Abstract
In this note, we prove that the convergence rate of the modified Gauss-Seidel (MGS) method with preconditional I+Sa is a monotonic function of preconditioning parameter a. Based on this result, to achieve better convergence rate we suggest proforming twice preconditoning when applying the MGS method to solve a linear system whose coefficient matrix is an irreducible non-singular M-matrix.
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