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Solving bordered linear systems
Štrausová, Jitka ; Janovský, Vladimír (advisor) ; Zítko, Jan (referee)
The comparison of two algorithms for solving bordered linear systems is considered. The matrix of this system consists of four blocks (matrices A,B,C,D), the upper left one is a sparse matrix A, which is ill-conditioned and structured. The other blocks (B,C,D) are dense. We say that the matrix A is bordered with the matrices B,C,D. It is desirable to preserve the block structure of the matrix and take advantage of sparsity and structure of the matrix A. The literature suggests to use two different algorithms: The first one is the method BEM for matrices with the borders of width equal to one. The recursive alternative for matrices with wider borders is called BEMW. The second algorithm is an iterative method. Both techniques are based on different variants of the block LU-decomposition.
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Aplication of Peano and Sard kernel for error term of quadrature and cubature formula
Valešová, Petra ; Zítko, Jan (referee) ; Kofroň, Josef (advisor)
In the present work we study the expressing of errors of quadrature and cubature formulae by Peano and Sard kernel. Firstly the Peano kernel of a quadrature formula and its generalization are defined, either are shown on the examples. Further the Peano kernel and its generalization are used for finding the optimal quadrature formula of Nikolskij's type. Furthermore there are the Sard kernels of a cubature formula for the square and the cube defined. There are in detail described Romberg's cubature formula and respective Sard kernels for both case. Further the Sard kernels of the trapezoidal rule and Romberg's cubature formula are used for estimation the error of these cubature formulae.
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Conditions for convergence of the restarted and augmented GMRES method
Nádhera, David ; Strakoš, Zdeněk (referee) ; Zítko, Jan (advisor)
The GMRES method is one of the most useful methods for solving a system of linear algebraic equations with nonsymmetric matrix. So on, many bounds for the residual norm have been derived, that can give us information about the convergence or possible stagnation of the method. A generalization of the GMRES method is the augmented GMRES method. In this paper we will analyze the implementation of augmented GMRES method proposed by Morgan. In these consequences we will be interested in how precise harmonic Ritz vectors approximate the eigenvectors belonging to the smallest in magnitude eigenvalues. We generalize some previous results concerning the convergence of restarted GMRES method for the case of augmented GMRES method. This is the rst contribution of the work. Another main point will be numerical testing and comparing of the bounds for restarted and augmented GMRES and an attempt to state a criterion, when it is suitable to stop the improvement of augmenting vectors, i. e. apply the augmented GMRES method without additional computations.
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