Assume I know eigen-pairs $(\epsilon_i,\vec\phi_i)$ for matrix operator $\hat H$ that $ \hat H \vec \phi_i = \epsilon_i \vec \phi_i $
Now I have slightly perturbed $\hat H' = \hat H + \hat R$ were perturbation $\hat R$ is small with respect to $\hat H$ (in some sence, e.g. Frobenius norm ?). Now I would like to find eigen-pairs of the perturbed $\hat H'$. $\hat H' \vec \phi'_i = \epsilon'_i \vec \phi'_i $
I expect there should exist some efficient iterative algorithm which use the knowledge of solution $ \hat H \vec \phi_i = \epsilon_i \vec \phi_i $ to obtain solution of $\hat H' \vec \phi'_i = \epsilon'_i \vec \phi'_i $ with modest computational effort ( measured in CPU flops ) especially if I work with sparse matrices.
I'm not expert in linear algebra, nor iterative methods. I'm just trying to familiarize myself with the field by googling. I downloaded some papers about Davidson / Arnoldi / Lanczos methods, but at first look I wasn't able to extract clear answer to even the very basic question if it is relevant to my problem.
I also read wikipage Eigenvalue perturbation but it doesn't concern with explicit numerical algorithm and it computational cost.
Mostly I'm interested in case when $\hat H$ and $\hat H'$ are hermitian (ground state Hamiltonian of some quantum system), but for more general understanding I do not want to limit the question on that case.