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Finite Integrals Numerical Computations
Mikulka, Jiří ; Šátek, Václav (referee) ; Kunovský, Jiří (advisor)
The application of the finite integral of multiple variable functions is penetrating into more and more industries and science disciplines. The demands placed on solutions to these problems (such as high accuracy or high speed) are often quite contradictory. Therefore, it is not always possible to apply analytical approaches to these problems; numerical methods provide a suitable alternative. However, the ever-growing complexity of these problems places too high a demand on many of these numerical methods, and so neither of these methods are useful for solving such problems. The goal of this thesis is to design and implement a new numerical method that provides highly accurate and very fast computation of finite integrals of multiple variable functions. This new method combines pre-existing approaches in the field of numerical mathematics.
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Effective Algorithms for High-Precision Computation of Elementary Functions
Chaloupka, Jan ; Kunovský, Jiří (referee) ; Šátek, Václav (advisor)
Nowadays high-precision computations are still more desired. Either for simulation on a level of atoms where every digit is important and inaccurary in computation can cause invalid result or numerical approximations in partial differential equations solving where a small deviation causes a result to be useless. The computations are carried over data types with precision of order hundred to thousand digits, or even more. This creates pressure on time complexity of problem solving and so it is essential to find very efficient methods for computation. Every complex physical problem is usually described by a system of equations frequently containing elementary functions like sinus, cosines or exponentials. The aim of the work is to design and implement methods that for a given precision, arbitrary elementary function and a point compute its value in the most efficent way. The core of the work is an application of methods based on AGM (arithmetic-geometric mean) with a time complexity of order $O(M(n)\log_2{n})$ 9(expresed for multiplication $M(n)$). The complexity can not be improved. There are many libraries supporting multi-precision atithmetic, one of which is GMP and is about to be used for efficent method implementation. In the end all implemented methods are compared with existing ones.
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Multiple Integrals
Valešová, Nikola ; Veigend, Petr (referee) ; Šátek, Václav (advisor)
The problem of definite integral and differential equation computation is still a significant part of many scientific branches and the solution of integral calculus tasks can be found in many industrial fields too. During the computation of such tasks, the accuracy and high-speed requirements are often confronted. These requirements are crucial during the process of the suitable method choice. The aim of this thesis is to propose, describe, implement and test a new numerical method, which combines the solution of definite integrals by transforming them into differential equations solved by the Taylor series with the traditional methods, which use the Newton-Cotes formulas. As a result, a new application has been developed, that provides fast results of definite two-dimensional integrals and reaches at least the precision of MATLAB. The major accomplishment of this thesis is the development of a new numerical method and its comparison to other established ways of computation.
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Analysis of Methods of Differences for Partial Differential Equations Solving
Zpěváková, Jana ; Zbořil, František (referee) ; Šátek, Václav (advisor)
In this thesis, we discuss the numerical solution of ordinary differential equation and numerical methods of solving partial differential equations. We propose and implement an application, that converts partial differential hyperbolic equation to a set of ordinary differential equations using finite difference method. After that, the system of equations is solved using the Taylor method programmed in Matlab environment. Finally, we compare the time complexity of proposed solution with parallel numerical computation.
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Partial Differential Equations Parallel Solutions
Nečasová, Gabriela ; Šátek, Václav (referee) ; Kunovský, Jiří (advisor)
This thesis deals with the topic of partial differential equations parallel solutions. First, it focuses on ordinary differential equations (ODE) and their solution methods using Taylor polynomial. Another part is devoted to partial differential equations (PDE). There are several types of PDE, there are parabolic, hyperbolic and eliptic PDE. There is also explained how to use TKSL system for PDE computing. Another part focuses on solution methods of PDE, these methods are forward, backward and combined methods. There was explained, how to solve these methods in TKSL and Matlab systems. Computing accuracy and time complexity are also discussed. Another part of thesis is PDE parallel solutions. Thanks to the possibility of PDE convertion to ODE systems it is possible to represent each ODE equation by independent operation unit. These units enable parallel computing. The last chapter is devoted to implementation. Application enables generation of ODE systems for TKSL system. These ODE systems represent given hyperbolic PDE.
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Limits and L'Hospital's rule
Ranšová, Kateřina ; Staněk, Jakub (advisor) ; Halas, Zdeněk (referee)
Title: Limits and l'Hospital's rule Author: Kateřina Ranšová Department: Department of Mathematics Education Supervisor: RNDr. Jakub Staněk, Ph.D., Department of Mathematics Education Abstract: The aim of this BA thesis is to introduce to the reader the concept of the limit of function and the means of its solution. The main impact of the thesis lies in a didactic approach and in the connection of a limit theory with its graphic representation and different methods of algebraic calculation. The text consists of eight chapters which can be divided into two parts according to their content. The first part explores the term "the limit of the function". Individual types of limits are then defined. To facilitate understanding, most of the definitions are accompanied by a particular example and a graphic representation . The first part is concluded by a unified definition which by means of the term "vicinity" summarizes all preceding types of limits. The other part deals with some basic methods of limits' calculation. Other topics include Taylor Series, l'Hospital's Rule and their applications to the limits. The core of the thesis is a comparison of calculation by means of l'Hospital's Rule and Taylor Series. The conclusion of the thesis presents some advantages and disadvantages of applying Taylor Series and...
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Calculation of Values of Trigonometric Functions
Uhlířová, Iva ; Halas, Zdeněk (advisor) ; Štěpánová, Martina (referee)
Title: Calculation of Values of Trigonometric Functions Author: Iva Uhlířová Department: Department of Mathematics Education Supervisor: Mgr. Zdeněk Halas, DiS., Ph.D. Abstract: This bachelor thesis deals with various calculation methods of how to calculate values of trigonometric functions (sine and tangent chiefly). These methods either were used in the past times or are still used nowadays. However, in this thesis, these methods are explained in a modern way in order to be easily understandable by such readers who have basic knowledge of calculus. In each chapter, there is only one method discussed. At first, lengths of chords are calculated and a table of them is constructed, based on Ptolemy's and Copernicus' methods. Then, al-Kashi's approximation method is interpreted elaborately. Furthermore, Newton's method of development of Taylor series for the sine function is explained in detail. Last but not least the CORDIC algorithm is discussed. In order to provide a better understanding, there are particular values calculated in each chapter. Keywords: Almagest, CORDIC, Taylor series
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Limits and L'Hospital's rule
Ranšová, Kateřina ; Staněk, Jakub (advisor) ; Halas, Zdeněk (referee)
Title: Limits and l'Hospital's rule Author: Kateřina Ranšová Department: Department of Mathematics Education Supervisor: RNDr. Jakub Staněk, Ph.D., Department of Mathematics Education Abstract: The aim of this BA thesis is to introduce to the reader the concept of the limit of function and the means of its solution. The main impact of the thesis lies in a didactic approach and in the connection of a limit theory with its graphic representation and different methods of algebraic calculation. The text consists of eight chapters which can be divided into two parts according to their content. The first part explores the term "the limit of the function". Individual types of limits are then defined. To facilitate understanding, most of the definitions are accompanied by a particular example and a graphic representation . The first part is concluded by a unified definition which by means of the term "vicinity" summarizes all preceding types of limits. The other part deals with some basic methods of limits' calculation. Other topics include Taylor Series, l'Hospital's Rule and their applications to the limits. The core of the thesis is a comparison of calculation by means of l'Hospital's Rule and Taylor Series. The conclusion of the thesis presents some advantages and disadvantages of applying Taylor Series and...
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Analysis of Methods of Differences for Partial Differential Equations Solving
Zpěváková, Jana ; Zbořil, František (referee) ; Šátek, Václav (advisor)
In this thesis, we discuss the numerical solution of ordinary differential equation and numerical methods of solving partial differential equations. We propose and implement an application, that converts partial differential hyperbolic equation to a set of ordinary differential equations using finite difference method. After that, the system of equations is solved using the Taylor method programmed in Matlab environment. Finally, we compare the time complexity of proposed solution with parallel numerical computation.
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Multiple Integrals
Valešová, Nikola ; Veigend, Petr (referee) ; Šátek, Václav (advisor)
The problem of definite integral and differential equation computation is still a significant part of many scientific branches and the solution of integral calculus tasks can be found in many industrial fields too. During the computation of such tasks, the accuracy and high-speed requirements are often confronted. These requirements are crucial during the process of the suitable method choice. The aim of this thesis is to propose, describe, implement and test a new numerical method, which combines the solution of definite integrals by transforming them into differential equations solved by the Taylor series with the traditional methods, which use the Newton-Cotes formulas. As a result, a new application has been developed, that provides fast results of definite two-dimensional integrals and reaches at least the precision of MATLAB. The major accomplishment of this thesis is the development of a new numerical method and its comparison to other established ways of computation.
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