A robust approximate inverse preconditioner based on the Sherman-Morrison formula∗

To solve a large, and sparse linear system Ax = b; (1) an approximate solution of (1) is usually obtained by using a preconditioned iterative Krylov subspace method [6]. In this work we focus on factorized approximate inverse preconditioners, in this class of preconditioners two matrices such that its product is an approximation of the inverse of A are computed and stored explicitely. Therefore, the application of the preconditioner reduces to a matrix-vector product which can be easily implemented on parallel computers. Recently a new framework for computing sparse approximate inverse preconditioners for nonsymmetric matrices has been presented in [2]. This algorithm, referred to as AISM, computes an approximate inverse of A using the Sherman-Morrison formula. The authors show that its computation is stable for M-matrices and H-matrices [3]. For symmetric positive definite matrices a question which remained open is how to exploit the symmetry in order to avoid the computation of two factors. In this work we answer this question and we present a modification of AISM which is well defined for this class of matrices. In addition, the new formulation seems to be more robust in the nonsymmetric case. The results of numerical experiments performed with the modified approximate inverse preconditioner based on the ShermanMorrison will be presented. We will refer to it as AISMr preconditioner. The matrices used in the test come from the Harwell Boeing collection [5] and Tim Davis’ collection [4]. ∗ Supported by Spanish DGI Grant MTM2004-02998.