A comparison theorem for the SOR iterative method (Q557697)

This paper is concerned with solving a linear system \(Ax=b\), where \(A\) is a non-singular \(M\)-matrix, by the Gauss-Seidel method using a preconditioner of the form \(P=I+S\), where \(S\) is composed of the (scaled) subdiagonal entries of \(A\). It is shown that the spectral radius of the resulting iteration matrix is smaller than the one corresponding to the successive overrelaxation (SOR) method, provided that the relaxation parameter \(\omega\) satisfies \(0 < \omega \leq 1\).