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