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Kachiashvili K.J., Melikdzhanian D.Yu., Prangishvili A.I. Computing Algorithms for Solutions of Problems in Applied Mathematics and their Standard Program Realization 2 Stochastic Mathematics

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Kachiashvili K.J., Melikdzhanian D.Yu., Prangishvili A.I. Computing Algorithms for Solutions of Problems in Applied Mathematics and their Standard Program Realization 2 Stochastic Mathematics
New York: Nova Publishers, 2015. — 366 p.
Contents:
List of Figures
List of Tables
Numerical Methods of Probability Theory and Mathematical Statistics
Methods of Combinatorics
Main Concepts and Theorems of Combinatorics
Algorithms for the Generation of Numerical Sequences
Discrete Probability Distributions
Simplest Discrete Distributions
Binomial Distribution
Geometric and Pascal Distributions
Hypergeometric Distribution
Poisson Distribution
Series Distribution
Connection between Different Discrete Distributions
Major Continuous Probability Distributions
Uniform Distribution
Exponential Distribution
Normal Distribution
Properties of the Normal Distribution Function
m-Dimensional Normal Distribution
Irregular Distributions
Triangular Distribution
Trapezoidal Distribution
Generalized Trapezoidal Distribution
Antimodal-I Distribution
Antimodal-II Distribution
Basic Probability Distributions Used in Mathematical Statistics
Chi-Square Distribution
Properties of the Chi-Square Distribution Function
Student’s Distribution
Properties of Student’s Distribution Function
Fisher’s Distribution
Properties of Fisher’s Distribution Function
Connection between Different Distributions
Additional Probability Distributions Used in Mathematical Statistics
Kolmogorov Distribution
Omega-Square Distribution
D-Distribution
Samples
Samples and Statistics
Variational Series
Statistical Estimates of Main Characteristics of a Random Variable
Sample Moments and Empiric Probabilities
Histogram and the Concepts Connected with It
Empiric Distribution Function
Statistical Estimates of Distribution Parameters
Methods of Obtaining of Estimators
Estimators of the Parameters of Some Special Probability Distributions
Confidence and Tolerance Intervals
Determination of Confidence Intervals
Tolerance Interval
Non-classical Method for Construction of the Confidence Interval for Expectation
Main Properties of the Functions ψN(h), ΨN (H), hN (α), HN (α)
Testing of Statistical Hypotheses
Scheme of Hypotheses Testing
Test of Identity of the Distributions of Two Random Variables
Statistics of Two Samples from Normal Sets
Statistics of the United Samples from Normal Sets
Testing the Normality of Probability Distribution
Identification of the Density of Probability Distribution
Testing of Some Other Hypotheses
Nonparametric Methods of Statistics
Test of Identity of the Distributions of Two Random Variables
Criteria of Signs and Sign Ranks
One-Factor Analysis
Two Factor Analysis
Two Factor Analysis of a Variance
Correlation Analysis
Identification of Regression Dependencies
Restoration of Some Special Types of Nonlinear Functional Dependencies
Geometric Regression
Exponential Regression
Logarithmic Regression
Geometric-Exponential Regression
Exponential Regression with a Free Term
Geometric Regression with a Free Term
Inverse Exponential Regression
Linear-Exponential Regression
Linear-Exponential Dependence with a Free Term
Product of Geometric Dependencies
Sum of Exponential Dependencies
Sum of Geometric Dependencies
Sum of Exponential Dependencies with a Free Term
Sum of Geometric Dependencies with a Free Term
Exponential-Sinusoidal Regression
Exponential-Sinusoidal Regression with a Free Term
Polynomial Regression
Geometric-Polynomial Regression
Exponential-Polynomial Regression
Logarithmic-Polynomial Regression
Periodic Regression
Linear Multiple Regression
Main Properties of Restored Dependencies
Some Additional Problems
Algebra of Integers
Performance of Arithmetic Operations with Rational and Complex Numbers
Search of Words and Phrases in a Dictionary
A Derivation of Formulae and Proofs of Theorems
Numerical Methods of Linear Algebra
Sweep Method
Seidel Method
Iterative Methods for the Determination of Eigenvalues and Eigen
vectors
Rotation Method
Basic Properties of the Eigenvalues and Eigenvectors of the Operators Sˆ and Sˆz Eigenvalues of the Operator Sˆ
Generalized Recurrence Relations for the Clebsch–Gordan Coefficients
Special Values of the Clebsch–Gordan Coefficients
Numerical Analysis of a Power Series and its Polynomials
Calculation of an Exponent
Calculation of a Power Function
Fundamental Theorem of Algebra
Euclidean Algorithm for the Finding of GCD{α(z), β(z)})
Lagrange Theorem Determining the Boundaries of Real Zeros
Sturm Theorem
Interpolational Formulas for Polynomials
Gr¨affe–Lobatchevsky Method
Properties of Orthogonal Polynomials
Sums of Partial Fractions
Solution of Nonlinear Equations and the Determination of Extremums
Principle of Contraction Mappings
Auxiliary Theorems for the Principle of Contraction Mappings
Iterative Sequence Having a Square-Law Convergence
Newton Method
Newton Method for Analytical Functions
Newton Method for a System of Equations
Interpolation and Approximation of Functions
Interpolation Error
Optimal Choice of Interpolation Nodes
Interpolation Error for Polynomials Expressed in Terms of Divided Differences
Explicit Expression for Interpolational Polynomial at Optimal Choice of Nodes
Interpolation Formulas for One-Parametric Families of Functions of Polynomials
Interpolation of the Function a xc · ( − b x)d
Cubic Spline Interpolation
Numerical Differentiation and Integration
Connection of Divided Differences with Derivatives
Quadrature Formulas of an Interpolational Type
Newton–Cotes Quadrature Formulas
Quadrature Formulas of a Gaussian Type
Modification of the Gauss Quadrature Formula
Calculation of the Values of Some Functions
Asymptotic Behavior of the Inverse Function of x ex
Determination of the Radius of Convergence of a Taylor Series of the Function P(s, z)
Main Inequalities for the Function P(s, z)
Values of the Function Ψ(x) for the Fractional Values of Argument
Derivation of the Basic Functional Equations for Π(z)
Gauss Multiplication Formula for Π(z)
Integral Representation for Π(z)
Integral Representation for Ψ(z)
Integral Representation for Ln Π(z)
Integral Representations of a Beta-Function
Asymptotic Expansion of the Function Ψ(z)
Properties of the Operator zdz
Coefficients of the Expansion of a Hypergeometric Function in a Power Series
Convergence of a Hypergeometric Series
Transformation of Argument for the Hypergeometric Function: z → 1/z
Transformation of Argument for the Kummer Function
Recurrence Formulas for Hypergeometric Functions one of the Parameters of Which Varies by Unit
Main Inequalities for Hypergeometric Functions
Asymptotic Expansion of Hypergeometric Function ()
Asymptotic Expansion of Hypergeometric Function ()
Asymptotic Expansion of Hypergeometric Function ()
Approximation of the Functions Jm(x) by Trigonometrical Sums
Numerical Methods for Solving Differential Equations
Error of Approximation of the Runge–Kutta Methods
Error of Approximation of m-Step Difference Methods
Derivation of Formulas for the Residuals of a Diffusion Equation
Numerical Methods Used in Geometry
Expression of the Matrix A in Terms of the Rotation Angle and the Axis of Rotation
Numerical Methods of Probability Theory and Mathematical Statistics
Asymptotic Expansions for the Function of Probabilities of Binomial Distribution
Asymptotic Expansion of Fisher’s Distribution Function
Remainders of the Expansions of a Kolmogorov Distribution Function Used for Small Values of the Argument
Nonclassical Method for the Construction of the Confidence Interval for the Expectation of a Random Variable
Special Values of the Function ψN(h)
Limit of hN (α)
Some Additional Problems
Decomposition of Natural Numbers on Prime Factors
Euclidean Algorithm for Integers
Solving of Linear Equations in Integers
Program Realization of Formulas and Algorithms
Numerical Methods of Linear Algebra
Problems which do not Use Iterative Algorithms
Iterative Methods for Solution of Linear Equations
Calculation of the Eigenvalues and Eigenvectors of Matrices
Additional Problems of Linear Algebra
Numerical Analysis of a Power Series and its Polynomials
Actions Over a Power Series
Actions with Polynomials
Calculation of Some Special Polynomials and Their Coefficients
Solution of Nonlinear Equations and Determination of Extremums
Interpolation and Approximation of Functions
Spline-Interpolation
Base Classes for the Approximation of Functions
Interpolation of the Nonlinear Functions of a Certain Class
Determination of the Boundaries for the Search of the Approximation Parameters for the Nonlinear Functions of a Certain Class
Restoration of Some Special Types of Functional Dependencies
Interpolation and Approximation of Functions by Means of Polynomials
Restoration of Periodic Dependence
Restoration of Dependence that is Linear with Respect to Parameters
Numerical Integration of Functions
Calculation of the Values of Some Functions
Calculation of the Values of Elementary Transcendental Functions and Gamma-Functions
Calculation of the Values of the Hypergeometric and Cylindrical
Functions of Real Variables by Elementary Methods
Numerical Solution of Differential Equations
Numerical Solution of Ordinary Differential Equations by the Runge–Kutta Method
Numerical Solving of One-Dimensional, Two-Dimensional, and Three-Dimensional Boundary Problems
Numerical Solving of One-Dimensional, Two-Dimensional, and Three-Dimensional Diffusion Equations and Wave Equations by Means of Explicit Schemes
Numerical Solving of One-Dimensional, Two-Dimensional and Three-Dimensional Diffusion Equations and Wave Equations by Means of Implicit Schemes
Numerical Methods Used in Geometry
Three-Dimensional Rotation Matrices
Description of Plane Curves by Splines
Numerical Methods of Probability Theory and Mathematical Statistics
Methods of Combinatorics
Functions of Statistical Distributions, Density Functions, and Quantiles
Statistical Estimates of Parameters
Testing of Statistical Hypotheses
Nonparametric Methods of Statistics
Some Additional Problems
Performance of Arithmetical and Algebraic Operations Over Integers, Rational and Complex Numbers
Search of Words and Phrases in a Dictionary
Used Designations
Objects of Mathematical Logic and General Algebra
Objects of Linear Algebra and Functional Analysis
Operations of Mathematical Analysis
Designations of Mathematical Functions and Constants
Geometrical Objects
Designations of Probability Theory and Mathematical Statistics
References
Index
About the Authors
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