Postoje dva pristupa rješavanja sustava linearnih jednadžbi Ax = b gdje za zadanu matricu \(A\in\mathbb{R}^{m\times n}\) i vektor \(b\in\mathbb{R}\) treba odrediti vektor \(x\in\mathbb{R}^n\). U prvi spadaju takozvane direktne metode kod kojih se matrica A prikazuje kao produkt jednostavnijih matrica, najćešće trokutastih. To omogućava rješavanje jednostavnijih linearnih sustava, te dolazak do egzaktnog rješenja, pri čemu je složenost, odnosno broj potrebnih operacija \(O(n^3)\). U drugi pristup spadaju iterativne metode, kod kojih je pogreška \( ||x^{k}− x||\) dovoljno mala nakon znatno manje operacija od \(O(n^3)\). One ne određuju rješenje direktno nego iterativno (postupno), a koriste se pri rješavanju velikih, rijetko popunjenih sustava linearnih jednadžbi, koji imaju puno nula. Takvim se metodama konstruira niz \(x^{(k)}\), \(k\in\mathbb{N}\) za koji vrijedi \(x^{(k)}\rightarrow x\) za \(k\rightarrow \infty\). Osnovne iterativne metode za rješavanje linearnih sustava jednadžbi su Jacobijeva metoda, te Gauss-Seidelova metoda. Metoda sukcesivne nadrelaksacije, kraće SOR metoda, je zapravo proširenje ili poboljšanje Gauss-Seidelove metode u smislu da se pomoćna nova aproksimacija \(x^{(k+1)}\) računa po Gauss-Seidelovoj metodi, a sve to sa svrhom poboljšanja konvergencije. Cilj modifikacije nije proširiti klasu matrica za koje iterativna metoda konvergira već ubrzati konvergenciju, što je i pokazano na primjerima.

There are two approaches to solve systems of linear equations Ax = b where for the matrix \(A\in\mathbb{R}^{m\times n}\) and vector \(b\in\mathbb{R}\) we have to determine vector \(x\in\mathbb{R}^n\). The first are so-called direct method in which the matrix A is presented as a product of simpler matrices, usually triangular. It allows us to solve simpler linear systems, and to come to the exact solution, where the complexity, or the number of necessary operations is \(O(n^3)\) . The second approach includes iterative methods, in which the error \( ||x^{k}− x||\) is small enough after much less operations then \(O(n^3)\) . Such methods do not define the solution directly, but iteratively, gradually, and we use them for solving big sparse systems of linear equations, who have a lot of zeros. With method like that, we construct sequence \(x^{(k)}\), \(k\in\mathbb{N}\) to which \(x^{(k)}\rightarrow x\) for \(k\rightarrow \infty\) The basic iterative methods for solving linear equation systems are Jacobi’s method, and Gauss-Seidel’s method. Successive over - relaxation method, shorter SOR method, is actually an extension or improvement of Gauss-Seidel method in that the auxiliary new approximation \(x^{(k+1)}\) is calculated by Gauss-Seidel method, and all that with purpose for improving the convergence. The goal of the modification was not to extend the class matrix for the iterative method converges, but speed up the convergence, as it is shown in examples.