How to find LU factors of a matrix when diagonals are changed Say I have $A=LU$ already factored into lower and upper triangular matrices $L$ and $U$. Now I want to work on the eigenvalue problem $A-\lambda I=A'=L'U'$ where prime indicates new matrices. Given $A$, $L$, $U$, and $\lambda$, is there a way to get $L'$ and $U'$ without refactoring $A'$ from the beginning?
Thanks!

To the best of my knowledge, there is currently no known algorithm for this without completely refactoring $A$. The dqds algorithm, promoted by Beresford Parlett, does what you want when $A$ is tridiagonal (not necessarily symmetric). This is probably too restrictive for what you want. Tridiagonal reduction for eigenvalue algorithms in the non-symmetric case are extremely finicky and tend to break down catastrophically without warning.
Note that what you are asking for is numerically problematic, since the LU factorization typically requires pivoting, and with a shift, the pivoting might need to be redone. Pivot updates have been implemented by Michael Saunders in his LUSOL and LUMOD packages (these are the only open source packages I know about).
