eigenvalues of many dense symmetric real matrix that are 'close' to each other

I have to find the eigenvalues of many dense symmetric real matrix that are 'close' to each other, i.e. they are not much different. Can I speed up eig or some other code if I know the spectral decomposition of A and want to find it for a nearby B. I.e. I have A = UDU' as the spectral decomposition and want to find it for B where

B-A is small. I know this can be done for eigs with 'restarts'. But what about finding all the eigenvalues with eig?

[David Goodmanson]

Hi Henry,

If the eigenvalues are not too closely spaced (no repeated ones either) then a simple first order approximation gives a quick look at how much the eigenvalues change. Let A1 = B-A. The diagonal elements of

E1 = U'*A1*U

are the shifts in the eigenvalues, to first order. Perturbation theory can provide results for higher orders, using increasingly complicated expressions.