Istituto di Informatica e Telematica     
Favati P., Meini B. On functional iteration methods for solving nonlinear matrix equations arising in queueing problems. Technical report, 1999.
The problem of the computation of the minimal nonnegative solution $G$ of the nonlinear matrix equation $X=sum_{i=0}^{+infty}X^iA_i$ is considered. This problem arises in the numerical solution of M/G/1 type Markov chains,where $A_i$, $ige0$, are nonnegative $ktimes k$ matrices such that $sum_{i=0}^{+infty}A_i$ is column stochastic. We analyze classical functional iteration methods, by estimating the rate of convergence, in relation with spectral properties of the starting approximation matrix $X_0$. Based on these new convergence results, we propose an effective method to choose a matrix $X_0$, which drastically reduces the number of iterations; the additional cost needed to compute $X_0$ is much less than the overall savings achieved by reducing the number of iterations.
Subject G.1.0 General
65U05 Numerical methods in probability and statistics

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