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Computations of Google's PageRank
Smejkalová, Barbora ; Tichý, Petr (advisor) ; Tůma, Miroslav (referee)
The thesis is concerned with numerical methods for solving the PageRank problem. The PageRank problem is formulated and mathematically described, based on intuitive observations called theses. We introduce and analyze two nu- merical methods for solving the resulting algebraic problems, namely the power method and the inner-outer method. The presented numerical experiments demonstrate and compare the behavior of the methods for various test matrices and input parameters. 1
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Algebraic view on the PCA method in selected applications
Hammerbauer, Tomáš ; Hnětynková, Iveta (advisor) ; Tichý, Petr (referee)
This thesis deals with describing algebraic and statistic view on Principal component analysis and the way of exporting important variables. Basic properties of the singular value decomposition are introduced and the best rank k aproximation of a matrix is de- rived. Then, a conection between PCA and singular value decomposition is described. At the end, PCA is ilustrated on two numerical experiments on image databases. It is shown, how we can aproximate images simillar to the elements of the database. Theo- retical foundations for the experiments are presented and then they are implemented in the Matlab enviroment. 1
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Incomplete Cholesky factorization
Hoang, Phuong Thao ; Tůma, Miroslav (advisor) ; Tichý, Petr (referee)
The thesis is about the incomplete Cholesky factorization and its va- riants, which are important for preconditioning a system with symmetric and positive definite matrix. Our main focus is on solving these systems, which arise in many technical applications and natural sciences, using preconditioned Con- jugate Gradients. Besides many other ways we can apply Cholesky factorization approximately, incompletely. In this thesis we study existence of the incomplete Cholesky factorization and we evaluate behaviour and potential of different vari- ants of the generic algorithm. 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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Optimization using derivative-free and metaheuristic methods
Henclová, Kateřina ; Tichý, Petr (advisor)
Evolutionary algorithms have proved to be useful for tackling many practical black-box optimization problems. In this thesis, we describe one of the most powerful evolutionary algorithms of today, CMA- ES, and apply it in novel way to solve the problem of tuning multiple coupled PID controllers in combustion engine models. Powered by TCPDF (www.tcpdf.org)
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Možnosti energetického využití odpadů z pálenic
Tichý, Petr
This thesis is focused on waste made by smal distilleries and utilization of this waste for energetic use. In thesis, there are described physical and chemical properties of each type of waste snd the possibility of their recycling or utilization as a raw material. Current situation and situation in ways of waste destruction were analyzed in a case study made with several chosen small distilleries. Based on this study, distillery waste is used as a raw material for energetic in a limited way. Next chapter describes technology used for separation of waste fraction and technology for re-using its energetic potential. Separate chapter consist from case study which deals with utilization of waste in a one concrete small distillery for energetic use.
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Numerical range of an interval matrix
Ivičič, Michal ; Hladík, Milan (advisor) ; Tichý, Petr (referee)
The numerical range of a matrix is a set of complex numbers that contains all the eigen- values of the matrix. It is used for instance to estimate a matrix norm. This thesis is about the numerical range of an interval matrix. In the theoretical part, we examine its properties. We prove for example that it is NP-hard to find out whether a given point lies in the numerical range. On an example, we show that field of values of an interval matrix is not necessarily convex. The thesis contains descriptions of two algorithms for visualization of the convex hull of the numerical range. Both of them are only suitable for matrices of small sizes due to high time complexity. Therefore we also present a polyno- mial algorithm for computing the upper bound of the numerical range. In the practical part, we implement the algorithms as functions in the Matlab language. 1
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Least-squares problems with sparse-dense matrices
Riegerová, Ilona ; Tůma, Miroslav (advisor) ; Tichý, Petr (referee)
Problém nejmenších čtverc· (dále jen LS problém) je aproximační úloha řešení soustav lineárních algebraických rovnic, které jsou z nějakého d·vodu za- tíženy chybami. Existence a jednoznačnost řešení a metody řešení jsou známé pro r·zné typy matic, kterými tyto soustavy reprezentujeme. Typicky jsou ma- tice řídké a obrovských dimenzí, ale velmi často dostáváme z praxe i úlohy s maticemi o proměnlivé hustotě nenulových prvk·. Těmi se myslí řídké matice s jedním nebo více hustými řádky. Zde rozebíráme metody řešení tohoto LS pro- blému. Obvykle jsou založeny na rozdělení úlohy na hustou a řídkou část, které řeší odděleně. Tak pro řídkou část m·že přestat platit předpoklad plné sloupcové hodnosti, který je potřebný pro většinu metod. Proto se zde speciálně zabýváme postupy, které tento problém řeší. 1
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Global krylov methods for solving linear algebraic problems with matrix observations
Rapavý, Martin ; Hnětynková, Iveta (advisor) ; Tichý, Petr (referee)
In this thesis we study methods for solving systems of linear algebraic equati- ons with multiple right hand sides. Specifically we focus on block Krylov subspace methods and global Krylov subspace methods, which can be derived by various approaches to generalization of methods GMRES and LSQR for solving systems of linear equations with single right hand side. We describe the difference in construction of orthonormal basis in block methods and F-orthonormal basis in global methods, in detail. Finally, we provide numerical experiments for the deri- ved algorithms in MATLAB enviroment. On carefully selected test problems we compare convergence properties of the methods. 1
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