Instability of the Sherman-Morrison formula and stabilization by iterative refinement
Behnam Hashemi, Yuji Nakatsukasa
TL;DR
The paper investigates the backward stability of the Sherman--Morrison formula for rank-one updates, identifying a stability source not tied to the usual conditioning of the update term and showing potential backward errors of $\mathcal{O}(\epsilon_M \kappa(A)^2)$ in some regimes. It then introduces iterative refinement (IR) inside the SM framework (SM-IR), proving that one IR step yields a backward-stable solution under mild verifiable conditions and preserving the algorithm's efficiency. Through extensive numerical experiments across diverse conditioning and norm scenarios, the authors demonstrate that SM-IR reliably delivers backward stability after a small number of IR steps, often matching or surpassing direct solvers in accuracy with minimal extra cost; they conjecture backward stability holds broadly when $\kappa(A)$ and $\kappa(A+uv^T)$ remain bounded away from $\epsilon_M^{-1}$. The work also connects to bordered-system formulations and existing backward-error analyses, outlining open problems for higher-rank updates and sharper universal bounds.
Abstract
Owing to its simplicity and efficiency, the Sherman-Morrison (SM) formula has seen widespread use across various scientific and engineering applications for solving rank-one perturbed linear systems of the form $(A+uv^T)x = b$. Although the formula dates back at least to 1944, its numerical stability properties have remained an open question and continue to be a topic of current research. We analyze the backward stability of the SM, demonstrate its instability in a scenario increasingly common in scientific computing and address an open question posed by Nick Higham on the proportionality of the backward error bound to the condition number of $A$. We then incorporate fixed-precision iterative refinement into the SM framework reusing the previously computed decompositions and prove that, under reasonable assumptions, it achieves backward stability without sacrificing the efficiency of the SM formula. While our theory does not prove the SM formula with iterative refinement always outputs a backward stable solution, empirically it is observed to eventually produce a backward stable solution in all our numerical experiments. We conjecture that with iterative refinement, the SM formula yields a backward stable solution provided that $κ_2(A), κ_2(A+uv^T)$ are both bounded safely away from $ε_M^{-1}$, where $ε_M$ is the unit roundoff.