Download Examination Questions “Numerical Methods of Algebra and Analysis

Examination Questions
“Numerical Methods of Algebra and Analysis ”
a second-year Bachelor, second half-year, credit 3
1. Matrices. Matrix Theory. Vector and Matrix Norms. Degree of matrix.
2. Matrix Norms and Absolute Values (Modulus).
3. Coordination Norm.
4. Matrix Condition Number.
5. Eigenvalue of Matrix.
6. Matrix Limits.
7. Matrix Series. Necessary condition of matrix series convergence.
8. Sufficient condition of matrix series convergence.
9. Degree Matrix Series. Remark. Example.
10. Degree Matrix Series. Adjunction.
11. Hamilton-Kelli equality.
12. Numerical Methods of Linear Algebraic Equations Systems (NMLAES). Problem 1.
Theorem 1.
13. NMLAES. Direct Methods. Gauss method.
14. NMLAES. Direct Methods. Square Roots Method.
15. NMLAES. Direct Methods. Cholesky Factorization. Remark 2.
16. NMLAES. Iterative methods: Jacobi method. Teorem1.
17. NMLAES. Iterative methods: Gauss-Seidel method.
18. NMLAES. Iterative methods: General schemes.
19. NMLAES. Iterative methods: The Sufficient Condition of convergence of the
Iterative Process. Theorem 2. Corollary 1, 2.
20. NMLAES. Iterative methods. The Estimation of the Error Vector of Iteration Process
21. NMLAES. Iterative methods. The Necessary and Sufficient Conditions of
convergence of Iterative Process of the LAES. Remark1. Corollary 3.
22. NMLAES. General Iterative Methods LAES. The principles of construction of the
Iterative Methods.
23. NMLAES. General Iterative Methods LAES. Stationary iteration process.
Progressive approximation. Step-by-step approximation. Cyclic iteration process.
Relaxation principle.
24. The property of symmetric matrices. Error function. Theorem 1, 2, 3, 4, 5, 6.
25. NMLAES. Optimization Method. Minimal Residual Method and Convergence.
26. NMLAES. Optimization Method. Method of Steepest Descent and Convergence.
Theorem 1.
27. General Iterative Methods of the Solutions Algebraic and Transcendental Equations.
Isolation of the Roots of Algebraic and Transcendental Equations (Root Finding).
Bisection method. Graphical method. Theorem 1, 1’, 2. Example 1, 2.
28. General Iterative Methods of the Solutions Algebraic and Transcendental Equations.
Idea of methods.
29. General Iterative Methods of the Solutions Algebraic and Transcendental Equations.
Contraction – Mapping Principle. Theorem 3 (Contraction – Mapping Principle).
30. General Iterative Methods of the Solutions Algebraic and Transcendental Equations.
Direct method. The Theorem 1 of the Contraction – Mapping Principle.
31. General Iterative Methods of the Solutions Algebraic and Transcendental Equations.
Iteration order.
1
32. Intersecting (Chord) Methods. Rate of convergence.
33. Newton (Tangent) Methods. Rate of convergence.
34. Mixed methods. Rate of convergence.
35. False Point Methods. Rate of convergence.
36. Stephenson Methods. Rate of convergence.
37. Wall Methods. Rate of convergence.
38. Newton – Kantorovich Methods. Theorem 1, 2.
39. Newton – Kantorovich Methods for operational equations.
40. Newton – Kantorovich Modifications Methods.
41. Matrix Eigenvalue Problems.
42. Characteristic polynomial. Theorem 1. Exercise 2. Fundamental Theorem of Algebra.
Problem 1 (Eigenvalues and Eigenvectors).
43. Spectral radius. Gelfand’s formula. Power Iterations.
44. Largest eigenvalue. Rayleigh quotient.
45. Smallest eigenvalue. Rayleigh quotient iterations.
46. Fast iteration process.
47. Function Interpolations. Vandermonde Interpolation. Theorem 1 (Taylor Series).
48. Theorem 2 (Weierstrass Approximation.)
49. Problem 1 (Polynomial Interpolation).
50. Linear interpolation. Theorem 3. Lagrange Interpolation.
51. Newton Interpolation. Divided differences. Theorem 4. Newton forward formula.
Newton backward formula.
52. Gauss interpolation formula. Sterling and Bessel Interpolations.
53. Errors of Polynomial Interpolation. Theorem 5. Exercise. Problem 2 (Polynomial
Approximation).
54. Integral Calculus. Integrals and Finite Sums. Newton-Cotes Integration Rules.
55. Trapezium formula. Simpson formula. Estimate of errors.
56. Gaussian Quadrature Rules. Estimate of errors.
57. Random Variable. Discrete Random Variable.
58. Probability Density and Distribution Functions (PDF). Expectation (Mean).
59. Variance and Transforms. Random Variables Modelling. Neumann Modelling.
Chebyshev Inequality.
60. Estimation of the Integral by Monte Carlo Methods. Essential Random Sample
Methods.
Lecturer, Doctor of Science, Professor
Kanat Shakenov
April 12, 2013
2