Stable computation with the fundamental matrix of a Markov chain

The short term behavior of a Markov chain can be inferred from its fundamental matrix F. One method of computing the parts of F that are needed is to compute Fy for a given vector y. It is shown that all forward stable algorithms that solve a particular least squares problem lead to forward stable algorithms for computing Fy. This in turn leads to a class of algorithms that compute Fy accurately whenever the underlying problem is well-conditioned. One algorithm from this class is based upon the Grassman-Taksar-Heyman variant of Gaussian elimination. Other such algorithms include one based upon orthogonal factorization and one based upon the conjugate gradient least squares algorithm.