On the Modified Methods for Irreducible Linear Systems with L-Matrices

Milaszewicz, [Milaszewic J.P, Linear Algebra. Appl. 93,1987, 161$-$170] presented new preconditioner for linear system in order to improve the convergence rates of Jacobi and Gauss-Seidel iterative methods. Li et al.,[Li Y., Li C., Wu S., Appl. Math.Comput. 186, 2007, 379-388] applied this preconditioner and provided convergence theorem for modified AOR method. Yun and Kim [Yun J.H., Kim S.W., Appl. Math. Comput. 201, 2008, 56-64] pointed out some errors in Li et al.'s theorem and provided some correct results for convergence of the preconditioned AOR method. In this paper, we analyze their convergence properly and propose a new theorem for irreducible modified AOR method. In particular, based on directed graph, we prove that the convergence theorem of Li et al. is true, without any additional assumptions.