Monotone convergence of iterative methods for singular linear systems

In this paper, monotonicity of iterative methods for solving general solvable singular systems is discussed. The monotonicity results given by Berman, Plemmons, and Semal are generalized to singular systems. It is shown that for an iterative method introduced by a nonnegative splitting of the coefficient matrix there exist some initial guesses such that the iterative sequence converges towards a solution of the system from below or from above. The monotonicity of the block Gauss–Seidel method for solving a p-cyclic system and Markov chains is considered.