I have a large, dense, symmetric, positive semi-definite $n \times n$ Matrix $\mathcal{M}$, which is constructed by a slight perturbation of a Matrix $\mathcal{M'}$ (with an symmetric, positive semi-definite matrix). I have the eigenvalues and eigenvectors of $\mathcal{M'}$ and I am wondering whether I can use them to (computationally) cheaply approximate the eigenvectors and eigenvalues of $\mathcal{M}$ (So in practice I care about the constant costs of the algorithms).
I am currently doing a few iteration of the simultaneous iteration method:
$\mathcal{V}',R \gets QR(\mathcal{M} \cdot \mathcal{V})$
but this is probably less than ideal.