7.9 MethodsofApproximatingFunctionsforthe
NumericalSolutionofDifferentialEquations . . . . . . . . . . . . . . . . 329
8 NumericalMethodsUsedinGeometry 333
8.1 Three-DimensionalRotationMatrixes . . . . . . . . . . . . . . . . . . . . 333
8.2 DescriptionofPlaneCurvesbySplines . . . . . . . . . . . . . . . . . . . 339
8.2.1 CurvilinearCoordinatesConnectedwiththePlaneCurve . . . . . . 339
8.2.2 UsingSpline–InterpolationforRepresentationofaCurve . . . . . . 341
References 349
About the Authors 367
Index 369