SPLITTINGS OF M-OPERATORS - IRREDUCIBILITY AND THE INDEX OF THE ITERATION OPERATOR

For the solution of linear systems of equations of the form Au = f by iterations, it is customary to consider a splitting A = M — N with the iterative process being uk+i = Tuk + M-1 f, T — M-1 N and tto is the initial guess. In this paper we extend the theory of Splittings to Banach spaces and in particular we generalize some results by Schneider, Rose and Szyld on splittings and irreducibility of M-matrices, and those by Schneider, Neumann and Plemmons on the indices if A and T. We present a new concept of cone-operator irreducibility which generalizes the usual concept of irreducibility. We also introduce the concept of G-compatible spittings for M-operators and show its relation to graph compatible splittings in the finite dimensional case. Most of our results are graph independent. © 1990, Taylor & Francis Group, LLC. All rights reserved.