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 to the time taken by one processor. Our study has also verified that the Sherman--Morrison preconditioner performs better than ILU or MR preconditioners. doi:10.1017/S1446181109000364