Derivative of Network Loglikelihood for brlen optimization

[lutteropp]

(from https://github.com/lutteropp/NetRAX/issues/43)

Use Newton-Raphson instead of Brent for optimizing branches (this will give us some niiiiiice additional speedup). This requires some Maths now, as I need to understand how to compute the derivative of network loglikelihood. Given the derivatives of the displayed tree loglikelihoods, this should be easy. I then need to understand how pll_modules computes likelihood derivatives for trees (something something sumtable), and how this can be generalized to networks.

Luckily, given the derivatives of displayed tree loglikelihoods, the derivative of the network loglikelihood is easy to compute (little mistake in the image, but not relevant here - the displayed tree loglikelihoods of course depend on the partition, also on brlens, model parameters, etc):

(EDIT: THIS IS WRONG, SEE LATER COMMENT)

Thus the main challenge here is to understand these sumtables used for tree loglh derivatives in pll_modules, being especially careful about how this works with virtual rerooting. And then I need to do this for the displayed trees displayed by the network and voilà, then we can switch from Brent brlen-opt to Newton-Raphson brlen-opt! :-)