Numeričko rješavanje linearnih matričnih jednadžbi

master's thesis

• Author: Jasmina Šestan
• Mentor: Zvonimir Bujanović (mentor)
• Committee member: Zvonimir Bujanović (predsjednik povjerenstva)
• Committee member: Boris Muha (član povjerenstva)
• Committee member: Josip Tambača (član povjerenstva)
• Committee member: Nenad Antonić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2016-04-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom smo radu dali pregled postojećih rezultata u području numeričkog rješavanja linearnih matričnih jednadžbi na primjeru Sylvesterove i Lyapunovljeve jednadžbe. Naveli smo zašto su od interesa u primijenjenoj matematici, posebno u analizi stabilnosti dinamičkih sustava. Razmotrili smo cesto korišten Bartels–Stewartov algoritam koji je pogodan za rješavanje jednadžbi malih dimenzija. Za matrične jednadžbe velikih dimenzija razmotrili smo iterativne načine rješavanja: metodu ADI iteracija i projekcijsku Krylovljevu metodu. Pritom smo uzeli u obzir strukture matrica koje se često pojavljuju u primjenama. Implementaciju ovih algoritama smo napravili u Pythonu.

Abstract (english)

In this work we reviewed main computational methods for solving linear matrix equations, particularly Sylvester and Lyapunov equations. We explained their role in applied mathematics, especially in stability analysis of linear dynamic systems. We described Bartels–Stewart algorithm which is frequently used for solving matrix equations of small dimensions. For large-scale matrix equations, we considered special structure of matrices which are often encountered in applications and we described iterative methods for finding solution: ADI method and projection (Krylov) method. We implemented listed algorithms using Python.

• Keywords: numeričko rješavanje linearnih matričnih jednadžbi; Sylvesterova jednadžba; Lyapunovljeva jednadžba; Bartels–Stewartov algoritam; metoda ADI; projekcijska Krylovljeva metoda; Python
• Keywords (english): computational methods for solving linear matrix equations; Sylvester equation; Lyapunov equation; Bartels–Stewart algorithm; ADI method; projection (Krylov) method; Python
• Language: croatian
• URN:NBN: urn:nbn:hr:217:559029
• Study programme: Title: Computer Science and Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra računarstva i matematike (magistar/magistra računarstva i matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-02-13 11:14:27