We can generalize x(k+1)1=(b1−a12x(k)2−a13x(k)3−a14x(k)4)/a11x(k+1)2=(b2−a21x(k+1)1−a23x(k)3−a24x(k)4)/a22x(k+1)3=(b3−a31x(k+1)1−a32x(k+1)2−a34x(k)4)/a33x(k+1)4=(b4−a41x(k+1)1−a42x(k+1)2−a43x(k+1)3)/a44, to the following form x(k+1)i=1aii(bi−∑j>iaijx(k)j−∑j<iaijx(k+1)j),i=1,2,…,n. The procedure is generally continued until the changes made by an iteration are below some tolerance.
The convergence properties of the Jacobi method and the Gauss-Seidel method are dependent on the matrix A. These methods converge when the matrix is symmetric positive-definite, or is strictly or irreducibly diagonally dominant. Both methods sometimes converge even if these conditions are not satisfied.