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I am trying to find the eccentric anomaly E of an orbit of a binary system knowing the mean anomaly M through Newtons solution:

E=M+e*sin(E)

where an iterative solution can be found.

I am using this to fit radial velocity data vs. time. But calculating E is hindering my progress (getting a good fit to data and length of time taken to calculate E). Is there another way to find E other than Newtons method?