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

Abstract

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 A −1 0 − A −1 , where A 0 is a nonsingular matrix whose inverse is known or easy to compute, can be factorized in the form U ΩV 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 = 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.Key words. nonsymmetric linear systems, factorized sparse approximate inverses, ShermanMorrison formula, preconditioned iterative methods

An overview of approximate inverse methods

“…Bru et al [6,7,8] proposed an alternative algorithm called NBIF to calculate the factors W and Z based on the Inverse of Sherman-Morrison (ISM)…”

Preconditioners based on the ISM factorization

“…The factorization was introduced in [8] and later developed in several works. During the process some changes in the original notation have been introduced to improve readability.…”

“…In [8] the dropping strategy was based on the size of entries and several results about the incomplete factorization are proved. The most important one is that the incomplete process can be applied without breakdown to M-matrices.…”

“…The Inverse Sherman-Morrison (ISM) decomposition introduced in [8] that can be used to derive preconditioners of both types: approximate inverse preconditioners and incomplete LU or Cholesky preconditioners. In the next section we will review the general framework of the factorization and some theoretical results about the decomposition.…”

Comparison of approximate inverse preconditioners for the fractional step Navier--Stokes equations

“…A number of studies developed sparse approximate preconditioners [5,6,7,8,9]. The most popular are the Sparse Approximate Inverse (spai) method, which is based on the minimisation of the Frobenius norm, and the factorised sparse Approximate Inverse (ainv) method, which is based on the biconjugation algorithm.…”