S/P Images of Upper Triangular M-Matrices

Preconditioning by approximate factorizations is widely used in iterative methods for solving linear systems such as those arising from the finite element formulation of many engineering problems. The influence of the ordering of the unknowns on their convergence behaviour has been the subject of recent investigations because of its particular relevance for the parallel implementation of these methods. Consistent orderings are attractive for parallel implementations and subclasses of these orderings have been shown to also enhance the convergence properties of the associated preconditioned iteration scheme, The present contribution is concerned with one such class of orderings, called SIP consistent orderings.. More precisely, we review here their known properties and we propose a new definition which enlarges their scope of application. A device, called SIP image of an upper triangular M-matrix, provides a criterion for checking S/P consistency and a means to compute a relevant parameter, called maximal reduction ratio. All known properties of S/P consistent orderings are generalized to the new definition.