I have a function $f:\mathbb{R}^n\to\mathbb{R}$ with an analytical expression. The function itself is very complicated, so I will omit the definition for brevity. It is a highly non-linear function that took me about 3 months to derive, and I was initially unsure if I would even be able to find the global minimum of the system (given the 'apparent' nonlinearity). However, my physical intuition of the system I am trying to model (network of inter-connected springs and pivoting surfaces) seemed to suggest that the starting points for all the variations of my simulation will always be within the convergence ball of the global minimum.
Only a week ago did I get to a point where I could implement and test the mathematics. I first implemented a steepest descent-type algorithm for a simplified test case with $n=6$. I stepped in the direction, $\mathbf{d}_i=-\nabla f$, and then took tiny steps until I found the minimum of $f$ in that direction. Repeating this procedure did in fact yield the global minimum, which was a definite confirmation of my physical intuition of the system. However, when I computed the Hessian of $f$, and attempted to minimise $f$ using Newton's method, I didn't get any sort of convergence. The minimisation routine is:
$$ \mathbf{x}_{i+1} = \mathbf{x}_{i} - \mathbf{H}(\mathbf{x}_{i})^{-1}\nabla f(\mathbf{x}_{i}) $$
I have independently verified the correctness of each term in the gradient and the Hessian via discrete means, i.e. by computing $f$ only. So I am at a loss for what to make of the non-convergence using Newton's method. Can anyone advise me on the following: Is it not to be expected that, if Steepest descent finds the global minimum, Newton's method should too? Can someone suggest how I can robustly check for the positive-definiteness of the Hessian, and thereby infer why Newton's method would not work in this case?