Low rank update of QR of inverse

Question

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:

1. Apply a symmetric rank one update $uu^\top$ to my inverse matrix $A^{-1}$
2. Compute the QR decomposition of my updated inverse

Right now, I'm using Sherman-Morrison-Woodbury formula to update my inverse:

1. 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:

2. 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.)

Answer 1

The update on the inverse is actually rank-one. You can group the terms as $(A^{-1}u)(A^{-1}u)^T$ because of the symmetry of $A$.