Comments on a Shifted Cyclic Reduction Algorithm for Quasi-Birth-Death Problems

Abstract

A shifted cyclic reduction algorithm has been proposed by He, Meini, and Rhee [SIAM J. Matrix Anal. Appl., 23 (2001), pp. 673--691] for finding the stochastic matrix G associated with discrete-time quasi-birth-death (QBD) processes. We point out that the algorithm has quadratic convergence even for null recurrent QBDs. We also note that the approximations (to the matrix G) obtained by their algorithm are always stochastic when they are nonnegative.