I have a Matrix, $A$ which is positive semidefinite. No consider, $B=A+\Delta$. I have Eigen decomposition of $A$ and $\Delta_{ij}<= \epsilon_1$, $\Vert \Delta \Vert_F <= \epsilon_2$.
Is there any method which can compute Eigen decomposition of $B$, starting from Eigen decomposition of $A$?
I assume that this will speed up the Eigen decomposition. Am I right? Isn't it an implementation of this in LAPACK/BLAS?
Yes, in principle you can apply the inverse subspace iteration to compute select eigenpairs of $B$ using eigenvalues of $A$ to initialize the search.
However, unless $A$ is such that you can solve linear systems $(A + \sigma I)x = b$ using $O(n)$, rather than the usual $O(n^3)$ arithmetic operations, you will not be able to do better than finding the eigendecomposition of $B$ from scratch using a the dense methods implemented in (SCA)LAPACK.
Even if you can solve the linear system in $O(n)$ time, you will discover that it is extremely difficult to get all $n$ eigenvalues using the inverse subspace iteration algorithm.
There is no algorithm implemented in (SCA)LAPACK which can exploit the eigendecomposition of $A$ to find the eigendecomposition of $B$. You will have to code the invers subspace iteration with Rayleigh-Ritz acceleration yourself.
Your notation implies that the perturbation is small, but you have not indicated if it is symmetric. You may very well end up in the situation were the perturbed matrix can not be diagonalized. There is a cure for this situation, but that is topic for another question.