A defect correction formula for quadratic matrix equations of the kind $A_1X^2+A_0X+A_{-1}=0$ is presented. This formula, expressed by means of an invariant subspace of a suitable pencil, allows us to introduce a modification of the Structure-preserving Doubling Algorithm (SDA), that enables refining an initial approximation to the sought solution. This modification provides substantial advantages, in terms of convergence acceleration, in the solution of equations coming from stochastic models, by choosing a stochastic matrix as the initial approximation. An application to solving random walks in the quarter plane is shown, where the coefficients $A_{-1},A_0,A_1$ are quasi-Toeplitz matrices of infinite size.

翻译：A_1X%2+A_0X+A_0X+A ⁇ ⁇ -1 ⁇ 0美元等量方程式的缺陷校正公式。该公式以合适的铅笔的无变子空间表示,使我们能够对结构保护多倍的Algorithm(SDA)进行修改,从而能够使初步近似精确到所寻求的解决方案。这一修改在趋同加速解决来自随机模型的方程式方面有很大的优势,通过选择一个随机方程式作为初始近似值。一个用于解决四分之一平面随机行走的应用程序被显示,其中系数$A ⁇ -1},A_0,A_1美元是无限大小的准托里茨矩阵。