Numeričko rješavanje linearnih sustava

master's thesis

• Author: Karla Mikec
• Mentor: Marko Erceg (mentor)
• Committee member: Marko Erceg (predsjednik povjerenstva)
• Committee member: Marko Vrdoljak (član povjerenstva)
• Committee member: Miljenko Huzak (član povjerenstva)
• Committee member: Nenad Antonić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-09-24, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Za rješavanje linearnih sustava jednadžbi postoji nekoliko metoda. Neke od njih su direktne (Gaussove eliminacije, LR faktorizacija) i daju točno rješenje do na pogrešku u zaokruživanju računala, ali nisu praktične za veće sustave jer imaju kubnu složenost. S druge strane su iterativne metode (Jacobijeva, Gauss-Seidelova) koje su brže i efikasne za veće sustave, ali daju aproksimaciju rješenja i u nekim slučajevima sporo konvergiraju prema egzaktnom rješenju. Odabir bolje metode ovisi o danom problemu. U ovom radu su obrađene neke direktne i iterativne metode, te je prezentirana njihova usporedba na primjeru rješavanja diferencijalne jednadžbe. Svi prezentirani algoritmi zapisani su u programskom jeziku Python 3.6.

Abstract (english)

There are several methods for solving linear systems of equations. Some of them are direct (Gaussian Elimination, LR Decomposition) and provide the exact solution neglecting the roundoff error caused by the computer, but they are not practical for solving large systems because their complexity is cubic. On the other side, there are iterative methods (Jacobi, Gauss-Seidel) which are faster and more efficient to use in large systems, but they give approximate solutions and their convergence to the exact solution can be slow. Choosing a better method depends on a given problem. This thesis deals with some of direct and iterative methods, and shows their comparison on the example of solving differential equations. All presented algorithms are written in a programming language Python 3.6.

• Keywords: linearni sustav jednadžbi; direktne metode rješavanja; iterativne metode rješavanja; Python
• Keywords (english): linear system of equations; direct solving methods; iterative solving methods; Python
• Language: croatian
• URN:NBN: urn:nbn:hr:217:967905
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-06-07 12:41:24