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Modern problems of computational mathematics

Обов’язкова дисципліна для ОП

Second

2023/2024

3 Semester

8

PLO1. Be able to use of in-depth professional knowledge and practical skills to optimize the design of models of any complexity, to solve specific problems of designing intelligent information systems of different physical nature. PLO10. Be able to build models of physical and production processes, design of storage and data space, knowledge base, using charting techniques and standards for information systems development.

To successfully learn the discipline “Modern problems of computational mathematics” the student should satisfy the following requirements. They have successfully passed the courses Calculus and Linear Algebra. They know (a) fundamentals of methods for solving systems of linear algebraic equations. They can (a) apply fundamentals of methods for solving systems of linear algebraic equations to solve practical problems. They should be able to (a) seek information in the Internet.

Block 1. Fundamentals of iterative methods Direct and iterative methods Types of convergence of iterative methods Basic iterative methods Jacobi and Gauss–Seidel methods Successive over-relaxation method Symmetric Successive over-relaxation method Control work Модуль 2. Acceleration procedures Polynomial acceleration Optimal Chebyshev acceleration Rate of convergence of optimal Chebyshev acceleration Chebyshev acceleration with estimations of eigenvalues Adaptive Chebyshev acceleration using special norms Computation of new parameters in adaptive Chebyshev acceleration using special norms Steepest descent method Arbitrary directions of descent Conjugate gradients method Procedures ORTHOMIN, ORTODIR, ORTHORES Versions of conjugate gradient method Technology for sparse matrices Applications of applied iterative methods Control work

1. Heygeman L., Yang D. Prikladnyie iteratsionnyie metodyi. — M.: Mir, 1986. 2. Golub Dzh., Van Loun. Matrichnyie vyichisleniya. — M.: Mir, 1999. 3. Ortega Dzh. Vvedenie v parallelnyie i vektornyie metodyi resheniya sistem. — M.: Mir, 1991. 4. Saad Yu. Iteratsionnyie metodyi dlya razrezhennyih lineynyih sistem. V 2-h tomah — M.: Izdat. Moskov. universiteta, 2013. 5. Kelley C.T. Iterative Methods for Linear and Nonlinear Equations. In: Frontiers in Applied Mathematics —. SIAM, Philadelphia, N 16, 1995. 6. Kelley C.T. Iterative Methods for Optimization. In: Frontiers in Applied Mathematics —. SIAM, Philadelphia, N 18, 1999. 7. Nurminskiy E. A. Chislennyie metodyi resheniya determinirovannyih i stohasticheskih minimaksnyih zadach. Kiev: Nauk. dumka, 1979. 8. Lyashko S. I. Generalized optimal control of linear systems with distributed parameters. Boston/Dordrecht/London: Kluwer Academic Publishers, 2002. 466 p. .

Lectures, independent work, recommended literature processing, homework.

Intermediate assessment: The maximal number of available points is 60. Test work no. 1: RN 1.1, RN 1.2 – 30/18 points. Test work no. 2: RN 1.1, RN 1.2 – 30/18 points. Final assessment (in the form of final test): The maximal number of available points is 40. The results of study to be assessed are RN 1.1, RN 1.2, RN 2.1, and RN 3.1. The form of final test: writing. The types of assignments are 3 writing assignments (2 theoretical and 1 practical).

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