Numerical solution of sparse singular systems of equations arising from ergodic markov chains

Abstract

The stationary probability distribution vector, x, associated with an ergodic finite Markov chain satisfies a homogeneous singular system of equations , where A is a real and generally unsymmetric square matrix of the form . Here I is the identity matrix and T is the chain's column stochastic matrix. In many applications A is very large and sparse, and in such cases it is desirable to exploit this property in computing x. In this paper we review some of the literature dealing with sparse techniques for solving the above system of equations, and in so doing attempt to present a variety of methods from a unified point of view