Two new modified Gauss–Seidel methods for linear system with M-matrices

In 2002, H. Kotakemori et al. proposed the modified Gauss–Seidel (MGS) method for solving the linear system with the preconditioner P=I+Smax [H. Kotakemori, K. Harada, M. Morimoto, H. Niki, A comparison theorem for the iterative method with the preconditioner (I+Smax) J. Comput. Appl. Math. 145 (2002) 373–378]. Since this preconditioner is constructed by only the largest element on each row of the upper triangular part of the coefficient matrix, the preconditioning effect is not observed on the nth row. In the present paper, to deal with this drawback, we propose two new preconditioners. The convergence and comparison theorems of the modified Gauss–Seidel methods with these two preconditioners for solving the linear system are established. The convergence rates of the new proposed preconditioned methods are compared. In addition, numerical experiments are used to show the effectiveness of the new MGS methods.