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Transformace Sylvestrovy matice a výpočet největšího společného dělitele dvou polynomů
Eckstein, Jiří ; Zítko, Jan (advisor) ; Tůma, Miroslav (referee)
In this thesis we study the computation of the greatest common divisor of two polynomials. Firstly, properties of Sylvester matrices are considered as well as their role in computation. We then note, that this approach can be naturally generalized for several polynomials. In the penultimate section, Bézout matrices are studied as an analogy to the Sylvester ones, providing necessary comparison. Extension for more than polynomials is presented here as well. Algorithms corresponding to the individual approaches are presented as well. Finally, the algorithms are implemented in MATLAB and are compared in numerical experiments. Powered by TCPDF (www.tcpdf.org)
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Transformace Sylvestrovy matice a výpočet největšího společného dělitele dvou polynomů
Eckstein, Jiří ; Zítko, Jan (advisor) ; Tůma, Miroslav (referee)
In this thesis we study the computation of the greatest common divisor of two polynomials. Firstly, properties of Sylvester matrices are considered as well as their role in computation. We then note, that this approach can be naturally generalized for several polynomials. In the penultimate section, Bézout matrices are studied as an analogy to the Sylvester ones, providing necessary comparison. Extension for more than polynomials is presented here as well. Algorithms corresponding to the individual approaches are presented as well. Finally, the algorithms are implemented in MATLAB and are compared in numerical experiments. Powered by TCPDF (www.tcpdf.org)
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Theory of Numbers in Ancient Greece
Smrčka, Zdeněk ; Bečvář, Jindřich (advisor) ; Halas, Zdeněk (referee)
Title: Theory of Numbers in Ancient Greece Author: Bc. Zdenek Smrcka Department: The Department of Mathematics Education Supervisor: doc. RNDr. Jindřich Bečvář, CSc. Abstract: The goal of this thesis is to write up clearly and comprehensibly numeric theoretical research and its results in Ancient Greece between 6 century before Christ and 4 century after Christ. In this thesis we try show examples use of Greece's Mathematics for improvement teaching in education and better understanding abstract thinking in Mathematics. We want so that students understand thinking and abilities Greece's mathematicians. We compare high school view on searching greatest common divisor and Euclidean algorithm. We present important Greece's knowledges as sieve of Eratosthenes, arithmetic of Diofantos etc.. Something of Greece's knowledges as Euclidean algorithm, sieve of Eratosthenes etc. are use of up to now. Keywords: Mathematics in Ancient Greece, figurate number, theory of numbers, Continual fraction, Euclidean algorithm
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Analýza výpočtu největšího společného dělitele polynomů
Kuřátko, Jan ; Zítko, Jan (advisor) ; Janovský, Vladimír (referee)
In this work, the analysis of the computation of the greatest common divisor of univariate and bivariate polynomials is presented. The whole process is split into three stages. In the first stage, data preprocessing is explained and the resulting better numerical behavior is demonstrated. Next stage is concerned with the problem of the computation of the numerical rank of the Sylvester matrix, from which the degree of the greatest common divisor is obtained. The last stage is the actual algorithm for calculating the greatest common divisor of two polynomials. Furthermore, the underlying theory behind the computation of the greatest common divisor is explained and illustrated on many examples. 1
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