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

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Let Ax = b be a large and sparse system of linear equations where A is a nonsingular matrix. An approximate solution is frequently obtained by applying preconditioned iterations. Consider the matrix B = A + P Q T where P , Q ∈ R n×k are full rank matrices. In this work, we study the problem of updating a previously computed preconditioner for A in order to solve the updated linear system Bx = b by preconditioned iterations. In particular, we propose a method for updating a Balanced Incomplete Factorization preconditioner. The strategy is based on the computation of an approximate Inverse Sherman-Morrison decomposition for an equivalent augmented linear system. Approximation properties of the preconditioned matrix and an analysis of the computational cost of the algorithm are studied. Moreover, the results of the numerical experiments with different types of problems show that the proposed method contributes to accelerate the convergence.

We consider the numerical solution of linear systems arising from computational electromagnetics applications. For large scale problems the solution is usually obtained iteratively with a Krylov subspace method. It is well known that for ill conditioned problems the convergence of these methods can be very slow or even it may be impossible to obtain a satisfactory solution. To improve the convergence a preconditioner can be used, but in some cases additional strategies are needed. In this work we study the application of spectral lowrank updates (SLRU) to a previously computed sparse approximate inverse preconditioner. The updates are based on the computation of a small subset of the eigenpairs closest to the origin. Thus, the performance of the SLRU technique depends on the method available to compute the eigenpairs of interest. The SLRU method was first used using the IRA's method implemented in ARPACK. In this work we investigate the use of a Jacobi-Davidson method, in particular its JDQR variant . The results of the numerical experiments show that the application of the JDQR method to obtain the spectral low-rank updates can be quite competitive compared with the IRA's method.

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