Semiconvergence of nonnegative splittings for singular matrices

In this paper, we discuss semiconvergence of the matrix splitting methods for solving singular linear systems. The concepts that a splitting of a matrix is regular or nonnegative are generalized and we introduce the terminologies that a splitting is quasi-regular or quasi-nonnegative. The equivalent conditions for the semiconvergence are proved. Comparison theorem on convergence factors for two different quasi-nonnegative splittings is presented. As an application, the semiconvergence of the power method for solving the Markov chain is derived. The monotone convergence of the quasi-nonnegative splittings is proved. That is, for some initial guess, the iterative sequence generated by the iterative method introduced by a quasi-nonnegative splitting converges towards a solution of the system from below or from above.