Note on functional iteration technique for M/G/1 type Markov chains

A thorough theoretical explanation of the numerical behaviour of functional iteration methods for the computation of the minimal nonnegative solution G of the matrix equation X = Sigma(infinity)(i=0)X(i)A(i), arising in the numerical solution of M/G/1 type Markov chains, is given in Meini (1997) [2]. In this note we add some more results. In particular, we show that an upper bound of the mean asymptotic convergence rate of the best functional iteration method is given in terms of the second largest modulus eigenvalue of G. (C) 2009 Elsevier Inc. All rights reserved.