Quadratic matrix equations of the kind $A_1X^2+A_0X+A_{-1}=X$ are encountered in the analysis of Quasi--Birth-Death stochastic processes where the solution of interest is the minimal nonnegative solution $G$. In many queueing models, described by random walks in the quarter plane, the coefficients $A_1,A_0,A_{-1}$ are infinite tridiagonal matrices with an almost Toeplitz structure. Here, we analyze some fixed point iterations, including Newton's iteration, for the computation of $G$ and introduce effective algorithms and acceleration strategies which fully exploit the Toeplitz structure of the matrix coefficients and of the current approximation. Moreover, we provide a structured perturbation analysis for the solution $G$. The results of some numerical experiments which demonstrate the effectiveness of our approach are reported.

中文翻译：

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求解在四分之一平面随机游动中产生的二次矩阵方程

$A_1X^2+A_0X+A_{-1}=X$ 类型的二次矩阵方程在准-出生-死亡随机过程的分析中遇到，其中感兴趣的解是最小非负解$G$。在许多排队模型中，由四分之一平面中的随机游走描述，系数 $A_1​​,A_0,A_{-1}$ 是具有几乎 Toeplitz 结构的无限三对角矩阵。在这里，我们分析了一些不动点迭代，包括牛顿迭代，用于计算 $G$，并介绍了有效的算法和加速策略，它们充分利用了矩阵系数和当前近似的 Toeplitz 结构。此外，我们为解决方案 $G$ 提供结构化扰动分析。一些数值实验的结果证明了我们的方法的有效性。