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MATHEMATICS RESEARCH DEVELOPMENTS COMPUTING ALGORITHMS FOR SOLUTIONS OF PROBLEMS IN APPLIED MATHEMATICS AND THEIR STANDARD PROGRAM REALIZATION PART 1 DETERMINISTIC MATHEMATICS K. J. KACHIASHVILI, D. YU. MELIKDZHANIAN AND A. I. PRANGISHVILI New York Copyright © 2015 by Nova Science Publishers, Inc. ISBN: (eBook) Contents List of Figures xi xiii xv List of Tables Introduction 1 Numerical Methods of Linear Algebra 1.1 General Properties of Linear Equations . . . . . . . . . 1.2 Solving Systems of Linear Equations Using the Cramer and Gaussian Methods . . . . . . . . . . . . . . . . . 1.3 Gaussian Algorithms . . . . . . . . . . . . . . . . . . 1.4 Solution of Linear Equations Containing Tridiagonal Matrixes . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Iterative Methods of Solution of Linear Equations . . . 1.6 Pseudoinverse Matrixes . . . . . . . . . . . . . . . . . 1.7 Eigenvalues and Eigenvectors of Linear Operators . . . 1.8 Characteristic Polynomials of Matrixes . . . . . . . . . 1.9 Numerical Methods of Determination of Eigenvalues and Eigenvectors of Matrixes . . . . . . . . . . . . . . 1.9.1 Iterative Methods . . . . . . . . . . . . . . . . 1.9.2 Rotation Method . . . . . . . . . . . . . . . . 1.10 Clebsch–Gordan Coefficients . . . . . . . . . . . . . . 1.10.1 Angular Momentum Operator . . . . . . . . . 1.10.2 Addition of Angular Momentum Operators . . 1.10.3 Properties of Clebsch–Gordan Coefficients . . . . . . . . . . . . . 1 1 . . . . . . . . . . . . . . . . . . . . . . 3 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 9 12 15 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 24 29 31 31 34 35 . . . . . . . . 43 43 52 52 53 55 60 64 64 . . . . . . 65 66 67 2 Numerical Analysis of Power Series and Polynomials 2.1 Actions with Power Series . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Some Properties of Polynomials and their Zeros . . . . . . . . . . . . . 2.2.1 Some Properties of Polynomials . . . . . . . . . . . . . . . . . 2.2.2 Zeros of Polynomials . . . . . . . . . . . . . . . . . . . . . . . 2.3 Division of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Expansion of Fractional Rational Functions into Partial Fractions . . . . 2.5 Polynomials with Real Coefficients . . . . . . . . . . . . . . . . . . . . 2.5.1 Elementary Properties of Polynomials with Real Coefficients . . 2.5.2 Properties of Zeros of Polynomials Influencing on Stability of Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Boundaries of Real Zeros of Polynomials with Real Coefficients 2.5.4 The Number Real Zeros of Polynomials with Real Coefficients . . . . . . . . . 2.5.5 Algorithm of Determination of Real Zeros of Polynomials with Real Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Restoration of Polynomial by its Zeros . . . . . . . . . . . . . . . . . . . . 2.6.1 Expressions for the Polynomial and its Coefficients . . . . . . . . . 2.6.2 Properties of Elementary Symmetric Functions . . . . . . . . . . . 2.7 Restoration of Polynomial by its Values in Given Points . . . . . . . . . . . 2.7.1 Expressions for the Polynomial and its Coefficients and Some Properties of the Auxiliary Functions . . . . . . . . . . . . . . . . . . . (m) 2.7.2 Main Properties of the Functions λjk (...). . . . . . . . . . . . . . 2.8 Determ