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Newton and numerical mathematics
Obrátil, Štěpán ; Nechvátal, Luděk (referee) ; Zatočilová, Jitka (advisor)
Topic of this bachelor thesis are Newton's methods for numerical solutions of various problems. Especially the problems of solving nonlinear equations and systems of nonlinear equations, as well as numerical integration are explained. The Newton's method for solving nonlinear equations is presented, as well as its many modifications and its generalisation for systems of nonlinear equations. Usefulness of methods is demonstrated on various examples. In the end, Newton-Cotes quadrature formulae for numerical integration are presented.
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The choice of the stopping criteria for Newton-like methods
Kurnas, Jakub ; Dolejší, Vít (advisor) ; Tichý, Petr (referee)
We formulate examples of partial differential equations which can be solved through their discretization and subsequent solution of derived algebraic system. A brief summary of Discontinuous Galerkin Discretization is given as well as definitions of algebraic and discretization errors. We derive the Newton method, which solves nonlinear algebraic systems by solving a sequence of linear problems, we modify the method and examine implementation options. We define stopping criteria for the Newton-like method using aforementioned errors and we explain how to keep accuracy of the solution of derived algebraic system and the original partial differential equation in balance. We present numerical experiments to illustrate theoretical background and mention several basic properties of the Newton- like method.
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Methods for the solution of nonlinear equations
Havelková, Eva ; Kučera, Václav (advisor) ; Tichý, Petr (referee)
The aim of this bachelor thesis is to present an overview of elementary numerical methods for solving nonlinear algebraic equations in one variable. Firstly, related concepts from numerical mathematics and mathematical analysis are explained. The main part of the thesis provides a detailed description of chosen iterative methods as well as the proofs of their orders of convergence. The methods covered are namely the bisection method, fixed-point iteration, regula falsi method, Newton's method, secant method and methods that are based on quadratic interpolation. The practical part of the thesis presents results of numerical experiments that were carried out with Matlab software on various types of nonlinear equations. These results are compared with the theory introduced in the preceding parts. The contribution of this thesis is to provide a comprehensive overview and comparison of the characteristics of basic methods for solving nonlinear equations based on a variety of literature. Powered by TCPDF (www.tcpdf.org)
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Newton and numerical mathematics
Obrátil, Štěpán ; Nechvátal, Luděk (referee) ; Zatočilová, Jitka (advisor)
Topic of this bachelor thesis are Newton's methods for numerical solutions of various problems. Especially the problems of solving nonlinear equations and systems of nonlinear equations, as well as numerical integration are explained. The Newton's method for solving nonlinear equations is presented, as well as its many modifications and its generalisation for systems of nonlinear equations. Usefulness of methods is demonstrated on various examples. In the end, Newton-Cotes quadrature formulae for numerical integration are presented.
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