On functional iteration methods for solving nonlinear matrix equations arising in queueing problems

Authors: Favati P.1; Meini B.2

Source: IMA Journal of Numerical Analysis, Volume 19, Number 1, January 1999 , pp. 39-49(11)

Publisher: Oxford University Press

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Abstract:

The problem of the computation of the minimal nonnegative solution G of the nonlinear matrix equation X = [sum ]+infini=0Xi Ai is considered. This problem arises in the numerical solution of M/G/1 type Markov chains, where Ai,i ges 0, are nonnegative k x k matrices such that [sum ]=infini=0A is column stochastic. We analyse classical functional iteration methods, by estimating the rate of convergence, in relation to the spectral properties of the starting approximation matrix X0. Based on these new convergence results, we propose an effective method to choose a matrix X0, which drastically reduces the number of iterations; the additional cost needed to compute X0 is much less than the overall savings achieved by reducing the number of iterations.

Document Type: Original article

Affiliations: 1: Istituto di Matematica Computazionale del C.N.R., via S. Maria 46, 56127 Pisa, Italy 2: Dipartimento di Matematica, via Buonarroti 2, 56127 Pisa, Italy

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