[SciPy-Dev] Draft Proposal

[Jennifer Janani]

Here is a draft proposal, suggestions are really welcome. I would like to
thank everyone especially Ralf and Pauli for guiding me till here.



 This is P.S.Janani (Jennifer), a undergraduate student, majoring in
Computer Science (2nd Year). A self found love for python and high level
math college courses naturally made me an ardent user of SciPy/NumPy.
Beyond this I have had formal C course for past 3 years, so I am
comfortable with that language too. I am comparitively new to open
sourcing, and after few collaborrated projects at college level
(PyCoursesync, Helios), SciPy is the first formal organization I have tried
contibuting to.

I am genuinely interested in contributing to SciPy this summer,
particularly to scipy.special

The tentative areas of work shall be: Gaussian Hypergeometric Functions
(2F1) and Spherical Harmonic Functions.

1. *Gaussian Hypergeometric functions:*

The implementation of the hypergeometric functions in scipy.special is
lacking in several respects, and is not optimal in many parameter
regimes. Gaussian
hypergeometric function being the most frequently used of hypergeometric
functions, focus shall be to seal the present loop holes.

1.1.*Handling the errors that arise when one or more of a,b,c is large:*

Here, We shall aim to overcome the lack of accuracy that occurs when
attempting to compute the Gauss hypergeometric function when one or more of
the values of |Re(a)|, |Re(b)| and |Re(c)| is large. At present
inaccuracies have been sighted at these parametric domains.

The possible approaches to overcome this can be the following:

1.To develop the present hyp2f1ra and expand the recurrence relation
method according
to the table given in Pg 52 of
http://people.maths.ox.ac.uk/porterm/research/pearson_final.pdf<http://bl-1.com/click/load/VmMBNlQyBzYAZ1w5U2Q-b0231>(cross
reference
http://www.ams.org/journals/mcom/2007-76-259/S0025-5718-07-01918-7/<http://bl-1.com/click/load/XWhdalYwAjMDZFUwCDw-b0231>
).

2.To <http://bl-1.com/click/load/AzYOOQZgVGUCZV04VWA-b0231> increase the
sensitivity of hys2f1 by decreasing the error tolerance, however this is
heavy process and to be avoided.

*1.**2.**Extending Gaussian Hypergeometric functions for complex a,b,c:*

At present SciPy's hyp2f1(a,b,c,z) entertains only z as complex, and not
a,b or c. The function is to be expanded to include complex inputs too,
with the domains of the parameters wide and accurate.

*1.**3.Sealing the unexpected and yet-unrecognized loopholes of hyp2f1: *

Almost all possible combinations of a,b,c,z (real and complex) have been
mentioned in Abramowitz and Stegan (
http://people.math.sfu.ca/~cbm/aands/page_560.htm<http://bl-1.com/click/load/U2YPOAdhADEFYlcyBDY-b0231>)
and . A testing on all the mentioned cases may be considered as a thorough
testing and also the many issues that have been reported scipy shall be
attempted to be resolved.


 *2.Harmonic Functions:*

*2.1.Improving spherical harmonic functions:*

The function for spherical harmonic function, sph_harm at present calls
lpmn thus evaluating all orders <N. On the other hand, lpmv avoids the
storage of values small N's by using recursion. Therefore an attempt can be
made to avoid the overheads by directly evaluating the function for given
degree and order.

*2.2. Implementing ellipsoidal harmonic functions:*

Further, we can introduce ellipsoidal harmonic functions. The thesis on O
pen-source

implementations of ellipsoidal harmonics (
http://arxiv.org/pdf/1204.0267v2.pdf<http://bl-1.com/click/load/BDEIP1I0UGFTNFQxCTo-b0231>),
presents implementations of ellipsoidal harmonic expansions for solving
problems of potential theory using separation of variables


 *TENTATIVE TIMELINE:*

The project is for the span of 90 days ie approx 12 weeks = 480 hrs.

Considering that it would be a bit slow in the starting, it would be most
probably in clean synchronization, with sub-proposal 1 ending almost during
mid-term evaluations. And the second part can be done post mid term
evaluations. The second half may be a bit busy with me new semester
beginning by August 1st.
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