Use a better algorithm to solve Kepler equation

[abhisrkckl]

Right now the Kepler equation is solved using the Newton-Raphson method.

PINT/src/pint/models/stand_alone_psr_binaries/binary_generic.py

Line 350 in 18c643a

def compute_eccentric_anomaly(self, eccentricity, mean_anomaly):

There are faster (and no less accurate) methods for doing this.
For example, Markley (1995) http://adsabs.harvard.edu/abs/1995CeMDA..63..101M

We should implement such a method instead of the Newton-Raphson method.

[dlakaplan]

This seems like a good student project

[AkashA98]

I am quite interested in this stuff... If this is not time-sensitive, I can take this up...

[scottransom]

I think this is a fine project, but we should temper our expectations! I suspect that the current solver isn't even close to the bottleneck in any of our calculations. N-R usually converges in only a couple iterations except in high eccentricity cases, I believe.

[dlakaplan]

Yes, a more thorough analysis of where time is spent (which a few people have started but not finished) could be helpful to really focus us.

[abhisrkckl]

Here is an implementation of this
https://pyastronomy.readthedocs.io/en/latest/pyaslDoc/aslDoc/keplerOrbitAPI.html