Computational optimal transport has been recently upset by the use of entropic methods combined with the Sinkhorn algorithm. This evolution also reached other types of transport, like the martingale optimal transport, using the ideas of the Sinkhorn algorithm via Bregman projections algorithm. We compare the existing methods and suggest to use a Newton algorithm that turns out to be much more effective and really fit to martingale optimal transport. We give convergence rates and tips to solve some classical problems like the lack of convex ordering.