[Numpy-discussion] Suggestions for GSoC Projects

[Jennifer stone]

In an attempt to analyze the accuracy of hyp2f1,
Different cases mentioned in Abramowitz (
http://people.math.sfu.ca/~cbm/aands/page_561.htm<http://bl-1.com/click/load/UmcPPANhUGxVMQNuUGU-b0231>
)
and also in the Thesis on 'Computation of Hypergeometric functions"
(http://people.maths.ox.ac.uk/porterm/research/pearson_final.pdf<http://bl-1.com/click/load/VmMBMlw-b0221ADxRNVI-b0169BDY-b0231>,
pg 65-66)
were tried out, and the function fails without warning when:
    c<0, c is not integral
   |c|>>|a| and |b|

For example:
sp.hyp2f1(10,5,-300.5,0.5)
>>-6.5184949735e+156
while the answer is

*-3.8520770815e+32*
this case appears to filter down to hys2f1 in the source code
(scipy.special.cephes.hyp2f1)

I tried the same input in mpmath to check if it works there:
hyp2f1(10,5,-300.5,0.5)
>>mpf('0.9211827166328477893913199888')
which is the solution when we apply power series expansion.

however MATLAB succeeds in giving the required solution.
Another interesting fact is that of the methods mentioned in the thesis:
Taylor series expansion, fraction method with double precision,
Gauss-Jacobi method and RK4), none succeeds in the given case.


I don't have any idea how the function itself is evaluated in the given
case.
Any leads on how it is done and how MATLAB executes it?



On Thu, Feb 20, 2014 at 1:16 AM, Jennifer stone <jenny.stone125 at gmail.com>wrote:

>
> If you are interested in the hypergeometric numerical evaluation, it's
>
>> probably a good idea to take a look at this recent master's thesis
>> written on the problem:
>>
>> http://people.maths.ox.ac.uk/porterm/research/pearson_final.pdf<http://bl-1.com/click/load/BDEKOABvUWAAYgNhATc-b0231>
>>
>> The thesis is really comprehensive and detailed with quite convincing
> conclusions on the methods to be used with varying a,b,x (though I am
> yet to read the thesis properly enough understand and validate each
> of the multitude of the cases for the boundaries for the parameters).
> It seems to be an assuring and reliable walk through for the project.
>
>
>> This may give some systematic overview on the range of methods
>> available. (Note that for copyright reasons, it's not a good idea to
>> look closely at the source codes linked from that thesis, as they are
>> not available under a compatible license.)
>>
>> It may well be that the best approach for evaluating these functions,
>> if accuracy in the whole parameter range is wanted, in the end turns
>> out to require arbitrary-precision computations.  In that case, it
>> would be a very good idea to look at how the problem is approached in
>> mpmath. There are existing multiprecision packages written in C, and
>> using one of them in scipy.special could bring better evaluation
>> performance even if the algorithm is the same.
>>
>
> Yeah, this seems to be brilliant idea. mpmath too, I assume, must have
> used some of the methods mentioned in the thesis. I ll look through the
> code and get back.
>
> I am still unaware of the complexity of project expected at GSoC. This
> project
> looks engaging to me. Will an attempt to improve both Spherical harmonic
> functions ( improving the present algorithm to avoid the calculation for
> lower n's and m's) and hypergeometric functions be too ambitious or
> is it doable?
>
> Regards
> Jennifer
>
>>  --
>> Pauli Virtanen
>>
>>
>
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