Rafael Bru

Models of parallel chaotic iteration methods

Abstract We consider two models of parallel multisplitting chaotic iterations for solving large nonsingular systems of equations Ax = b. In the first model each processor can carry out an arbitrary number of local iterations before the next global approximation to the solution is formed. In the second model any processor can update the global approximation which resides in the central processor at any time. This model is a generalization of a sequential iterative scheme due to Ostrowski called the free steering group Jacobi iterative scheme and a chaotic relaxation point iterative scheme due to Chazan and Miranker. We show that when A is a monotone matrix and all the splittings are weak regular, both models lead to convergent schemes.

Structural properties of positive linear time-invariant difference-algebraic equations☆

In this paper we analyze positive difference-algebraic equations. Conditions regarding some of the matrices involved in the solution of this kind of systems are described. Geometrical conditions are used to characterize positive structural properties.

Canonical forms for positive discrete-time linear control systems☆

Abstract In this paper, the properties of reachability, controllability and essential reachability of positive discrete-time linear control systems are studied. These properties are characterized in terms of the directed graph of the state matrix. From these characterizations canonical forms of those properties are deduced.

Preconditioning Sparse Nonsymmetric Linear Systems with the Sherman--Morrison Formula

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 A0-1 - A-1, where A0 is a nonsingular matrix whose inverse is known or easy to compute, can be factorized in the form

Additive Schwarz Iterations for Markov Chains

U\Omega V^T

High order backward discretization of the neutron diffusion equation

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 A0=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.

Group inverse and group involutory Matrices

A convergence analysis is presented for additive Schwarz iterations when applied to consistent singular systems of equations of the form

Extensions of Jordan Bases for Invariant Subspaces of a Matrix

Ax=b

Parallel chaotic extrapolated Jacobi method

. The theory applies to singular

Subdirect sums of nonsingular M-matrices and of their inverses

M