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National Repository of Grey Literature 2 records found  Search took 0.00 seconds. 
The Lanczos method in finite precision arithmetic
 Šimonová, Dorota ; Tichý, Petr (advisor) ; Hnětynková, Iveta (referee)
In this thesis we consider the Lanczos algoritm and its behaviour in finite precision. Having summarized theoretical properties of the algorithm and its connection to orthogonal polynomials, we recall the idea of the Lanczos method for approximating the matrix eigenvalues. As the behaviour of the algorithm is strongly influenced by finite precision arithmetic, the linear independence of the Lanczos vectors is usually lost after a few iterations. We use the most im- portant results from analysis of the finite precision Lanczos algorithm according to Paige, Greenbaum, Strakos and others. Based on that, we study formulation and properties of the mathematical model of finite presicion Lanczos computati- ons suggested by Greenbaum. We carry out numerical experiments in Matlab, which support the theoretical results.
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Special eigenvalue problems for symmetric sparse matrices related to electronic structure calculations
 Novák, Matyáš ; Tůma, Miroslav (advisor) ; Plešek, Jiří (referee)
Ab-initio methods for calculating electronic structure represent an important field of material physics. The aim of this theses - within the project focused on developing the new method for calculating electronic states in non-periodic structures based on density functional theory, pseudopotentials, and finite elements methods - is to convert Kohn-Sham equations into the form suitable for discretisation, to suggest apropriate method for solving generalized eigenproblem resulting from this discretisation and to implement an eigenvalue solver (or to modify existing one). The thesis describes a procedure for converting the many-particle Schrödinger equation into generalized rank-k-update eigenvalue problem and discusses various methods for its solution. Eigensolver Blzpack, which makes use of the block Lanczos method, has been modified, integrated into the Sfepy framework (a tool for the finite element method calculation) and resulting code has been successfully tested.
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