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Iterative methods, Gauss-Seidel's method

We can generalize x(k+1)1=(b1a12x(k)2a13x(k)3a14x(k)4)/a11x(k+1)2=(b2a21x(k+1)1a23x(k)3a24x(k)4)/a22x(k+1)3=(b3a31x(k+1)1a32x(k+1)2a34x(k)4)/a33x(k+1)4=(b4a41x(k+1)1a42x(k+1)2a43x(k+1)3)/a44, to the following form x(k+1)i=1aii(bij>iaijx(k)jj<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.