Convergent Regular Splittings for Singular M-Matrices

A classic theorem of numerical linear algebra (Varga’s Theorem) says that any regular splitting of an M-matrix, A, is convergent. However when A is a singular M-matrix, such as might arise in queuing networks, this general theorem needs some modification. We define a class of block splittings, called R-regular splittings, and show that any R-regular splitting is convergent. Furthermore, given a block splitting like block Gauss-Seidel but not quite as special, we show how to tinker slightly to make the splitting convergent. We show also that some natural splittings require no tinkering at all. In an algorithmic context, our results indicate how to choose and order the blocks in a block iterative method for solving a singular M-matrix system of linear equations so as to insure convergence of the method.