A chaotic asynchronous algorithm for computing the fixed point of a nonnegative matrix of unit spectral radius

Abstract

Given a nonnegative, irreducible matrix P of spectral radius unity, there exists a positive vector π such that π = πP. If P also happens to be stochastic, then π gives the stationary distribution of the Markov chain that has state-transition probabilities given by the elements of P. This paper gives an algorithm for computing π that is particularly well suited for parallel processing. The main attraction of our algorithm is that the timing and sequencing restrictions on individual processors are almost entirely eliminated and, consequently, the necessary coordination between processors is negligible and the enforced idle time is also negligible. Under certain mild and easily satisfied restrictions on P and on the implementation of the algorithm, x(.), the vectors of computed values are proved to converge to within a positive, finite constant of proportionality of π. It is also proved that a natural measure of the projective distance of x(.) from π vanishes geometrically fast, and at a rate for which a lower bound is given. We have conducted extensive experiments on random matrices P, and the results show that the improvement over the parallel implementation of the synchronous version of the algorithm is substantial, sometimes exceeding the synchronization penalty to which the latter is always subject.