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Iterative methods, Gauss-Seidel's method

Our 4×4 matrix problem x(k+1)1=(b1a12x(k)2a13x(k)3a14x(k)4)/a11x(k+1)2=(b2a21x(k)1a23x(k)3a24x(k)4)/a22x(k+1)3=(b3a31x(k)1a32x(k)2a34x(k)4)/a33x(k+1)4=(b4a41x(k)1a42x(k)2a43x(k)3)/a44, can be rewritten as 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, which allows us to utilize the preceding solution (forward substitution). This improves normally the convergence behavior and leads to the Gauss-Seidel method!