A comparison theorem for the SOR iterative method

In 1997, Kohno et al. have reported numerically that the improving modified Gauss–Seidel method, which was referred to as the IMGS method, is superior to the SOR iterative method. In this paper, we prove that the spectral radius of the IMGS method is smaller than that of the SOR method and Gauss–Seidel method, if the relaxation parameter ω ∈ ( 0 , 1 ] . As a result, we prove theoretically that this method is succeeded in improving the convergence of some classical iterative methods. Some recent results are improved.