Consistency and Convergence of the Parallel Multisplitting Method for Singular M-Matrices

O’Leary and White have suggested a parallel multisplitting iteration scheme for solving a non-singular linear system $Ax = b$. Among other things they have shown that when A has a nonnegative inverse and the multisplitting is weak regular, then the iteration converges to the solution from any initial vector. The extension of this result to the case where A is a singular M-matrix is discussed. Problems of solvability, consistency, and convergence arise and their resolution is considered.