|
|
Solution of difference equations and relation with Z-transform
Klimek, Jaroslav ; Smékal, Zdeněk (referee) ; Růžičková,, Miroslava (referee) ; Diblík, Josef (advisor)
This dissertation presents the solution of difference equations and focuses on a method of difference equations solution with the aid of eigenvectors. The first part reminds the basic terms from area of difference equations such as dynamic of difference equations and linear difference equations of first order and higher order. Then the second section recalls also the system of difference equations including the fundamental matrix and general solution description. Afterthat, the method of solving the difference equations with a variation of constants and transform of scalar equations to the system are shown. The second part of the dissertation analyses some known algorithms and methods for the solution of linear difference equations. The Z-transform, its importance and usage for finding the solution of difference equation is recalled. Then the discrete analogue of Putzer's algorithm is mentioned because this algorithm was often used to check the results obtained by the newly described algorithm in further parts of this thesis. Also some ways of the system matrix power are stated. The next section then describes the principle of Weyr's method which is the basic point for further development of the theory including the presentation of the research results gained by Jiří Čermák in this area. The third part describes own solution of the difference equations system via eigenvectors based on the principle of Weyr's method for differential equations. The solution of system of linear homogeneous difference equtions with constant coefficients including the proof is presented and this solution is then extended to nonhomogeneous systems. Consequently to the theory, the influence of a nulity and the multiplicity of roots on the form of the solution is discussed. The last section of this part shows the implementation of the algorithm in Matlab program (for basic simpler cases) and its application to some cases of difference equations and systems with these equations. The final part of the thesis is more practical and it presents the usage of the designed algorithm and theory. Firstly, the algorithm is compared with Z-transform and the method of variation of constants and it is illustrated how to obtain the same results by using these three approaches. Then an example of current response solution in RLC circuit is demonstrated. The continuous case is solved and then the problem is transferred to discrete case and solved with the Z-transform and the method of eigenvectors. The obtained results are compared with the result of the continuous case.
|
|
|
Multipoint identification of position and orientation of object
Řičánek, Dominik ; Ligocki, Adam (referee) ; Burian, František (advisor)
The goal of this paper is to find a good method to determine the pose of two, mutually rotated, objects and to try and implement it first in C++ without the usage of any external libraries and then in Kuka Robot Language (KRL). First we are going to look at two different approaches to solving this problem: The Iterative Closest Point algorithm (ICP) and the Kabsch Algorithm. From them one is going to be chosen and a program will be build around it. Following its implementation, the algorithm’s precision is going to be tested. Finally KRL will be briefly introduced and the various problems involving transition of the algorithm from C++ to KRL will be talked about.
|
|
|
Mathematical methods in some ranking models
Pažourek, Lubomír ; Kureš, Miroslav (referee) ; Čermák, Jan (advisor)
The bachelor thesis deals with the mathematical essence of some ranking methods. Their unifying element is the so-called Perron-Frobenius theorem for non-negative and irreducible matrices, which formulates the conditions for the existence of a positive eigenvalue and a positive eigenvector of the given matrix. The goal of the thesis consists in providing an overview of the necessary theoretical results, explaining their application within some ranking methods and performing simulations during the evaluation of some competitions.
|
|
|
Multipoint identification of position and orientation of object
Řičánek, Dominik ; Ligocki, Adam (referee) ; Burian, František (advisor)
The goal of this paper is to find a good method to determine the pose of two, mutually rotated, objects and to try and implement it first in C++ without the usage of any external libraries and then in Kuka Robot Language (KRL). First we are going to look at two different approaches to solving this problem: The Iterative Closest Point algorithm (ICP) and the Kabsch Algorithm. From them one is going to be chosen and a program will be build around it. Following its implementation, the algorithm’s precision is going to be tested. Finally KRL will be briefly introduced and the various problems involving transition of the algorithm from C++ to KRL will be talked about.
|
|
|
Eigenvalues of Matrices and Their Localization
Borzíková, Žofia ; Škorpilová, Martina (advisor) ; Halas, Zdeněk (referee)
The diploma thesis is concerned with the topic of eigenvalues of matrices and their lo- calization in the complex plane. First introducing general theorems concerning eigenvalues, eigenvalues for special classes of matrices are then discussed. After presenting the theory of Jordan and Weyr canonical forms, the connection and relation of these two forms is also explained. The estimates of the localizations of the eigenvalues follows from Gershgo- rin's theorem. This text might be used as a didactic material for college-level students of mathematics, thanks to its form having theoretical parts accompanied by examples with commented solutions. It may also be used as a source of information for anyone interested in extending their knowledge of linear algebra. 1
|
|
|
Affine mappings and transformations in the plane with solved examples
Barborka, Lukáš ; Zamboj, Michal (advisor) ; Jančařík, Antonín (referee)
Analytical geometry widely uses the apparatus of linear algebra, it is, of course, its natural application. The aim of this thesis is the theoretical interconnection, for many students still abstract, bases of the linear algebra with their practical application in the analyti- cal geometry, especially in affine transformations and their use in the solved examples in the plane. This thesis is intended to put concepts known from the course of Linear algebra (homomorphism, eigenvalues/eigenvectors, orthogonal matrices, transition matri- ces...) into context with practical using in the analytical geometry, whether in the form of proofs of important theorems using the linear algebra and arithmetic apparatus, or the following solved examples. The aim of the examples is to provide some insight or guidance on the solution of the same or analogous tasks. The theory and examples are in some cases supplemented with illustrations for better clarity. The work is divided into several parts for greater clarity. The introduction is repeated important concepts of linear algebra such as group, field, vector space, Euclidean space, linear mapping (homomorphism), change of coordinates matrix, eigenvalue/eigenvector of the matrix. It also switches to affine point space, affine coordinate system, transformation equation for...
|
|
|
Affine mappings and transformations in the plane with solved examples
Barborka, Lukáš ; Tůmová, Veronika (advisor) ; Zamboj, Michal (referee)
Analytical geometry widely uses the apparatus of linear algebra, it is, of course, its natural application. The aim of this thesis is the theoretical interconnection, for many students still abstract, bases of the linear algebra with their practical application in the analyti- cal geometry, especially in affine transformations and their use in the solved examples in the plane. This thesis is intended to put concepts known from the course of Linear algebra (homomorphism, eigenvalues/eigenvectors, orthogonal matrices, transition matri- ces...) into context with practical using in the analytical geometry, whether in the form of proofs of important theorems using the linear algebra and arithmetic apparatus, or the following solved examples. The aim of the examples is to provide some insight or guidance on the solution of the same or analogous tasks. The theory and examples are in some cases supplemented with illustrations for better clarity. The work is divided into several parts for greater clarity. The introduction is repeated important concepts of linear algebra such as group, field, vector space, Euclidean space, linear mapping (homomorphism), change of coordinates matrix, eigenvalue/eigenvector of the matrix. It also switches to affine point space, affine coordinate system, transformation equation for...
|
|
| |
|
|
Solution of difference equations and relation with Z-transform
Klimek, Jaroslav ; Smékal, Zdeněk (referee) ; Růžičková,, Miroslava (referee) ; Diblík, Josef (advisor)
This dissertation presents the solution of difference equations and focuses on a method of difference equations solution with the aid of eigenvectors. The first part reminds the basic terms from area of difference equations such as dynamic of difference equations and linear difference equations of first order and higher order. Then the second section recalls also the system of difference equations including the fundamental matrix and general solution description. Afterthat, the method of solving the difference equations with a variation of constants and transform of scalar equations to the system are shown. The second part of the dissertation analyses some known algorithms and methods for the solution of linear difference equations. The Z-transform, its importance and usage for finding the solution of difference equation is recalled. Then the discrete analogue of Putzer's algorithm is mentioned because this algorithm was often used to check the results obtained by the newly described algorithm in further parts of this thesis. Also some ways of the system matrix power are stated. The next section then describes the principle of Weyr's method which is the basic point for further development of the theory including the presentation of the research results gained by Jiří Čermák in this area. The third part describes own solution of the difference equations system via eigenvectors based on the principle of Weyr's method for differential equations. The solution of system of linear homogeneous difference equtions with constant coefficients including the proof is presented and this solution is then extended to nonhomogeneous systems. Consequently to the theory, the influence of a nulity and the multiplicity of roots on the form of the solution is discussed. The last section of this part shows the implementation of the algorithm in Matlab program (for basic simpler cases) and its application to some cases of difference equations and systems with these equations. The final part of the thesis is more practical and it presents the usage of the designed algorithm and theory. Firstly, the algorithm is compared with Z-transform and the method of variation of constants and it is illustrated how to obtain the same results by using these three approaches. Then an example of current response solution in RLC circuit is demonstrated. The continuous case is solved and then the problem is transferred to discrete case and solved with the Z-transform and the method of eigenvectors. The obtained results are compared with the result of the continuous case.
|
guest ::
