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.

中文翻译：

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M/G/1型马尔可夫链的快速不动点迭代族

我们考虑计算非线性矩阵方程$X=\sum _{i=-1}^\infty A_iX^{i+1}$的最小非负解$G$的问题，其中$A_i$，对于$ i\geqslant -1$ 是非负方阵，使得 $\sum _{i=-1}^\infty A_i$ 是随机的。这个方程是分析 M/G/1 型马尔可夫链的基础，因为矩阵 $G$ 提供了感兴趣的概率度量。介绍了用于$G$ 数值计算的新定点迭代族，其中包括经典迭代。详细的收敛分析证明，新类中的迭代比经典迭代收敛得更快。数值实验证实了我们扩展的有效性。