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Banded matrix solvers and polynomial Diophantine equations
Hromčík, Martin ; Hurák, Zdeněk ; Frízel, R. ; Šebek, M.
Numerical procedures and codes for linear Diophantine polynomial equations are proposed in this paper based on the banded matrix algorithms and solvers. Both the scalar and matrix cases are covered. The algorithms and programs developed are based on the observation that a set of constant linear equations resulting from the polynomial problem features a special structure. This structure, known as Sylvester, or block Syelvester in the matrix case, can in turn be accommodated in the banded matrix framework.
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Výpočet determinantu rozměrné matice Sylvesterova typu
Kujan, Petr ; Hromčík, M. ; Šebek, Michael
Tato práce je věnována různým metodám výpočtu determinantu velké n-D polynomiální matice se speciální strukturou. Aplikace řešení této úlohy se vyskytují v teorii n-D systémů (např. test soudělnosti dvou n-D polynomů) nebo v teorii algebraických rovnic. Konkrétně se řešení této úlohy věnuje profesor Chiasson v praktické úloze výpočtu optimálních spínacích úhlů vícehladinového konvertoru. Zde se pomocí výpočtu determinantů speciálních matic řeší soustava polynomiálních rovnic.
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Numerical algorithms for polynomial matrices
Hromčík, Martin ; Šebek, Michael
This report is devoted to new numerical methods for computations with polynomials and polynomial matrices that are encountered when solving the problems of control systems design via the algebraic methods. A distinguishing feature ofour approach is the extensive employment of the discrete Fourier transform tech-niques, namely of the famous Fast Fourier Transform routine and its relation to polynomial interpolation and Z-transform.
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Polynomial Toolbox
Hromčík, M. ; Šebek, M. ; Ježek, Jan
Quite recently the polynomial design methods found a new great field of application outside the control area: the algebraic approach have been used successfully in signal processing and mobile communications. In contrast to the control systems synthesis, polynomials and polynomial matrices with complex coefficients are often required when designing filters, equalizers, decouplers and other components of mobile phones for instance.
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New algorithm for spectral factorization and its application in signal processing
Ježek, Jan ; Hromčík, Martin ; Šebek, Michael
In this report a new algorithm is presented for the spectral factorization of a two-sided symmetric polynomial. The method is based on the discrete Fourier transform theory (DFT) and its relationship to the Z-transform. Involving DFT computational techniques, namely the famous fast Fourier transform routine (FFT), brings high computational efficiency and reliability. The power of the proposed procedure is employed in a particular practical signal processing application.
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Numerical algorithms for polynomial matrices
Hromčík, Martin ; Šebek, Michael
This report is devoted to new numerical methods for computations with polynomials and polynomial matrices that are encountered when solving the problems of control systems design via the algebraic methods. A distinguishing feature of our approach is the extensive employment of the discrete Fourier transform techniques, namely of the famous Fast Fourier Transform routine and its relation topolynomial interpolation and Z-transform.
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Polynomial toolbox and control education
Hromčík, Martin ; Šebek, Michael
In this report we give our experience with employing thePolynomial Toolbox for Matlab version 2 at the lectures and labs of Algebraic Design Methods course taught at the Czech Technical University in Prague. Practical examples illustrating the Toolbox performance and contribution are also included. Though we have been using the tool for teaching purposes for arelatively short time, our experience is positive and we find the Polynomial Toolbox a very useful software tool for teachers.
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Fast Fourier Transform and robustness analysis with respect to parametric uncertainties
Hromčík, Martin ; Šebek, M.
In this paper two new numerical algorithms based on the Fast Fourier Transform techniques (FFT) are used to solve the structured robustness analysis problem in the case of one parameter entering polynomially. Both scalar and matrix cases are considered. The employed algorithms are namely numerical routines for the computation of one- and two-dimensional polynomial matrix determinants, based on the one- and two-dimensional FFT's respectively.
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Polynomial toolbox 2.5 and systems with parametric uncertainties
Šebek, Michael ; Hromčík, Martin ; Ježek, Jan
Polynomial Toolbox is a new MATLAB toolbox for systems, signals and control analysis and design using polynomial methods. See www.polyx.cz for more details. This paper demonstrates on several examples how the Polynomial Toolbox solves various robust control problems for plants with parametric uncertainties. Both classical problems such as one-parameter, interval or polytopic uncertainty are discussed as well as cases with complicated uncertainty structure.
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