Preconditioning sparse nonsymmetric linear systems with the Sherman-Morrison formula

Let Ax = b be a large, sparse, nonsymmetricsystem of linear equations. A new sparse approximate inverse preconditioning technique for such a class of systems is proposed. We show how the matrix A(0)(-1) - A(-1), where A(0) is a nonsingular matrix whose inverse is known or easy to compute, can be factorized in the form UOmegaV(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 A(0) = sI(n), where I-n 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.