National Repository of Grey Literature 6 records found  Search took 0.00 seconds. 
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.
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.
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.
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.
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.
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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