The convergence of the modified Gauss–Seidel methods for consistent linear systems

In this paper we present a convergence analysis for the modified Gauss–Seidel methods given in Gunawardena et al. (Linear Algebra Appl. 154–156 (1991) 125) and Kohno et al. (Linear Algebra Appl. 267 (1997) 113) for consistent linear systems. We prove that the modified Gauss–Seidel method converges for some values of the parameters in the preconditioned matrix.