Convergence of parallel multisplitting iterative methods for M-matrices

Abstract

Despite the major recent activity in parallel processing, few effective new algorithms designed exclusively for multiprocessors have been put forth. One such is the multisplitting iterative algorithm suggested by O'Leary and White. Although they have given some sufficient conditions for convergence, a general convergence theory has not been developed even for the classical situation where the coefficient matrix A is an M-matrix or is symmetric positive definite. In this paper we study the M-matrix case in detail. The multisplitting process for A ∈ R n , n is recast as an ordinary iterative process for a certain block matrix A ∈R kn, kn , where k is the number of processors, and standard convergence results are used to develop a convergence theory for multisplitting iterative methods where A is an M-matrix. Comparison results between multisplitting methods are established in terms of monotonic norms and, for the case where A is irreducible, in terms of the asymptotic convergence rate. A key observation here is that in certain cases the rate of global convergence of these parallel iterative methods is inherent in the splitting of A and is independent of the manner in which the work is distributed among the processors. Thus in general one can distribute the work for load balancing purposes without affecting the convergence rate.