A comparison theorem for the iterative method with the preconditioner (I+Smax)

In 1991, Gunawardena et al. (Linear Algebra Appl. 154–156 (1991) 123) have reported the modified Gauss–Seidel method with a preconditioner (I+S). In this article, we propose to use a preconditioner (I+Smax) instead of (I+S). Here, Smax is constructed by only the largest element at each row of the upper triangular part of A. By using the lemma established Neumann and Plemmons (Linear Algebra Appl. 88/89 (1987) 559), we get the comparison theorem for the proposed method. Simple numerical examples are also given.