An Approximate Matrix Inversion Procedure by Parallelization of the Sherman–morrison Formula

Abstract

The Sherman-Morrison formula is one scheme for computing the approximate inverse preconditioner of a large linear system of equations. However, parallelizing a preconditioning approach is not straightforward as it is necessary to include a sequential process in the matrix factorization. In this paper, we propose a formula that improves the performance of the Sherman-Morrison preconditioner by partially parallelizing the matrix factorization. This study shows that our parallel technique implemented on a PC cluster system of eight processing elements significantly reduces the computational time for the matrix factorization compared with the time taken by a single processor. Our study has also verified that the Sherman-Morrison preconditioner performs better than ILU or MR preconditioners.

Two level parallel preconditioning derived from an approximate inverse based on the Sherman--Morrison formula

“…For these reasons, the aism is not the favored method for computing large sparse linear systems of equations (1). Recently, in order to reduce computation time, Moriya et al [15] proposed a partially parallelized aism based on a technique developed by Naik [19], in which vectors were distributed in all processors that were making computations and communications were carried out in turns.…”

“…Parameter s in the aism was set to 1.5 × A ∞ or 15 × A ∞ . The default value was 1.5× A ∞ [3,15]. The tolerance of U and V was tolU = tolV = tol, where tol was set to 0.1 or 0.01.…”