On the convergence of asynchronous iteration methods for nonlinear paracontractions and consistent linear systems
Matthias Pott · 2010
In [6] the authors introduced the concept of paracontracting operators for fixed point problems and their solution with asynchronous iteration methods. We will extend their results essentially with respect to the criterion of contraction for the pool of operators, from which a common fixed point is searched and a more general kind of asynchronous iterations, which we will call confluent. As an application of this theory, we will consider asynchronous iteration methods for consistent linear systems Ax = b ; where A is a singular M-matrix. In [7] Lubachevski and Mitra investigated asynchronous iteration methods for the Perron-vector problem, e.g. determining a positive solution of Tx = ae(T ) x ; where T is an irreducible nonnegative matrix and its spectral radius is known. We will show that we can extend their results by using the developed theory of paracontractions and confluence to the reducible-affine case which arises, if we have those kind of linear systems. Key words: Asynchron...