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.

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In this note preconditioners for the Conjugate Gradient method are studied to solve the Newton system with a symmetric positive definite Jacobian. In particular, we define a sequence of preconditioners built by means of BFGS rank-two updates. Reasonable conditions are derived which guarantee that the preconditioned matrices are not far from the identity in a matrix norm. Some notes on the implementation of the corresponding inexact Newton method are given and some numerical results on a number of model problems illustrate the efficiency of the proposed preconditioners

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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