I am dealing with a non-trivial Riemannian metric $H$ defined on a particular subset of Euclidean space ($E \subset \mathbb{R}^n$). I was able to show the Riemannian manifold $(E,H)$ is geodesically complete. I am interested in finding a geodesic that connects arbitrary points. I think this is accomplished via the shooting method...
I was wondering if there were any libraries, preferably in Matlab or Python, dedicated to numerically computing geodesics on manifolds. I rather not re-invent the wheel and instead take advantage of some (hopefully) library with a dedicated community for fixing bugs and adding cool features I haven't thought of.
ManOpt is a great library, but the Manifold object assumes that you have the exponential map readily available. In my case, I do not. I only have a closed-form expression of the Riemannian metric.
I attempted at doing this myself but ran into many difficulties. First, the resulting geodesic ODE seems to be stiff, and hence Ode45 isn't useful. I found that other ODE solvers for stiff equations work nice (to an extent...), but frankly, I'm dealing with stuff I just don't understand. I want a library written by people who do understand these issues. Not to mention, trying to implement the shooting method for solving 2-point boundary problems for the geodesic will be quite difficult.
I would appreciate any help with this.