Find all points where the gradient of a high-dimensional function are equal to zero in some domain, numerically

Question

I was wondering if anybody was aware of a numerical method to find all points where the gradient of a high-dimensional function are equal to zero in some domain.

Thanks

Answer 1

I don't know of any methods that are specially designed for gradients. However, what I would try is, find all the first partial derivatives of the function (or at least, find a way to calculate them numerically) and then apply the generalized Newton's method to the system of equations generated by setting each partial derivative equal to 0. This page gives an example of doing so in the context of finding relative extrema. Using Newton's method should work even if you can only find numerical approximations to the first partial derivatives, as long as you are able to do so at arbitrary points.