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Polynomial Equations Roots
Tomšík, Filip ; Kopřiva, Jan (referee) ; Kunovský, Jiří (advisor)
Bachelor´s thesis purpose was been study solution algebraic and differential equation. We were studying Bairstow method, which is the most conducive to solution homogenous differential equation higher order. Implementation Bairstow method and her connection with Gauss elimination method. In the end we are performed tests on rate calculation and accuracy.
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Numerical Methods Accuracy vs Precision of Arithmetic
Kluknavský, František ; Šátek, Václav (referee) ; Peringer, Petr (advisor)
Thesis is dedicated to evaluation of roundoff impact on numerical integration methods accuracy and effectivity. Contains theoretical expectations taken from existing literature, implementation of chosen methods, experimental measurement of attained accuracy under different circumstances and their comparison with regard to time complexity. Library contains Runge-Kutta methods to order 7 and Adams-Bashforth methods to order 20 implemented using C++ templates which allow optional arbitrary-precision arithmetic. Small models with known analytic solution were used for experiments.
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Semi - analytical computations and continuous systems simulation
Kopřiva, Jan ; Kubátová, Hana (referee) ; Novitzká,, Valerie (referee) ; Kunovský, Jiří (advisor)
The thesis deals with speedup and accuracy of numerical computation, especially when differential equations are solved. Algorithms, which are fulling these conditions are named semi-analytical. One posibility how to accelerate computation of differential equation is paralelization. Presented paralelization is based on transformation numerical solution into residue number system, which is extended to floating point computation. A new algorithm for modulo multiplication is also proposed. As application applications in solution of differential calculus are the main goal it is discussed numeric integration with modified Euler, Runge - Kutta and Taylor series method in residue number system. Next possibilities and extension for implemented residue number system are mentioned at the end.
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Advanced Electronic Circuits Simulation Methods
Kocina, Filip ; Kozek, Martin (referee) ; Kyncl, Jan (referee) ; Kunovský, Jiří (advisor)
Disertační práce se zabývá simulací elektronických obvodů. Popisuje metodu kapacitorové substituce (CSM) pro převod elektronických obvodů na elektrické obvody, jež mohou být následně řešeny pomocí numerických metod, zejména Moderní metodou Taylorovy řady (MTSM). Tato metoda se odlišuje automatickým výběrem řádu, půlením kroku v případě potřeby a rozsáhlou oblastí stability podle zvoleného řádu. V rámci disertační práce bylo autorem disertace vytvořeno specializované programové vybavení pro řešení obyčejných diferenciálních rovnic pomocí MTSM, s mnoha vylepšeními v algoritmech (v porovnání s TKSL/386). Tyto algoritmy zahrnují zjednodušování obecných výrazů na polynomy, paralelizaci nezávislou na integrační metodě atp. Tento software běží na linuxovém serveru, který komunikuje pomocí protokolu TCP/IP. Toto vybavení bylo úspěšně použito pro simulaci VLSI obvodů, jejichž řešení pomocí CSM bylo značně rychlejší a spotřebovávalo méně paměti než state-of-the-art SPICE.
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Continuous models in biology
Kozák, Michal ; Stará, Jana (advisor) ; Kučera, Milan (referee)
This Bachelor Thesis is devoted to study of conditions guaranteeing that the modelled biological system is stable from the point of view of surviving of species. First, we give a short survey of various concepts of ecological stability (persistence, permanence) and then we concentrate on permanence. The models we study are described in terms of semidynamical systems on metric spaces. In this framework we define permanence of a semidynamical system. Main part of the thesis are theorems giving sufficient conditions for permanence or non- permanence by adapting the method of Average Lyapunov Function. In the last chapter a model of aquatic population interacting with a polluted environment is considered and its permanence proved under certain conditions on coefficients. The aim of the theses is to present a survey of these notions. Moreover, the contri- bution of theses is the proof of non-permanence theorem whose part was known for difference equations, only. 34
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Filippov dynamical systems with applications
Šimonová, Dorota ; Janovský, Vladimír (advisor) ; Ratschan, Stefan (referee)
The thesis is motivated by problems of contact mechanics with friction. At the beginning we describe a class of piecewise smooth systems with discontinuous vector field called Filippov systems. We also show how to solve them. The rest of this thesis is focused on applications, especially dry friction model and finite element model of Coulomb friction with one contact point. We propose a technique for simulation of the second mentioned model which combines sovling methods for Filippov systems and impact oscillators. Powered by TCPDF (www.tcpdf.org)
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Method of Green function for boundary value problems for ODR
Héda, Ivan ; Rokyta, Mirko (advisor) ; Pražák, Dalibor (referee)
The main aim of this work is to summarize the basic knowledge of method which is using Green's functions for solving boundary value problems for linear diffe- rential equations. These functions will be defined and, with some not very strong presumptions, uniquely constructed. This method is primarily derived for solving problems with homogenous boundary conditions. However it will be shown that there is no more presumtions needed to use this method to solve problems with non-homogenous linear boundary conditions. As a main consequence of preceding existence and uniqueness of solution for relatively wide class of linear boundary problems will be provided. 1
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Mathematical models of ecosystems
Scholle, David ; Janovský, Vladimír (advisor) ; Kofroň, Josef (referee)
This work is about models of population growth in different situations. At first, we will examine amount of spiders and their prey in the region of Langa Astigiana, based on models of dynamical systems. We will also consider the usage of spraying of near vineyards and effect of this on the ecosystem. The aim of this work is also to check the possibility of periodical cycles, and thus also of the Hopf Bifurcation, appearing. Next part talks about the model of a beehive and examines the influence of insecticides on the population of bee drones and worker bees. The aim of the last chapter is to examine the effectivity and possible impact of human intervention in the region of Šumava forest. The model will check the necessity of such action against parasites. The software used for these tasks will be mainly the continuation toolbox MatCont, which is a part of the program MatLab.
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Transformations of ODEs into gradient systems in stationary points
Bílý, Michael ; Bárta, Tomáš (advisor) ; Spurný, Jiří (referee)
This bachelor thesis follows article by Bárta, Chill a Fašangová [1]. It is proven there that every ordinary differential equation with a strict Lyapunov function is in fact a gradient system for certain Riemannian metric on the set of all non-equilibrium points. We will try to determine necessary and sufficient conditions for this Riemannian metric to have continuous extension to isolated equilibrium point so that the ODE is gradient system on the whole domain. Powered by TCPDF (www.tcpdf.org)
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Advanced Electronic Circuits Simulation Methods
Kocina, Filip ; Kozek, Martin (referee) ; Kyncl, Jan (referee) ; Kunovský, Jiří (advisor)
Disertační práce se zabývá simulací elektronických obvodů. Popisuje metodu kapacitorové substituce (CSM) pro převod elektronických obvodů na elektrické obvody, jež mohou být následně řešeny pomocí numerických metod, zejména Moderní metodou Taylorovy řady (MTSM). Tato metoda se odlišuje automatickým výběrem řádu, půlením kroku v případě potřeby a rozsáhlou oblastí stability podle zvoleného řádu. V rámci disertační práce bylo autorem disertace vytvořeno specializované programové vybavení pro řešení obyčejných diferenciálních rovnic pomocí MTSM, s mnoha vylepšeními v algoritmech (v porovnání s TKSL/386). Tyto algoritmy zahrnují zjednodušování obecných výrazů na polynomy, paralelizaci nezávislou na integrační metodě atp. Tento software běží na linuxovém serveru, který komunikuje pomocí protokolu TCP/IP. Toto vybavení bylo úspěšně použito pro simulaci VLSI obvodů, jejichž řešení pomocí CSM bylo značně rychlejší a spotřebovávalo méně paměti než state-of-the-art SPICE.
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