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Re: Numerical accuracy of Hypergeometric2F1
*To*: mathgroup at smc.vnet.net
*Subject*: [mg55733] Re: Numerical accuracy of Hypergeometric2F1
*From*: Paul Abbott <paul at physics.uwa.edu.au>
*Date*: Tue, 5 Apr 2005 03:20:52 -0400 (EDT)
*Organization*: The University of Western Australia
*References*: <d2qj7t$p0o$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
In article <d2qj7t$p0o$1 at smc.vnet.net>,
"Christos Argyropoulos M.D." <chrisarg at fuse.net> wrote:
> Hi,
> Recently I ran into a problem in applied statistics which required
> evaluation of a specific Hypergeometric2F1 functional i.e.
> Hypergeometric2F1[k+1/2,1,3/2,x] where Element[k,Integers], k>0 ,
> Element[x, Reals] and 0<=x<=1.
> It appears that for "large" values of k , Mathematica returns the wrong
> answer .
> If one re-writes the hypergeometric as a polynomial (using the distant
> neighbor relation http://functions.wolfram.com/07.23.17.0007.01), get
> rid of the ratio of Pochhamer functions and then examine the numerical
> values given by the two formulations diverge.
> \!\(fpol[x_, k_] :=
> Module[{}, coeff = Table[1, {k + 1}];
> For[i = 2, i \[LessEqual] k + 1, \(i++\),
> coeff[\([i]\)] =
> coeff[\([i -
> 1]\)]\ \(\(-k\) + i - 1\)\/\(i - k - 1/2\)]; \
> {Hypergeometric2F1[k + 1/2, 1, 3/2,
> x], \(1\/\(2 k -
> 1\)\) \(\[Sum]\+\(i = 1\)\%\(k + 1\)\((\((1 -
> x)\)\^\(-i\)\ \
> coeff[\([i]\)])\)\)}]\)
>
> In[4]:=
> SetPrecision[fpol[0.7,100],16]
> Out[4]=
> \!\({\(-2.5109638442451095387366941924357`15.9546*^56\),
> 2.0578481017803705921545468593643`15.9546*^51}\)
No! You are applying SetPrecision to an expression that has been
computed using machine precision. Instead you want to compute
fpol[SetPrecision[0.7,20], 100]
or
fpol[0.7`20, 100]
or
N[fpol[7/10,100], 20]
This has nothing to do with any specific Hypergeometric2F1. It is a
general property of the way Mathematica's works with machine and
arbitrary precision expressions.
Cheers,
Paul
--
Paul Abbott Phone: +61 8 6488 2734
School of Physics, M013 Fax: +61 8 6488 1014
The University of Western Australia (CRICOS Provider No 00126G)
35 Stirling Highway
Crawley WA 6009 mailto:paul at physics.uwa.edu.au
AUSTRALIA http://physics.uwa.edu.au/~paul
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