A Note on the Stability of the Sherman–Morrison–Woodbury Formula

Abstract. We study the numerical stability of the Sherman–Morrison–Woodbury (SMW) identity. Let [Formula: see text], and assume [Formula: see text] and [Formula: see text] both have full column rank. We explore error bounds for the SMW identity when we are only able to compute approximate inverses. For both forward and backward errors, we present upper bounds as a function of the two-norm error of the approximate inverses. We verify with numerical experiments that, in certain cases, our bounds accurately capture the behavior of the errors. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SIMAX and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/LinkaiMa/SMW . [Formula: see text]