Re: [Bug-gsl] Re: [Help-gsl] Hypergeometric function

[Imad-Eddine Srairi]

Hi,

I could not manage to fix this problem yet, but I have tried to narrow 
down its location. We are here in a special case where conditions of [1] 
are fulfilled i.e.:

a = -0.5 , b =  1.5 , c =  1.0

and : c = 0.5 * a + 0.5 * b + 0.5.

It is easy to check the two results returned by the test case attached 
to this issue against:

M_SQRTPI * gsl_sf_gamma(1.0) / gsl_sf_gamma(0.25) / gsl_sf_gamma(1.25);

to find that the correct answer is the positive one (the one returned 
for (1/2 - epsilon)) and that the error lies in the (1/2 + epsilon) case.

The result returned in this case is produced by call to this function

        return hyperg_2F1_reflect(a, b, c, x, result);

which seems to rely on this reference:

/* Do the reflection described in [Moshier, p. 334].
 * Assumes a,b,c != neg integer.
 */

I have no access to this book, but I am currently trying to match the 
code against the special cases described in [2].

The result is computed in these lines:

    sgn_2 = ( GSL_IS_ODD(intd) ? -1.0 : 1.0 );
    result->val  = F1.val + sgn_2 * F2.val;
    result->err  = F1.err + F2. err;
    result->err += 2.0 * GSL_DBL_EPSILON * (fabs(F1.val) + fabs(F2.val));
    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
    return stat_F2;

with sgn_2 = 1.0, F1.val = 0.0 and F2.val = -0.538171 .

Hope this helps,

imad

[1] http://dlmf.nist.gov/15.4#E28
[2] http://dlmf.nist.gov/15.8#SS1.p1

On 01/31/2011 05:46 PM, Brian Gough wrote:
> At Thu, 27 Jan 2011 20:38:28 +0100,
> Paola Alpresa Gutiérrez wrote:
> > In a general case of gsl_sf_hyperg_2F1(-nu,nu+1,1,x): the function sign is
> > correct for x<1/2, but it is minus the function for x>=1/2.
> >
> > There is a discontinuity in z=0 or x=1/2, if I am correct. I compared it
> > with Mathematica.
>
>
> Thanks for the bug report, I've confirmed that there is a sign problem there.