Convergence of generalized relaxed multisplitting methods for symmetric positive definite matrices

Abstract

Some generalized relaxed parallel multisplitting methods for the solution of a symmetric positive definite linear system Ax= b are considered. In this paper, in particular: (1) The diagonally compensated reduction (cf. [Numer. Linear Algebra Appl. 1 (1994) 155–177]) is applied to the multisplitting methods for an s.p.d. (symmetric positive definite) matrix. Here the s.p.d. matrix A need not be assumed in a special form (e.g., the dissection form [Linear Algebra Appl. 119 (1989) 141–152]). (2) We investigate several different variants of relaxed multisplitting methods. (3) We come to the conclusion that these relaxed methods converge if the relaxation parameter is from an interval (0, ω 0) with ω 0>1. (4) We establish some comparison results between multisplittings in terms of the asymptotic convergence rate.