Given a nonnegative, irreducible matrix P with spectral radius one, this paper presents an algorithm for finding a positive vector x such that x = xP . The algorithm is well suited for parallel processing. The chaotic nature of the computations along with some very minor synchronization requirements make this approach particularly attractive. Under certain conditions on P and on the algorithm, the sequence of vectors is proved to converge. Results are given for a simulated multiprocessor with shared memory.

This paper is very well written. Particular care is taken to explain how this work fits in with others on the subject. Results of Chazan and Miranker [1] are cited which indicate that asynchronous algorithms cannot be used for this problem. The assumptions which make this setting different than that of earlier works are carefully explained. Examples are used to illustrate the theory throughout, and the results of the numerical experiments are thoroughly discussed. This is a valuable work for those interested in multiprocessor stategies for this matrix problem.