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Numerical accuracy of Hypergeometric2F1
*To*: mathgroup at smc.vnet.net
*Subject*: [mg55717] Numerical accuracy of Hypergeometric2F1
*From*: "Christos Argyropoulos M.D." <chrisarg at fuse.net>
*Date*: Mon, 4 Apr 2005 00:59:23 -0400 (EDT)
*Sender*: owner-wri-mathgroup at wolfram.com
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}\)
but
In[5]:=
SetPrecision[fpol[0.7,10],16]
Out[5]=
{57439.31727367684,57439.31727367681}
or graphically ...
ListPlot3D[Table[Apply[(#1/#2)&,fpol[x,k]],{x,0.,1.,.1},{k,1,100}]]
The problems do not seem to be caused by numerical errors secondary to
direct evaluation of high - order polynomials in the equivalent
expression since, using the Horner formulation for the polynomial gives
essentially identical results
\!\(fhorn[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\/\(2 k - 1\)\)
Module[{y = 1/\((1 - x)\), res = 1}, res = 0;
For[i = k + 1, i \[GreaterEqual] \ 1\ , \(i--\),
res = \((res + coeff[\([i]\)])\)\ y\ ]; res]}]\)
In[6]:=
SetPrecision[fhorn[0.7,10],16]
Out[6]=
{57439.31727367684,57439.31727367682}
and
In[7]:=
SetPrecision[fhorn[0.7,100],16]
Out[7]=
\!\({\(-2.5109638442451095387366941924357`15.9546*^56\),
2.0578481017803822228995099773782`15.9546*^51}\)
and graphically :
ListPlot3D[Table[Apply[(#1/#2)&,fhorn[x,k]],{x,0.,1.,.1},{k,1,100}]]
Has anyone else run into similar problems with the Hypergeometric2F1 for
large values of one of the parameters? Is there a quick and dirty way of
knowing when to mistruct the hypergeometric numerical answer returned by
Mathematica ?
Christos Argyropoulos MD PhD
University of Cincinnati,
College of Medicine
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