Sherman-Morrison 公式及其改进稳定性修正的误差界
Error bounds for the Sherman-Morrison formula and its modification with improved stability
- University of Leicester（莱斯特大学）
- University of Oxford（牛津大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文提出改进的 Sherman-Morrison 方法 MSM，通过内置自校正增强数值稳定性，并推导出更强的误差界及可事后计算的增长因子，实验验证其向后和向前稳定性。
中文摘要 AI 辅助
众所周知，Sherman-Morrison (SM) 公式在数值上不稳定。在最近的工作中，我们引入了 SMIR，一种结合迭代精化以增强 SM 向后误差的算法。在本文中，我们采取不同的途径：改编 Govaerts 最初为一般边界线性系统设计的算法，我们开发了一种改进的 Sherman-Morrison (MSM) 方法，其内置的自校正特性使其具有惊人的鲁棒性。标准 SM 需要求解两个 $n\ imes n$ 线性系统，而 MSM 需要三个；相比之下，SMIR 需要 $2+k$ 次求解，其中 $k$ 是 IR 步数，当 IR 收敛缓慢时，$k$ 可能远大于三。随后，我们推导了 SM 和 MSM 的向后和向前误差界。这里建立的 SM 向后误差界比 [Hashemi \& Nakatsukasa 2026] 中证明的更强：它考虑了每一次舍入误差，并且对电容的大小没有任何条件限制。从这些界中，我们提取了可在事后廉价计算的增长因子，可用于在给定问题上认证 SM 和 MSM 的向后和向前稳定性。在我们的实验中，MSM 始终产生向后稳定的解，因此也被观察到是向前稳定的。关于 MSM 稳定性（或不稳定性）的形式化证明仍然是一个开放问题。
英文摘要
It is known that the Sherman--Morrison (SM) formula is not numerically stable. In recent work, we introduced SMIR, an algorithm that incorporates iterative refinement to enhance the SM backward error. In this paper we take a different route: adapting an algorithm of Govaerts, originally designed for general bordered linear systems, we develop a modified Sherman--Morrison (MSM) method whose built-in self-correction makes it surprisingly resilient. Whereas standard SM requires the solution of two $n\times n$ linear systems, MSM requires three; by contrast, SMIR requires $2+k$ solves, where $k$ is the number of IR steps and can be substantially larger than three, when IR converges slowly. We then derive backward and forward error bounds for both SM and MSM. The SM backward error bound established here is stronger than the one proved in [Hashemi \& Nakatsukasa 2026]: it accounts for every rounding error and holds with no conditions on the size of the capacitance. From these bounds we extract growth factors that are cheap to compute a posteriori and can be used to certify the backward and forward stability of SM and MSM on a given problem. In our experiments, MSM consistently produces backward stable solutions, and is therefore observed to be forward stable as well. A formal proof of the stability --- or instability --- of MSM remains an open problem.
