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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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Elementary functions and domain
Vitásek, Tomáš ; Pilous, Derek (advisor) ; Jančařík, Antonín (referee)
The aim of my bachelor thesis is to show the teachers and the students of educational mathematics the relation between the elementary functions and their domains and the methods of creating problems focused on finding the domain. In the chapter called "Zobrazení", I try to define and explain the terminology closely connected to elementary functions and their doma- ins which I use in the following parts. Crucial chapters are "Definiční obory elementárních funkcí" and "Návrhy úloh". In the first of those two chapters I thoroughly explain the algorithm of solving problems focused on finding the domain of elementary functions and I discuss what the domains of various elementary functions are like. In the second of those two chapters, I suggest functions fitting a concrete domain. The aim is to suggest functions in a way that a student taking their maturita exam in mathematics at high school would be able to solve the problem of finding the domain. Keywords: elementary function, domain, function composition.
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Inversion of elementary functions
Vitásek, Tomáš ; Pilous, Derek (advisor) ; Kvasz, Ladislav (referee)
The main point of the bachelor thesis is to show mathematics students and te- achers the relation between elementary function and its inverse function, the approach how to find the inverse function and possible processes in designing inverse function tasks. In the chapter "Zobrazení", I define and explain ne- cessary terms, which are used in various parts of the thesis. Further, I discuss basic features of inverse functions. In the chapter "Inverzní funkce k elemen- tárním funkcím" first I discuss the invertibility of elementary functions, and then I describe the process of solving inverse function tasks in detail. In the chapter "Návrhy úloh" I show the proces of designing the task with defined difficulty, using the previously described knowledge. Keywords: elementary function, invertible function, inverse function.
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Monotonicity of functions which can be expressed using elementary functions
Peltan, Libor ; Bárta, Tomáš (advisor) ; Pyrih, Pavel (referee)
For certain types of functions expressible with formula (equivalently: functions from classes closed to arithmetic operations) under stated assumptions, we prove monotonicity at some neighbourhood of +∞. They are: formulas containing exp, log, sin, arctan, etc. with constrainted domain of these functions; power series with cofinite many coefficients positive; various classes of functions expressible with formulas with the requirement of preserving monotony in summation, or multiplication, or the monotony resulting from having a finite number of zero points; and finally formulas with square root. 1
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Problem posing for limits of functions
Prskavcová, Pavla ; Pilous, Derek (advisor) ; Kvasz, Ladislav (referee)
Title: Problem posing for limits of functions Author: Pavla Prskavcová Department: Department of Mathematics and Mathematical Education Supervisor: Mgr. Derek Pilous, Ph.D. The aim of the thesis is to describe methods of limit problems solving and of posing new problems that are solvable by those methods. The first part of the thesis contains the necessary theory (definitions and theorems). In the second part of the thesis there are described the standard methods used for evaluating containing limits. The third part suggest ways of posing problems solvable by using elementary methods, based on the analysis of methods describes in the se- cond part of the thesis (without usage of the differential calculus). Keywords: limit, problem solving, problem posing, elementary function.
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On impossibility of elementary integration
Zelina, Michael ; Pražák, Dalibor (advisor) ; Kazda, Alexandr (referee)
This work is devoted to studying of problem of (non)existence an elementary pri- mitive function of a given function. In the first place, we introduce the structure of a differential field and then we find a suitable way of formalizing the concept of the elementary function. This tools opens up the possibility to formulate and prove the crucial theorem which says what form an elemental primitive function must necessarily have if it exists. Then we use it to find the conditions for the existence of an ele- mentary integrals of two functions in a special but still quite general form. By using these conditions, we will show the nonelementarity of a number of more or less known integrals. 3
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Problem posing for limits of functions
Prskavcová, Pavla ; Pilous, Derek (advisor) ; Kvasz, Ladislav (referee)
Title: Problem posing for limits of functions Author: Pavla Prskavcová Department: Department of Mathematics and Mathematical Education Supervisor: Mgr. Derek Pilous, Ph.D. The aim of the thesis is to describe methods of limit problems solving and of posing new problems that are solvable by those methods. The first part of the thesis contains the necessary theory (definitions and theorems). In the second part of the thesis there are described the standard methods used for evaluating containing limits. The third part suggest ways of posing problems solvable by using elementary methods, based on the analysis of methods describes in the se- cond part of the thesis (without usage of the differential calculus). Keywords: limit, problem solving, problem posing, elementary function.
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Inversion of elementary functions
Vitásek, Tomáš ; Pilous, Derek (advisor) ; Kvasz, Ladislav (referee)
The main point of the bachelor thesis is to show mathematics students and te- achers the relation between elementary function and its inverse function, the approach how to find the inverse function and possible processes in designing inverse function tasks. In the chapter "Zobrazení", I define and explain ne- cessary terms, which are used in various parts of the thesis. Further, I discuss basic features of inverse functions. In the chapter "Inverzní funkce k elemen- tárním funkcím" first I discuss the invertibility of elementary functions, and then I describe the process of solving inverse function tasks in detail. In the chapter "Návrhy úloh" I show the proces of designing the task with defined difficulty, using the previously described knowledge. Keywords: elementary function, invertible function, inverse function.
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