Let M(A) denote the comparison matrix of a square H-matrix A, that is, M(A) is an M -matrix. H-matrices such that their comparison matrices are non-singular are well studied in the literature. In this paper, we study characterizations of H-matrices with singular or nonsingular comparison matrix. In particular, we analyze the case when A is irreducible and then give insights into the reducible case. The spectral radius of the Jacobi matrix of M(A) and the generalized diagonal dominance property are used in the characterizations. Finally, from these characterizations, a partition of the general H-matrix set in three classes is obtained.

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

Abstract. In this paper we improve the BIF algorithm which computes simultaneously the LU factors (direct factors) of a given matrix, and their inverses (inverse factors). This algorithm was introduced in [R. Bru, J. Marín, J. Mas and M. Tůma, SIAM J. Sci. Comput., 30 (2008Comput., 30 ( ), pp. 2302Comput., 30 ( -2318. The improvements are based on a deeper understanding of the Inverse Sherman-Morrison (ISM) decomposition and they provide a new insight into the BIF decomposition. In particular, it is shown that a slight algorithmic reformulation of the basic algorithm implies that the direct and inverse factors influence numerically each other even without any dropping for incompleteness. Algorithmically, the nonsymmetric version of the improved BIF algorithm is formulated. Numerical experiments show very high robustness of the incomplete implementation of the algorithm used for preconditioning nonsymmetric linear systems.

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Abstract. New preconditioning strategies for solving m × n overdetermined large and sparse linear least squares problems using the conjugate gradient for least squares (CGLS) method are described. First, direct preconditioning of the normal equations by the balanced incomplete factorization (BIF) for symmetric and positive definite matrices is studied, and a new breakdown-free strategy is proposed. Preconditioning based on the incomplete LU factors of an n × n submatrix of the system matrix is our second approach. A new way to find this submatrix based on a specific weighted transversal problem is proposed. Numerical experiments demonstrate different algebraic and implementational features of the new approaches and put them into the context of current progress in preconditioning of CGLS. It is shown, in particular, that the robustness demonstrated earlier by the BIF preconditioning strategy transfers into the linear least squares solvers and the use of the weighted transversal helps to improve the LU-based approach.