Postoje dva pristupa rješavanju sustava linearnih jednadžbi Ax = b gdje za zadanu matricu \(A\in \mathbb{R}^{m \times n}\) i vektor \(b \in \mathbb{R}^{m}\) treba odrediti vektor \(x \in \mathbb{R}^{n}\) . U prvi spadaju takozvane direktne metode kod kojih se matrica A prikazuje kao produkt jednostavnijih matrica, najčešće trokutastih. To omogućava rješavanje jednostavnijih linearnih sustava, te dolazak do egzaktnog rješenja pri čemu je složenost, odnosno broj potrebnih aritmetičkih operacija za dolazak do egzaktnog rješenja \(O(n^{3})\) U drugi pristup spadaju iterativne metode koje u pravilu ne mogu izračunati u potpunosti točan rezultat u konačnom broju koraka, već su konstruirane tako da počinju s aproksimacijom rješenja i svakim svojim korakom smanjuju odstupanje, tj. udaljenost od točnog rješenja, odnosno pogrešku u aproksimaciji rješenja. Kod njih je pogreška u izračunu aproksimacije rješenja dovoljno mala već nakon znatno manje operacija od \(O(n^{3})\) . Dakle, one ne određuju rješenje direktno nego iterativno (postupno), a koriste se pri rješavanju velikih, rijetko popunjenih sustava linearnih jednadžbi. Takvim se metodama konstruira niz \(x^{(k)}\) , k ∈ N za koji vrijedi \(x^{(k)}\rightarrow x\) za \(k\rightarrow \infty\). Najčešće korištene iterativne metode su Jacobijeva, Gauss-Seidelova te metoda konjigiranih gradijenata.

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 \(b \in \mathbb{R}^{m}\) we have to determine vector \(x \in \mathbb{R}^{n}\) . The first are so-called direct methods 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 required arithmetic operations for getting exact solution is \(O(n^{3})\) . The second approach includes iterative methods that can not in generally calculate a completely accurate solution in the final number of steps, but are designed to begin with the approximation of the solution and with each step to reduce the deviation, ie. the distance from the exact solution, or the error in the approximation of the solution. The error in calculated approximation of the solution is small enough even after significantly less operations than \(O(n^{3})\) . Such methods do not obtain the solution directly, but iteratively (gradually), and they are usually used to solve big sparse systems of linear equations. With method like that, we construct sequence niz \(x^{(k)}\) , k ∈ N to which it applies \(x^{(k)}\rightarrow x\) for \(k\rightarrow \infty\) . The most commonly used iterative methods are Jacobi method, Gauss-Seidel method, and the method of conjugated gradients.