= Dokchitser Project for Sage Days 11 =


TODO:

    1. Modularize the code (break into the following classes and sort dependencies)
        * LSeries (Is already in there. Have to free it from the other 3 modules)
        * GammaFactor (Done and documented)
        * IncompleteGeneralizedGammaFactor (To be done)
        * InverseMellinGammaFactor (To be done)

    2. 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
    3. Doctest everything, making sure it all works 100% and fix issues with coercion, complex inputs, etc., as they are systematically uncovered.
    4. Possibly maybe change from digits to bits prec.
    5. Optimize.


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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]]