Comparison of algorithms for calculation of the greatest common divisor of several polynomials

summary:The computation of the greatest common divisor (GCD) has many applications in several disciplines including computer graphics, image deblurring problem or computing multiple roots of inexact polynomials. In this paper, Sylvester and Bézout matrices are considered for this purpose. The computation is divided into three stages. A rank revealing method is shortly mentioned in the first one and then the algorithms for calculation of an approximation of GCD are formulated. In the final stage the coefficients are improved using Gauss-Newton method. Numerical results show the efficiency of proposed last two stages

Interpolace hladkých funkcí pomocí kvadratických a kubických splinů

In this thesis, we study properties of cubic and quadratic spline interpolation. First, we define the notions of spline and interpolation. We then merge them to study cubic and quadratic spline interpolations. We go through the individual spline interpolation types, show an algorithm for constructing selected types and sum up their basic properties. We then present a computer program based on the provided algorithms. We use it to construct spline interpolations of some sample functions and we calculate errors of these interpolation and compare them with theoretical estimates

Transformace Sylvestrovy matice a výpočet největšího společného dělitele dvou polynomů

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

Transformace Sylvestrovy matice a výpočet největšího společného dělitele dvou polynomů

V této diplomové práci se zabýváme výpočtem největšího společného dělitele dvou polynomů. V první řadě studujeme vlastnosti Sylvestrových matic a jakým způsobem je lze využít pro daný záměr. Dále si všimneme, že výsledky lze přirozeně zobecnit i pro více polynomů. V předposlední části se zabýváme využitím Bézoutových matic ke stejnému účelu, abychom získali srovnání s maticemi Sylvestrovými. I zde výsledek rozšíříme pro víc než dva polynomy. Ke všem přístupům jsou prezentovány algoritmy. Na závěr algoritmy implementujeme v prostředí MATLAB a jednotlivé algoritmy porovnáme v numerických experimentech. Powered by TCPDF (www.tcpdf.org)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)Department of Numerical MathematicsKatedra numerické matematikyMatematicko-fyzikální fakultaFaculty of Mathematics and Physic

Transformace Sylvestrovy matice a výpočet největšího společného dělitele dvou polynomů

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

Mechanism of the sulphurisation of phosphines and phosphites using 3-amino-1,2,4-dithiazole-5-thione (Xanthane Hydride)

Contrary to a previous report, the sulfurisation of phosphorus(III) derivatives by 3-amino-1,2,4-dithiazole-5-thione (xanthane hydride) does not yield carbon disulfide and cyanamide as the additional reaction products. The reaction of xanthane hydride with triphenyl phosphine or trimethyl phosphite yields triphenyl phosphine sulfide or trimethyl thiophosphate, respectively, and thiocarbamoyl isothiocyanate which has been trapped with nucleophiles. The reaction pathway involves initial nucleophilic attack of the phosphorus at sulfur next to the thiocarbonyl group of xanthane hydride followed by decomposition of the phosphonium intermediate formed to products. The Hammett -values for the sulfurisation of substituted triphenyl phosphines and triphenyl phosphites in acetonitrile are –1.0. The entropies of activation are very negative (–114 ± 15 J mol–1 K–1) with little dependence on solvent which is consistent with a bimolecular association step leading to the transition state. The negative values of S and values indicate that the rate limiting step of the sulfurisation reaction is formation of the phosphonium ion intermediate which has an early transition state with little covalent bond formation. The site of nucleophilic attack has been also confirmed using computational calculation

1,2,4-Dithiazole-5-ones and 5-thiones as Efficient Sulfurizing Agents of Phosphorus (III) Compounds – A Kinetic Comparative Study

The sulfurization efficiency of 25 3-substituted-1,2,4-dithiazole-5-ones and 5-thiones towards triphenyl phosphite in acetonitrile, DCM, THF and toluene at 25 °C was evaluated. All the 1,2,4-dithiazoles are much better sulfurizing reagents than commercially available agents (PADS, TETD, Beaucage's reagent). The most efficient sulfurizing agents in all solvents are 3-phenoxy (4), 3-phenylthio (5) and 3-ethoxy-1,2,4-dithiazole-5-one (1) whose reactivity is at least two orders of magnitude higher than that of other 1,2,4-dithiazoles. Contrary to a previous report, the sulfurization with 1 does not yield carbonylsulfide and ethyl cyanate as the additional reaction products but unstable ethoxythiocarbonyl isocyanate which has been trapped with 4-methoxyaniline. Similar trapping experiments have proven that the site of attack is at the sulfur adjacent to the C[double bond, length as m-dash]O group for compounds 4 and 5. The reaction pathway involves rate-limiting initial nucleophilic attack of the phosphorus at sulfur followed by decomposition of the phosphonium intermediate to the corresponding phosphorothioate and isocyanate/isothiocyanate species. The existence of the phosphonium intermediate during sulfurization of triphenyl phosphine with 3-phenyl-1,2,4-dithiazole-5-thione (7a) was proven using kinetic studies. From the Hammett and Brønsted correlations and from other kinetic measurements it was concluded that the transition-state structure is almost apolar for the most reactive 1,2,4-dithiazoles whereas a polar structure resembling a zwitter-ionic intermediate may be more appropriate for the least reactive 1,2,4-dithiazoles. The extent of P–S bond formation and S–S bond cleavage is very similar in all reaction series but it gradually decreases with the reactivity of the 1,2,4-dithiazole derivatives