I have a saddle point of a function defined on a $64 \times 64 \times 64$ grid, i.e. a function of $64^3$ variables. I would like to compute the lowest-lying eigenvalues of the Hessian to see how they depend on the parameters of the function.
The function is analytically twice differentiable, but computation of the complete Hessian is impossible due to memory constraints (the full matrix would be around 8TB). The Hessian is sparse, but it is not a simple sparsity pattern such as a tridiagonal matrix. I also know that the Hessian has several zero eigenvalues.
Is it possible to calculate the $n$ smallest eigenvalues without computing the full Hessian? Either an analytical method or a numerical approximation would be fine.