On parallel multisplitting methods for symmetric positive semidefinite linear systems

Abstract

In this paper we propose some parallel multisplitting methods for solving consistent symmetric positive semidefinite linear systems, based on modified diagonally compensated reduction. The semiconvergence of the parallel multisplitting method is discussed. The results here generalize some known results for the nonsingular linear systems to the singular systems. It is well known that if B is nonsingular, then a splitting B = M − N is convergent if and only if (M −1 N )<1. However, for the singular matrix B we have (M −1 N ) = 1, so that we can only require the semiconvergence of the splitting. By [24, Lemma (7-6.13), Exercise (6-4.9); 25, 30-32] the splitting B = M − N is semiconvergent if and only if the iteration matrix T = M −1 N satisfies

Semiconvergence of parallel multisplitting methods for symmetric positive semidefinite linear systems

“…In 14, we give a modified diagonal compensated reduction method, which is analogous to the method of diagonal compensated reduction proposed in 13.…”

“…In 13 a diagonally compensated reduction method for singular systems is proposed. Recently, we present in 14 a modified diagonal compensated reduction method. Using this idea, we investigate semiconvergence of the parallel multisplitting method.…”

On parallel multisplitting methods for non-Hermitian positive definite linear systems

“…Later, this technique was further studied by many authors; see e.g. [3], [14], [15], [23], [34], [35], [37], [16], [7], [8], [22], [12], [27], [28], [29], [20], [9], [39], [40], [41] [32], [33], [10], etc.…”

“…Conditions on the splittings (1.2) and on the weighting matrices which ensure the convergence of Algorithm 1.1 in some important cases where given by O'Leary and White [24], Nabben [22], Neumann and Plemmons [23], Frommer et al [14], [15], Song et al [27], [28], [29], Li et el [20], Hadjidimos and Yeyios [16], Cao and Song [7], etc. They showed that Algorithm 1.1 (semi)converges when • A is Hermitian (or symmetric) positive definite and the splittings (1.2) are P −regular; • A is monotone and the splittings (1.2) are (weak) regular;…”