Bini, Dario A.; Latouche, Guy; Meini, Beatrice A family of fast fixed point iterations for M/G/1-type Markov chains. (English) Zbl 1498.65056 IMA J. Numer. Anal. 42, No. 2, 1454-1477 (2022). Summary: We consider the problem of computing the minimal non-negative solution \(G\) of the nonlinear matrix equation \(X=\sum_{i=-1}^\infty A_iX^{i+1}\) where \(A_i\), for \(i\geqslant -1\), are non-negative square matrices such that \(\sum_{i=-1}^\infty A_i\) 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. Cited in 1 Document MSC: 65F45 Numerical methods for matrix equations 65C40 Numerical analysis or methods applied to Markov chains Keywords:nonlinear matrix equations; fixed point iterations; non-negative matrices; convergence analysis; Markov chains PDFBibTeX XMLCite \textit{D. A. Bini} et al., IMA J. Numer. Anal. 42, No. 2, 1454--1477 (2022; Zbl 1498.65056) Full Text: DOI arXiv