Metode ADI za reševanje Sylvestrove enačbe : magistrsko delo

Šnajder, Tadej (author), Plestenjak, Bor (mentor)

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.

• Language: Slovenian
• Keywords: metoda ADI, Sylvestrova enačba, enačba Ljapunova, razpršene matrike, Smithova metoda, metoda ADI nizkega ranga s faktorji Choleskega, faktorizirana metoda ADI, iterativne metode
• Work type: Master's thesis/paper
• Typology: 2.09 - Master's Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2018
• PID: 20.500.12556/RUL-105882
• COBISS.SI-ID: 18512473
• Publication date in RUL: 22.12.2018

Secondary language

• Language: English
• Title: ADI methods for solving Sylvester equation

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.

• Keywords: ADI method, Sylvester equation, Lyapunov equation, sparse matrices, Smith method, low rank Cholesky ADI method, factored ADI method, iterative methods