The inverse of a large matrix can often be accurately approximated by a polynomial of degree significantly lower than the order of the matrix. The iteration polynomial generated by a run of the GMRES algorithm is a good candidate, and its approximation to the inverse often seems to track the accuracy of the GMRES iteration. We investigate the quality of this approximation through theory and experiment, noting the practical need to add copies of some polynomial terms to improve stability. To mitigate storage and orthogonalization costs, other approaches have appeal, such as polynomial preconditioned GMRES and deflation of problematic eigenvalues. Applications of such polynomial approximations include solving systems of linear equations with multiple right-hand sides (where the solutions to subsequent problems come simply by multiplying the polynomial against the new right-hand sides) and variance reduction in multilevel Monte Carlo methods.

翻译：大型矩阵的逆通常可以通过一个阶数远低于矩阵阶数的多项式进行精确逼近。由GMRES算法运行生成的迭代多项式是一个优良候选，其对逆矩阵的逼近精度往往与GMRES迭代的收敛精度保持同步。我们通过理论分析与实验验证探究了该逼近方法的有效性，并指出实际应用中需要添加某些多项式项的副本以提升数值稳定性。为降低存储与正交化计算成本，其他方法如多项式预处理GMRES及问题特征值收缩技术也展现出应用潜力。此类多项式逼近方法可应用于多右端项线性方程组的求解（后续问题的解仅需将多项式与新右端项相乘即可获得），以及多级蒙特卡洛方法中的方差缩减。