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discrepancy in Hypergeometric1F1 results

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  • Subject: [mg21071] discrepancy in Hypergeometric1F1 results
  • From: "Atul Sharma" <atulksharma at>
  • Date: Sun, 12 Dec 1999 23:51:36 -0500 (EST)
  • Sender: owner-wri-mathgroup at

In the Mathematica Journal (Volume 5), Abad and Sesma described an approach
to overcoming the long CPU times needed to evaluate Hypergeometic1F1[a,b,z]
when |a z | is not <= 1. Their implementation in terms of Bessel functions
and Buchholz polynomials is slower than the built-in Hypergeometic1F1 when
|a z| <= 1, but provides real speed efficiencies  for less favorable
arguements. They include a table of timings and results comparing the two
approaches using Mathematica 2.2 for b=6.8, z=1.2, and varying values of a.

When I run the same routines in Mathematica 3.0. or 4.0, I get different
results for the 'conventional' approach with the built-in functions,
although the Buchholz polynomial calculations agree.

I guess my question really is which result should I believe, or should I
give up in trying to interpret the results of evaluations for unfavorable
arguments? Ideally, is there a way to identify which results are suspect? I
guess it's also possible that the agreement between the latest version and
the Buchholz polynomial form reflects adoption of the later algorithm in the
newer versions of Mathematica, but I'm not sure where to find details as to
the internal algorithm used for this purpose.

The three discrepancies occured at a = -10^3, -10^4, and -10^6 with the
following results
    Mathematica 2.2                                Mathematica 4.0
    -113216                                        - 1.0097 10^-7
    -7.15 10^66                                    -2.73 10^-11
    -3.9 10^913                                     2.67 10^-17


A. Sharma
Atul Sharma MD, FRCP(C)
Pediatric Nephrologist,
McGill University/Montreal Children's Hospital

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