A class of iteration methods based on the moser formula for nonlinear equations in Markov chains

Abstract

Many stochastic models in queueing, inventory, communications, and dam theories, etc., result in the problem of numerically determining the minimal nonnegative solutions for a class of nonlinear matrix equations. Various iterative methods have been proposed to determine the matrices of interest. We propose a new, efficient successive-substitution Moser method and a Newton-Moser method which use the Moser formula (which, originally, is just the Schulz method). These new methods avoid the inverses of the matrices, and thus considerable savings on the computational workloads may be achieved. Moreover, they are much more suitable for implementing on parallel multiprocessor systems. Under certain conditions, we establish monotone convergence of these new methods, and prove local linear convergence for the substitution Moser method and superlinear convergence for the Newton-Moser method.