A note on the preconditioner

Kotakemori et al. [H. Kotakemori, K. Harada, M. Morimoto, H. Niki, A comparison theorem for the iterative method with the preconditioner ( I + S max ) , Journal of Computational and Applied Mathematics 145 (2002) 373–378] have reported that the convergence rate of the iterative method with a preconditioner P m = ( I + S m ) was superior to one of the modified Gauss–Seidel method under the condition. These authors derived a theorem comparing the Gauss–Seidel method with the proposed method. However, through application of a counter example, Wen Li [Wen Li, A note on the preconditioned GaussSeidel (GS) method for linear systems, Journal of Computational and Applied Mathematics 182 (2005) 81–91] pointed out that there exists a special matrix that does not satisfy this comparison theorem. In this note, we analyze the reason why such a to counter example may be produced, and propose a preconditioner to overcome this problem.