System of linear equations, iterative methods

Linearne jednadžbe su važan dio naše svakodnevice i s njima se možemo susresti u svim granama znanosti, stoga ih je potrebno razumjeti te biti u mogućnosti što točnije i efikasnije rješavati. Tu dolazimo do primjene iteracijskih metoda, koje su se pokazale kao odličan način za to. Uz njihove mane kao što su približno točno rješenje, nepoznat broj koraka u početku, mogućnost divergencije, imamo i brojne pozitivne strane kao što je da se lako programiraju, imaju mnoštvo jednostavnih koraka, zaobilaze dugotrajno računanje s kojim bi se susreli kod direktnih metoda i tome slično. U ovom radu prikazan je način rješavanja sustava linearnih jednadžbi iterativnim metodama: Jacobijevom metodom, Gauss-Seidelovom metodom te metodom relaksacije. Na primjeru je napravljena usporedba kako se na istom problemu ponaša Jacobijeva, a kako Gauss-Seidlova metoda.Linear equations are an important part of our everyday life and we can encounter them in almost every branch of science. Therefore, it is necessary to understand linear equations to a great extent and be able to solve them as accurately and efficiently as possible. Iterative methods have been proven to be an excellent way of dealing with system of linear equations. Although iterative methods have flaws like giving approximation of exact solution, unknown number of steps in the beginning and the possibility of divergence they also have many advantages: they are easily programmable, made of many simple steps, they avoid long lasting calculations that are characteristic for direct methods etc. In this paper, the method of solving a system of linear equations by iterative methods is presented: Jacobi method, Gauss-Seidel method and relaxation method. On the example, a comparison is made of how Jacobi's and Gauss-Seidel's methods behave on the same problem