National Repository of Grey Literature 39 records found  1 - 10nextend  jump to record: Search took 0.00 seconds. 
Numerical comparison of the CGLS and LSQR algorithms
Mrňák, Petr ; Tichý, Petr (advisor) ; Tůma, Miroslav (referee)
This bachelor thesis deals with the introduction of two mathemati- cally equivalent algorithms, CGLS and LSQR, which can be viewed as versions of the method of conjugate gradients applied to a system of normal equations. This thesis is devoted to their comparison both from a theoretical point of view (showing the relations between vectors and coefficients) and from a practical point of view (the behaviour of both algorithms in finite precision arithmetic). 1
Computation of roots of polynomials using comrade matrices
Novák, Martin ; Tichý, Petr (advisor) ; Papež, Jan (referee)
The bachelor thesis describes the relationship between the roots of the polynomial and the eigenvalues of the companion matrix, which is formed from the coefficients of the given polynomial. For numerical computing, it can be better to express the polynomial in basis of some orthogonal polynomials. After that, the coefficients can be used to form the comrade matrix. A similar relationship between roots of the polynomial and eigenvalues of the comrade matrix holds. We show that comrade matrices are non-derogatory matrices. The thesis contains numerical experiments programmed in MATLAB. 1
Numerical comparison of the CGLS and LSQR algorithms
Mrňák, Petr ; Tichý, Petr (advisor) ; Tůma, Miroslav (referee)
This bachelor thesis deals with the introduction of two algorithms, namely LSQR and CGLS, and then their comparison in the field of theory and the field of practi- cal application and computation. First, it is important to lay the foundations for these algorithms by using conjugate gradients and Lanczos tridiagonalisation. Both algorithms are theoretically equivalent, but in practice it is necessary to distinguish between them which is more appropriate for a given calculation. 1
Analysis of Krylov regularization methods for image deblurring problems
Machalová, Markéta ; Hnětynková, Iveta (advisor) ; Tichý, Petr (referee)
The diploma thesis deals with the construction and properties of image deblurring problems along with approaches to their solution. We focus on Krylov subspace methods LSQR, GMRES and RRGMRES, which are known for their regularization properties. We analyze the convergence behavior of the methods, the time efficiency and the quality of the approximate solution. Next, we present block Krylov subspace methods, which are not well explored in the field of image processing. These methods solve a system of linear equations with a multiple right-hand side and were created by the generalizing Krylov subspace methods, which are used for solving linear equations with a vector right-hand side. Finally, we perform numerical experiments investigating the influence of various factors on the results of image deblurring and the time complexity of individual methods, and we compare block and non-block methods. 1
Deflated Conjugate Gradient Method
Piskalla, Adam ; Papež, Jan (advisor) ; Tichý, Petr (referee)
Conjugate gradient method is one of the basic iterative methods for solving systems of linear algebraic equations with a symmetric positive definite matrix. We present two different derivations of the method and show some its properties. In situations where the method converges slowly or almost stagnates, techniques that transform the original system are usually used to speed up the convergence. Among them there is a precon- ditioning, for which we briefly present the basic idea and algorithm of preconditioned conjugate gradients. We then focus in more detail on the so-called deflation. We present the context in which it has been described in the literature, and comment on various approaches to the derivation of the deflated CG algorithm. We explain the principle of deflation and derive thoroughly the algorithm, describing steps that are not explic- itly stated or discussed in detail in the literature. On simple numerical experiments we illustrate the effect of the deflation on the convergence rate. 1
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
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
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
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.
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)

National Repository of Grey Literature : 39 records found   1 - 10nextend  jump to record:
See also: similar author names
13 TICHÝ, Petr
10 Tichý, Pavel
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