Owing to its simplicity and efficiency\, the Sherman-Morris on (SM) formula has seen widespread use across various scientific and engi neering applications for solving rank-one perturbed linear systems of the form $(A + u v^T) x = b$. Although the formula dates back at least to 1944 \, its numerical stability properties have remained an open question and c ontinue to be a topic of current research.
We analyse the backward s tability of the SM formula and show\, both theoretically and through numer ical experiments\, that it is unstable in a scenario that is increasingly common in scientific computing. We then address an open question posed by Nick Higham regarding the proportionality of the SM backward error bound t o the condition number of A.
Finally\, we integrate fixed-precision iterative refinement (IR) into the SM framework to develop the SM-IR algor ithm. We prove that\, under reasonable assumptions\, IR enhances the backw ard error of the SM formula and present practical evidence supporting the eventual backward stability of SM-IR. Since SM-IR reuses previously comput ed decompositions\, it retains the efficiency of the original SM algorithm . This is joint work with Yuji Nakatsukasa (University of Oxford).
LOCATION:Centre for Mathematical Sciences\, MR14 END:VEVENT END:VCALENDAR