How can I compute the partial derivatives of the dominant eigenvalue and eigenvectors of a real symmetric matrix $\mathbf{A}$?

In particular, given

$ \mathbf{v}^* = \arg\max_{\mathbf{v}} \mathbf{v}^{\top}\mathbf{A}\mathbf{v},\quad \text{subject to } ~ \|\mathbf{v}\|_2 = 1, $

how can I find $\partial v_{i}^*/\partial A_{jk}$?

I know the power method is the usual way to compute the dominant eigenvalue and eigenvector. Is their any similar algorithm for computing the gradient?

Unlike another question, I am interested in an efficient computational solution rather than an analytical one.