Dokchitser Project for Sage Days 11

TODO:

1. Stuff still computed using gp:
   • Delta polynomials in _recursions_at_infinity (search for comment below)
   • _without_gp (gamma_series) has this line
     • sinser = sage_eval(rs(gp_eval('Vec(sin(Pi*(%s)))'%(z0+x))))
   • init_Ginf: still uses pari (see below)
   • Ginf: still uses pari to evaluate continued fraction
2. Doctest everything, making sure it all works 100% and fix issues with coercion, complex inputs, etc., as they are systematically uncovered.
3. Possibly maybe change from digits to bits prec.
4. Optimize.

From Jen:

Here's the version (closest to Dokchitser's original pari code) that still uses continued fraction approximation:

http://sage.math.washington.edu/home/jen/sage-3.0.5-x86_64-Linux/l4.py

(needs gamma_series.py to run:

http://sage.math.washington.edu/home/jen/sage-3.0.5-x86_64-Linux/gamma_series.py)

The version with Pade approximation (l5.py) has a negligible speedup but only really works for low precision. I'm not sure if Pade gives us a means of computing bounds (I think Mike Rubinstein said that continued fractions won't). Also, l4.py doesn't work for imaginary inputs yet - some coercion with SymbolicRing that I didn't try.

Dokchiter's Paper: attachment:dokchitser.pdf