magistrsko delo
Tadej Šnajder (Author), Bor Plestenjak (Mentor)

Abstract

V magistrskem delu smo se osredotočili na reševanje Sylvestrove enačbe in enačbe Ljapunova, kot poseben primer Sylvestrove enačbe, z metodami ADI. Če dimenzije matrik v Sylvestrovi matrični enačbi niso prevelike, jo lahko rešimo s pomočjo direktnih algoritmov, kot je na primer Bartels-Stewartova metoda. Ko imamo v Sylvestrovi enačbi razpršene matrike velikih dimenzij, namesto direktnih algoritmov raje uporabimo iteracijske metode, med katere spadajo tudi metode ADI. V magistrskem delu so najprej predstavljene povezave med teorijo upravljanja linearnih kontrolnih sistemov in enačbo Ljapunova, kot poseben primer Sylvestrove enačbe. Hkrati so navedene tudi predpostavke, ki jih uporabljamo v magistrskem delu. Sledi predstavitev Smithove metode, metode ADI in nekaj njenih najpomembnejših razširitev. Nato je predstavljen problem izbire premikov, ki vplivajo na hitrost konvergence metod ADI, podane so ocene za konvergenco metod ADI ter nekateri pristopi, s katerimi rešujemo problem izbire premikov. Predstavljene so tudi implementacije metod ADI v Matlabu. Narejena je bila primerjava premikov in primerjava metod na nekaterih primerih iz spletne zbirke Slicot.

Keywords

metoda ADI;Sylvestrova enačba;enačba Ljapunova;razpršene matrike;Smithova metoda;metoda ADI nizkega ranga s faktorji Choleskega;faktorizirana metoda ADI;iterativne metode;

Data

Language: Slovenian
Year of publishing:
Typology: 2.09 - Master's Thesis
Organization: UL FMF - Faculty of Mathematics and Physics
Publisher: [T. Šnajder]
UDC: 512.64
COBISS: 18512473 Link will open in a new window
Views: 594
Downloads: 180
Average score: 0 (0 votes)
Metadata: JSON JSON-RDF JSON-LD TURTLE N-TRIPLES XML RDFA MICRODATA DC-XML DC-RDF RDF

Other data

Secondary language: English
Secondary title: ADI methods for solving Sylvester equation
Secondary abstract: In master's thesis we focused in solving the Sylvester equation and the Lyapunov equation, as a special case of the Sylvester equation, by using ADI methods. If the matrix dimensions in the Sylvester matrix equation are not too large, then it can be solved by means of direct algorithms, such as the Bartels-Stewart method. When we are solving Sylvester equation with sparse matrices of large dimensions, iterative methods, such as ADI methods, are preferred over direct algorithms. In the thesis the connections between the theory of linear control systems and the Lyapunov equation, as a special case of the Sylvester equation, are first presented.At the same time, the assumptions used in the thesis are also presented. Then the Smith method, the ADI method and some of the most important extensions of the ADI method are presented. Next, the selection of shifts, which determine the rate of convergence of ADI methods,is presented. Some approaches to select the shifts are given. Implementations of algorithms from the previous chapters in Matlab are presented. Comparison of shifts and comparison of methods was obtained for some test cases from the online benchmark collection Slicot.
Secondary keywords: ADI method;Sylvester equation;Lyapunov equation;sparse matrices;Smith method;low rank Cholesky ADI method;factored ADI method;iterative methods;
Type (COBISS): Master's thesis/paper
Study programme: 0
Thesis comment: Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Finančna matematika - 2. stopnja
Pages: 50 str.
ID: 10995515