Reduced rank extrapolation (RRE) is an acceleration method typically used to accelerate the iterative solution of nonlinear systems of equations using a fixed-point process. In this context, the iterates are vectors generated from a fixed-point mapping function. However, when considering the iterative solution of large-scale matrix equations, the iterates are low-rank matrices generated from a fixed-point process for which, generally, the mapping function changes in each iteration. To enable acceleration of the iterative solution for these problems, we propose two novel generalizations of RRE. First, we show how to effectively compute RRE for sequences of low-rank matrices. Second, we derive a formulation of RRE that is suitable for fixed-point processes for which the mapping function changes each iteration. We demonstrate the potential of the methods on several numerical examples involving the iterative solution of large-scale Lyapunov and Riccati matrix equations.

翻译：降秩外推法（RRE）是一种加速方法，通常用于通过不动点过程加速非线性方程组的迭代求解。在此背景下，迭代向量是由不动点映射函数生成的。然而，当考虑大规模矩阵方程的迭代求解时，迭代项是由不动点过程生成的低秩矩阵，且该过程的映射函数通常在每次迭代中都会发生变化。为了加速此类问题的迭代求解，我们提出了RRE的两种新颖推广。首先，我们展示了如何有效计算低秩矩阵序列的RRE。其次，我们推导了一种适用于映射函数每次迭代都发生变化的RRE公式。我们通过多个涉及大规模Lyapunov和Riccati矩阵方程迭代求解的数值算例，展示了这些方法的潜力。