Preconditioning Sparse Nonsymmetric Linear Systems with the Sherman--Morrison Formula

Abstract

Let Ax=b be a large, sparse, nonsymmetric system of linear equations. A new sparse approximate inverse preconditioning technique for such a class of systems is proposed. We show how the matrix A0-1 - A-1 , where A0 is a nonsingular matrix whose inverse is known or easy to compute, can be factorized in the form $U\Omega V^T$ using the Sherman--Morrison formula. When this factorization process is done incompletely, an approximate factorization may be obtained and used as a preconditioner for Krylov iterative methods. For A0 =sIn, where In is the identity matrix and s is a positive scalar, the existence of the preconditioner for M-matrices is proved. In addition, some numerical experiments obtained for a representative set of matrices are presented. Results show that our approach is comparable with other existing approximate inverse techniques.