A SPARSE APPROXIMATE INVERSE PRECONDITIONER FOR NONSYMMETRIC LINEAR SYSTEMS ∗ MICHELE BENZI † AND MIROSLAV T ˚ UMA ‡ SIAM J. SCI.COMPUT. c  1998 Society for Industrial and Applied Mathematics Vol. 19, No. 3, pp. 968–994, May 1998 012 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 non- symmetric 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 precon- ditioner 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. Key words. preconditioning, approximate inverses, sparse linear systems, sparse matrices, incomplete factorizations, conjugate gradient–type methods AMS subject classifications. 65F10, 65F35, 65F50, 65Y05 PII. S1064827595294691 1. Introduction. In this paper we consider the solution of nonsingular linear systems of the form Ax = b, (1) where the coefficient matrix A ∈ R n×n is large and sparse. In particular, we are con- cerned with the development of preconditioners for conjugate gradient–type methods. It is well known that the rate of convergence of such methods for solving (1) is strongly influenced by the spectral properties of A. It is therefore natural to try to transform the original system into one having the same solution but more favorable spectral properties. A preconditioner is a matrix that can be used to accomplish such a trans- formation. If G is a nonsingular matrix which approximates A −1 (G ≈ A −1 ), the transformed linear system GAx = Gb (2) will have the same solution as system (1) but the convergence rate of iterative methods applied to (2) may be much higher. Problem (2) is preconditioned from the left, but right preconditioning is also possible. Preconditioning on the right leads to the transformed linear system AGy = b. (3) Once the solution y of (3) has been obtained, the solution of (1) is given by x = Gy. The choice between left or right preconditioning is often dictated by the choice of the * Received by the editors November 10, 1995; accepted for publication (in revised form) June 25, 1996. http://www.siam.org/journals/sisc/19-3/29469.html † Scientific Computing Group (CIC-19), MS B256, Los Alamos National Laboratory, Los Alamos, NM 87545 (benzi@lanl.gov). The work of this author was supported in part by a grant under the scientific cooperation agreement between Italy’s CNR and the Czech Academy of Sciences. ‡ Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod vod´arenskou vˇ eˇ z´ ı 2, 182 07 Prague 8 - Libeˇ n, Czech Republic (tuma@uivt.cas.cz). The work of this author was supported in part by grants GA CR 201/93/0067 and GA AS CR 230401 and by NSF grant INT-9218024. 968