On the convergence of asynchronous iteration methods for nonlinear paracontractions and consistent linear systems

Abstract

Elsner et al. (L. Elsner, I. Koltracht, M. Neumann, Numer. Math. 62(1992) 305–319) introduced the concept of paracontracting operators for fixed point problems and their solution with asynchronous iteration methods. Their results are extended with respect to the properties of the pool of operators from which a common fixed point is searched. Also, a more general kind of asynchronous iterations than theirs is presented. As an application of this theory, asynchronous iteration methods for consistent linear systems Ax = b are considered, where A is a singular M-matrix. Lubashevski and Mitra (B. Lubashevski, D. Mitra, J. ACM 33(1) (1986) 130–150) investigated asynchronous iteration methods for the Perron-vector problem, i.e. determining a positive solution of Tx = p( T) x, where T is an irreducible nonnegative matrix and its spectral radius is know. Their result can be fortified and extended to the reducible-affine case by using a quite different approach, the developed theory of paracontractions and confluence.