四分之一平面上随机游动中二次矩阵方程的求解
Solving quadratic matrix equations arising in random walks in the quarter plane
摘要：
在拟死亡随机过程的分析中，遇到了一类二次矩阵方程$A_1X^2+A_0X+A_{-1}=X$，其解是最小非负解$G$ .在四分之一平面上的随机游动描述的许多排队模型中，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，计算的$G$，并引入有效的算法和加速策略，充分利用Toeplitz结构的矩阵系数和当前的近似。此外，我们提供了一个结构化的扰动分析的解决方案$G$。一些数值实验的结果表明，我们的方法的有效性报告。
英文摘要：
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.
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