Suppose that $B$ is a Hermitian matrix with one known eigenpair $(\lambda,v)$. (assume its the smallest or largest pair, if you like). Form the rank one update $B+\rho bb^{T}$.

Now I'm interested in a (possibly approximate) formula for the updated eigenpair $(\widetilde{\lambda},\widetilde{v})$, that depends only on the entries of $B$ and $\lambda,v$. There are such formulas derived in a classic 1978 paper by Bunch, Nielsen and Sorensen (building on an earlier classic work by Golub), but they involve the full eigenbasis of $B$.

Any pointers?

P.S. (Assume $\lambda,\widetilde{\lambda}$ are simple, to keep things simple).