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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 1 Deterministic 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 1 Deterministic Mathematics
New York: Nova Publishers, 2015. — 377 p.
Contents:
Introduction
List of Figures
List of Tables
Numerical Methods of Linear Algebra
General Properties of Linear Equations
Solving Systems of Linear Equations Using the Cramer and Gaussian Methods
Gaussian Algorithms
Solution of Linear Equations Containing Tridiagonal Matrixes
Iterative Methods of Solution of Linear Equations
Pseudoinverse Matrixes
Eigenvalues and Eigenvectors of Linear Operators
Characteristic Polynomials of Matrixes
Numerical Methods of Determination of Eigenvalues and Eigenvectors of Matrixes
Iterative Methods
Rotation Method
Clebsch–Gordan Coefficients
Angular Momentum Operator
Addition of Angular Momentum Operators
Properties of Clebsch–Gordan Coefficients
Numerical Analysis of Power Series and Polynomials
Actions with Power Series
Some Properties of Polynomials and their Zeros
Some Properties of Polynomials
Zeros of Polynomials
Division of Polynomials
Expansion of Fractional Rational Functions into Partial Fractions
Polynomials with Real Coefficients
Elementary Properties of Polynomials with Real Coefficients
Properties of Zeros of Polynomials Influencing on Stability of Dynamical Systems
Boundaries of Real Zeros of Polynomials with Real Coefficients
The Number Real Zeros of Polynomials with Real Coefficients
Algorithm of Determination of Real Zeros of Polynomials with Real Coefficients
Restoration of Polynomial by its Zeros
Expressions for the Polynomial and its Coefficients
Properties of Elementary Symmetric Functions
Restoration of Polynomial by its Values in Given Points
Expressions for the Polynomial and its Coefficients and Some Properties of the Auxiliary Functions
Main Properties of the Functions λ( jk m)()
Determination of Zeros of Polynomials by Means of Explicit Expressions
Approximate Solution of Algebraic Equations by Gr¨affe–Lobatchevsky Method
Calculation of Some Special Polynomials and Their Coefficients
Binomial Coefficients
Polynomials of Type (ξ + z)n and Analogous Polynomials of Several Variables
Polynomials of Type ξ ± zn
Pochhammer Symbol
Main Properties of Stirling Numbers
Bernoulli Polynomials and Euler Polynomials
Main Properties of Bernoulli Numbers and Euler Numbers
Calculation of Values of Classical Orthogonal Polynomials
General Properties of Orthogonal Polynomials
Jacobi Polynomials
Laguerre Polynomials
Hermite Polynomials
Legendre Polynomials
Tchebyshev Polynomials
Some Functions Connected with Orthogonal Polynomials
Algorithms
Sums Containing Polynomials and Fractional Rational Functions
Solution of Nonlinear Equations and Determination of Extremums
Auxiliary Theorems for Numerical Solution of Equations
Numerical equations
General Properties of Numerical Equations
Numerical Solution of Equations Containing Real Variables
Numerical Solution of Equations Containing Complex Variables
Numerical Solution of Systems of Equations
Maximums and minimums
Conditions of Existence of Maximums and Minimums for Functions of One Real Variable
Additional Quadrature Formulas
Calculation of Values of Some Functions
Main Transcendental Mathematical Constants
Solution of Transcendental Equations of Special Types
Equations Containing Linear-Exponential or Geometric Exponential Dependence
Equations Containing Product of Geometrical Dependences
Equations Containing Sum of Exponents
Calculation of Values of Gamma-Function and Connected with it Functions
Main Properties of Considered Functions
Representation of the Functions in the Forms of Convergent Series and Integrals
Asymptotic Expansions
Riemann Zeta Function and Functions Connected with it
Calculation of Values of the Functions
Hypergeometric Functions
Elementary Properties of Hypergeometric Function
Differential Equations
Power Series
Functional Equations and Limits
Functional Equations for Hypergeometric Functions Satisfying
Second-Order Differential Equations
Differentiation and Integration Formulas
Integral Representations
Inequalities for Hypergeometric Functions
Cylindrical Function
Use of Hypergeometric Functions for Solving the Linear Differential Equations
Differential Equations of any Order N >
Second-Order Differential Equations
Reduction Formulas for Hypergeometric Function
Reduction Formulas Generally
Reduction Formulas for the Function F(γ, z)
Reduction Formulas for the Function F(α, α, z)
Reduction Formulas for the Kummer Hypergeometric Function
Reduction Formulas for the Gauss Hypergeometric Function
Asymtotic Expansion of Hypergeometric Functions in Terms of Parameters
Asymptotic Expansions
Main Properties of Functions hr(z)
Main Properties of Functions χr(λ, z)
Main Properties of Functions Ujk(λ), Vjk(λ), Wjk(λ) and Υjk()
Asymptotic Expansion of Hypergeometric Functions Satisfying the
Second-Order Differential Equations
Control Examples
Final Remarks
Elementary Methods of Calculation of Values of
Hypergeometric and Cylindrical Functions
Calculation of Values of Hypergeometric Functions by Means of Power
Series and Recurrence Relations
Description of the Method
Determination of Parameter ξ
Final Remarks
Numerical Methods for Solving Differential Equations
Numerical Solution of Ordinary Differential Equations by Runge–Kutta
Method
Numerical Solving of Ordinary Differential Equations by Multistep Difference Methods
One-dimensional Boundary Problems
Boundary Problems of General Form
Multidimensional Boundary Problems of Special Type
Diffusion Equation
Explicit Scheme
Classical Difference Scheme in the General Form
Method of Decomposition of the Operator
Wave Equation
Estimation of Derivatives of Unknown Function
Methods of Approximating Functions for the
Numerical Solution of Differential Equations
Numerical Methods Used in Geometry
Three- imensional Rotation Matrixes
Description of Plane Curves by Splines
Curvilinear Coordinates Connected with the Plane Curve
Using Spline–Interpolation for Representation of a Curve
References
About the Authors
Index
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