20.500.12556/RUL-184576 Kaczmarzova metoda za reševanje sistemov enačb delo diplomskega seminarja Kaczmarz method for solving systems of equations V delu diplomskega seminarja obravnavamo Kaczmarzovo iterativno metodo za reševanje sistemov linearnih enačb. Predstavimo izvirno ciklično metodo in dokažemo njeno konvergenco, nato pa opišemo več novejših različic, ki se razlikujejo po načinu izbiranja vrstic matrike, kot so naključna, požrešna in bločna Kaczmarzova metoda ter metodi z množico kandidatov in s povprečenjem. Posebno pozornost namenimo uporabi metode v računalniški tomografiji, kjer problem rekonstrukcije slike prevedemo na sistem linearnih enačb in izpeljemo algebraično rekonstrukcijsko tehniko. Obravnavamo tudi vpliv šuma, parameter relaksacije in polkonvergenco. V delo vključimo več primerov, implementiranih v jeziku MATLAB, s katerimi primerjamo konvergenco obravnavanih različic. In this diploma seminar thesis, we study the Kaczmarz iterative method for solving systems of linear equations. We present the original cyclic method and prove its convergence, and then describe several newer variants that differ in the way the rows of the matrix are selected, such as the randomized, greedy, and block Kaczmarz methods, as well as the methods with a selectable set and with averaging. We pay special attention to the application of the method in computed tomography, where we reduce the problem of image reconstruction to a system of linear equations and derive the algebraic reconstruction technique. We also address the effect of noise, the relaxation parameter, and semiconvergence. We include several examples implemented in MATLAB, with which we compare the convergence of the discussed variants. iterativna metoda sistem linearnih enačb Kaczmarzova metoda stopnja konvergence računalniška tomografija šum iterative method system of linear equations Kaczmarz method rate of convergence computed tomography noise true false false Slovenski jezik Angleški jezik Diplomsko delo/naloga 2026-07-10 08:15:07 2026-07-10 08:15:15 2026-07-15 09:42:51 0000-00-00 00:00:00 2026 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.6 160637 284850179 22044.pdf 22044.pdf 1 0733F1AC5BBEE5757C38871CA0A49E02 7c56079402f4cf566e8d6cd70ad0b82c6dc1e573a6a35f86c3f5d2912389f123 97629438-7c26-11f1-9b0d-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=237466 Fakulteta za matematiko in fiziko 0 0 0