1. Description<br />Given a square system of n linear equations with unknown x:<br />where:<br />Then A can be decomposed into a lower triangular component L*, and a strictly upper triangular component U:<br />The system of linear equations may be rewritten as:<br />The Gauss–Seidel method is an iterative technique that solves the left hand side of this expression for x, using previous value for x on the right hand side. Analytically, this may be written as:<br />However, by taking advantage of the triangular form of L*, the elements of x(k+1) can be computed sequentially using forward substitution:<br />Note that the sum inside this computation of xi(k+1) requires each element in x(k) except xi(k) itself.<br />The procedure is generally continued until the changes made by an iteration are below some tolerance.<br />