Abstract. This paper is concerned with a new approach to preconditioning for large, sparse linear systems. A procedure for computing an incomplete factorization of the inverse of a nonsymmetric matrix is developed, and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient-type methods. Some theoretical properties of the preconditioner are discussed, and numerical experiments on test matrices from the Harwell-Boeing collection and from Tim Davis's collection are presented. Our results indicate that the new preconditioner is cheaper to construct than other approximate inverse preconditioners. Furthermore, the new technique insures convergence rates of the preconditioned iteration which are comparable with those obtained with standard implicit preconditioners.

Abstract. This paper is concerned with a new approach to preconditioning for large, sparse linear systems. A procedure for computing an incomplete factorization of the inverse of a nonsymmetric matrix is developed, and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient-type methods. Some theoretical properties of the preconditioner are discussed, and numerical experiments on test matrices from the Harwell-Boeing collection and from Tim Davis's collection are presented. Our results indicate that the new preconditioner is cheaper to construct than other approximate inverse preconditioners. Furthermore, the new technique insures convergence rates of the preconditioned iteration which are comparable with those obtained with standard implicit preconditioners.

A number of recently proposed preconditioning techniques based on sparse approximate inverses are considered. A description of the preconditioners is given, and the results of an experimental comparison performed on one processor of a Cray C98 vector computer using sparse matrices from a variety of applications are presented. A comparison with more standard preconditioning techniques, such as incomplete factorizations, is also included. Robustness, convergence rates, and implementation issues are discussed.

Abstract. We present a variant of the AINV factorized sparse approximate inverse algorithm which is applicable to any symmetric positive definite matrix. The new preconditioner is breakdownfree and, when used in conjunction with the conjugate gradient method, results in a reliable solver for highly ill-conditioned linear systems. We also investigate an alternative approach to a stable approximate inverse algorithm, based on the idea of diagonally compensated reduction of matrix entries. The results of numerical tests on challenging linear systems arising from finite element modeling of elasticity and diffusion problems are presented.Key words. sparse linear systems, finite element matrices, preconditioned conjugate gradients, factorized sparse approximate inverses, incomplete conjugation, stabilized AINV, diagonally compensated reduction AMS subject classifications. Primary, 65F10, 65N22, 65F50; Secondary, 15A06 PII. S10648275993569001. Introduction. We consider the solution of sparse linear systems Ax = b, where A is a symmetric and positive definite (SPD) matrix, by the preconditioned conjugate gradient method. In the last few years there has been considerable interest in explicit preconditioning techniques based on directly approximating A −1 with a sparse matrix M ; see, e.g., [7] Although the main motivation for the development of sparse approximate inverse preconditioners comes from parallel processing, it is becoming clear that these techniques are also of interest because of their robustness. Sparse approximate inverses are often applicable to difficult problems where other preconditioners may break down [4]. For instance, incomplete factorization preconditioners, while widely popular and fairly robust, are not always reliable, in that the incomplete factorization process may