Two-Stage and Multisplitting Methods for the Parallel Solution of Linear Systems

Abstract

Two-stage and multisplitting methods for the parallel solution of linear systems are studied. A two-stage multisplitting method is presented that reduces to each of the others in particular cases. Conditions for its convergence are given. In the particular case of a multisplitting method related to block Jacobi, it is shown that it is equivalent to a two-stage method with only one inner iteration per outer iteration. A fixed number of iterations of this method, say, p, is compared with a two-stage method with p inner iterations. The asymptotic rate of convergence of the first method is faster, but, depending on the structure of the matrix and the parallel architecture, it takes more time to converge. This is illustrated with numerical experiments.