I am in a situation where as part of a sort of inverse power method scheme, I want to very often perform the following step:
- Apply a symmetric rank one update $uu^\top$ to my inverse matrix $A^{-1}$
- Compute the QR decomposition of my updated inverse
Right now, I'm using Sherman-Morrison-Woodbury formula to update my inverse:
- Update $A^{-1}$ with Sherman-Morrison-Woodbury:
- Let $a = 1 / (1 + u^\top A^{-1} u)$
- Update $A^{-1} \gets A^{-1} - a A^{-1} u u^\top A^{-1}$
and then I'm recomputing the QR from scratch:
- Recompute QR: $QR = A^{-1}$.
However, I'm aware that it's possible to efficiently update a QR decomposition with a low-rank update. Unfortunately, as we can see above the Sherman-Morrison update looks to be full rank, so we can't make use of the QR update.
I'm doubtful, but maybe there is some approach I am missing that can avoid the hit of recomputing the whole QR?
(Also: there are a lot of other questions here about low rank updates of different factorizations here, and I didn't see anything along these lines. Also, I have looked through Matrix Computations without success.)