We consider the problem of computing the minimal non-negative solution G of the nonlinear matrix equation X=∑∞i=−1AiXi+1 where Ai, for i⩾−1, are non-negative square matrices such that ∑∞i=−1Ai is stochastic. This equation is fundamental in the analysis of M/G/1-type Markov chains, since the matrix G provides probabilistic measures of interest. A new family of fixed point iterations for the numerical computation of G, which includes the classical iterations, is introduced. A detailed convergence analysis proves that the iterations in the new class converge faster than the classical iterations. Numerical experiments confirm the effectiveness of our extension.
A family of fast fixed point iterations for M/G/1-type Markov chains
Bini, Dario A;Latouche, Guy;Meini, Beatrice
2022-01-01
Abstract
We consider the problem of computing the minimal non-negative solution G of the nonlinear matrix equation X=∑∞i=−1AiXi+1 where Ai, for i⩾−1, are non-negative square matrices such that ∑∞i=−1Ai is stochastic. This equation is fundamental in the analysis of M/G/1-type Markov chains, since the matrix G provides probabilistic measures of interest. A new family of fixed point iterations for the numerical computation of G, which includes the classical iterations, is introduced. A detailed convergence analysis proves that the iterations in the new class converge faster than the classical iterations. Numerical experiments confirm the effectiveness of our extension.File | Dimensione | Formato | |
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