[Bug-gsl] [bug #28718] gsl_sf_hyperg_1F1(-1.5, 1.5, -100) catastrophic cancellation in series

[Brian Gough]

URL:
  <http://savannah.gnu.org/bugs/?28718>

                 Summary: gsl_sf_hyperg_1F1(-1.5,1.5,-100) catastrophic
cancellation in series
                 Project: GNU Scientific Library
            Submitted by: bjg
            Submitted on: Sat 23 Jan 2010 02:09:25 PM GMT
                Category: Accuracy problem
                Severity: 3 - Normal
        Operating System: 
                  Status: Confirmed
             Assigned to: bjg
             Open/Closed: Open
                 Release: 1.13
         Discussion Lock: Any

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Details:

From: Lorenzo Moneta <address@hidden>
To: <address@hidden>
CC: Weibin Li <address@hidden>, Brian Gough
<address@hidden>
Subject: Re: [Bug-gsl] gsl_sf_hyperg_1F1
Date: Thu, 21 Jan 2010 16:12:45 +0100

Hello, 

 I have found a similar problem, probably related to the inaccuracy in the
series calculation of 
the hypergeometric function for 
gsl_sf_hyperg_1F1( a < 0, b > 0, x << 0). 

For example : 

  gsl_sf_hyperg_1F1(-1.5, 1.5, - 99.9)  returns the correct value of 
4.55768964383819423e+02

while gsl_sf_hyperg_1F1(-1.5, 1.5, -100.) returns -1.13760e+20 when the
correct value is 456.44  (from Mathematica)

I think the Kummer formula (    1F1(a,b,x) = exp(-x) * 1F1(b-a,b,-x) could be
used for larger value of |x| ( at least works for |x| = 700), while 
the current cut-off is for |x| < 100. 

 Best Regards

 Lorenzo





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