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Teorie a aplikace krylovovských metod v souvislostech
Gergelits, Tomáš ; Strakoš, Zdeněk (advisor) ; Hnětynková, Iveta (referee)
Title: Krylov subspace methods: Theory, applications and interconnections Author: Tomáš Gergelits Department: Department of Numerical Mathematics Supervisor: prof. Ing. Zdeněk Strakoš, DrSc. Abstract: After recalling of properties of Chebyshev polynomials and of sta- tionary iterative methods, this thesis is focused on the description of Conjugate Gradient Method (CG), the Krylov method of the choice for symmetric positive definite matrices. Fundamental difference between stationary iterative methods and Krylov subspace methods is emphasized. CG is derived using the minimiza- tion of the quadratic functional and the relationship with several other fields of mathematics (Lanczos method, orthogonal polynomials, quadratic rules, moment problem) is pointed out. Effects of finite precision arithmetic are emphasized. In compliance with the theoretical part, the numerical experiments examine a bound derived assuming exact arithmetic which is often presented in literature. It is shown that this bound inevitably fails in practical computations. The thesis is concluded with description of two open problems which can motivate further research. Keywords: Krylov subspace methods, convergence behaviour, numerical stabil- ity, spectral information, convergence rate bounds
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