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The choice of the step in trust region methods
Rapavý, Martin ; Tichý, Petr (advisor) ; Kučera, Václav (referee)
The main goal of this thesis is the choice of steps in trust region methods for finding a minimum of a given function. The step corresponds to the problem of finding a minimum of a model function on a trust region. We characterize a solu- tion of this problem (Moré-Sorensen theorem) and consider various techniques for approximating a solution of this problem (the Cauchy point method, the dogleg method, the conjugate gradients method). In the case of the first two techniques we prove convergence of the optimization method. Finally, the above techniques are tested numerically in MATLAB on properly chosen functions and initial data. We comment on advantages and disadvantages of considered algorithms. 1
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Field of values of a matrix: Theory and computation
Vacek, Lukáš ; Tichý, Petr (advisor) ; Tůma, Miroslav (referee)
The field of values of a matrix A is a convex set in the complex plane assigned to A. It is important in matrix analysis, especially in invetigation of properties of nonnormal matrices and matrix polynomials, in study of the con- vergence of iterative methods applied to these matrices, in the estimation of ma- trix function norms, etc. This thesis summarizes theory about the field of values of a matrix, formulates open problems and explaines the main idea of the basic numerical method for its computation. In numerical experiments the standart algorithmic realization of method is compared with alternative approaches that use power method, Lanczos algorithm and Chebfun.
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Numerical computation with functions using Chebfun
Lébl, Matěj ; Tichý, Petr (advisor) ; Hnětynková, Iveta (referee)
Goal of this work is to introduce Chebfun software and show ideas behind it. In the first chapter we summarize the theory of polynomial interpolation with focus on the Chebyshev interpolants. In the second chapter we introduce Chebfun software, its basic commands and principles of constructing interpolants. The third chapter is devoted to demonstrate theorems from the first chapter and to show practical applications of Chebfun when finding roots of a function and solving differential equations. Powered by TCPDF (www.tcpdf.org)
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Methods for the solution of nonlinear equations
Havelková, Eva ; Kučera, Václav (advisor) ; Tichý, Petr (referee)
The aim of this bachelor thesis is to present an overview of elementary numerical methods for solving nonlinear algebraic equations in one variable. Firstly, related concepts from numerical mathematics and mathematical analysis are explained. The main part of the thesis provides a detailed description of chosen iterative methods as well as the proofs of their orders of convergence. The methods covered are namely the bisection method, fixed-point iteration, regula falsi method, Newton's method, secant method and methods that are based on quadratic interpolation. The practical part of the thesis presents results of numerical experiments that were carried out with Matlab software on various types of nonlinear equations. These results are compared with the theory introduced in the preceding parts. The contribution of this thesis is to provide a comprehensive overview and comparison of the characteristics of basic methods for solving nonlinear equations based on a variety of literature. Powered by TCPDF (www.tcpdf.org)
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Multilevel methods and adaptivity
Vacek, Petr ; Strakoš, Zdeněk (advisor) ; Tichý, Petr (referee)
After introduction of the model problem we derive its weak formulation, show the existence and the uniqueness of the solution, and present the Galerkin finite element method. Then we briefly describe some of the stationary iterative methods and their smoothing property. We present the most common multigrid schemes, i.e. two-grid correction scheme, V-cycle scheme, and the full multigrid algorithm. Then we perform numerical experiment showing the differences between the use of the direct and iterative coarsest grid solver in V-cycle scheme and experiment considering a perturbation of the correction vector simulating a fault of a computational device. Powered by TCPDF (www.tcpdf.org)
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Double Bass As A Solo Instrument In Jazz
Tichý, Petr ; HONZÁK, Jaromír (advisor) ; Novotný, Marek (referee)
This thesis maps the historical development of double-bass playing, and the expression techniques employed therein, from accompaniment in early jazz bands to solo performances. Among others, the thesis describes the circumstances of and the reasons behind the creation of the first double-bass solo album, which was recorded by Barre Phillips in December 1968.
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Support the publication of open data in municipalities
Tichý, Petr ; Chlapek, Dušan (advisor) ; Kučera, Jan (referee)
This thesis deals with so-called open data. Specifically, it aims to support the publication of open data in municipalities. The work has three defined objectives - to analyze the issue of open and linked data, to analyze and select data suitable for opening, and to create an application to facilitate the opening and visualization of data. Each goal provides some partial benefit, but the main and the biggest benefit is the verification of standardized process that allow for opening and publication of data in a small village. The structure of the work is partly based on defined goals. It consists of an introduction, theoretical part, practical part and conclusion. The introductory part deals with basic introduction to the work, defining goals and present searches resources. In the theoretical part basic concepts of open and linked data are defined, then presents the benefits and risks associated with the opening of data are presented. Then, a standardized process of opening and publication of data is introduced as well as the evaluation of the current situation of open data in the Czech Republic and in other countries. The practical part sets objectives subdivided to the described process of opening and publication of data in the selected municipality and describes the process of designing and creating applications using open data from the previous process. The conclusion summarizes the whole work, and discusses whether and how the objectives are reached.
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