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Numerical errors on computing the hypergeometric function of the second

From: Pulong Ma
Subject: Numerical errors on computing the hypergeometric function of the second kind
Date: Wed, 29 Apr 2020 05:01:01 +0000


I was trying to use the hypergeometric function of the second kind from the GSL 
library, and I found that the current implementation has serious numerical 

The routines gsl_sf_hyperg_U_e and gsl_sf_hyperg_U_e10_e are not able to give 
correct answers when the argument a is chosen to be some numbers say {10, 11, 
12, ..} in the short program. Instead, it outputs Inf’s in the short program 
below. I think this error is probably due to the numerical in-accuracy of the 
current implementation for the hypergeometric function of the second kind in 
the header file.

I found that in MATLAB and Mathematica, I could get correct answers. I am 
wondering whether there is any way to fix this issue.

  *   GSL version: 2.6
  *   Hardware and OS: Mac OS High Sierra
  *   Compiler: g++ 5.1.0
  *   short program

// g++ -Wall -o Ubug.o Ubug.cpp -I/usr/local/include -lgsl -lcblas


#include <gsl/gsl_sf_hyperg.h>
#include <gsl/gsl_errno.h>

using namespace std;

double HypergeometricU(double a, double b, double x){

  gsl_sf_result_e10 result;
  //gsl_sf_hyperg_U_e(a, b, x, &result);
  gsl_sf_hyperg_U_e10_e(a, b, x, &result);

  return result.val;

int main(){
double a1[9] = {8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0};
double a2[9] = {8.5, 9.5, 10.5, 11.5, 12.5, 13.5, 14.5, 15.5, 16.5};

double b=0.5, z=0.1;

    double out1[9]={9999.0};
    double out2[9]={9999.0};


    for(int k=0; k<9; k++){
    out1[k] = HypergeometricU(a1[k],b,z);
    out2[k] = HypergeometricU(a2[k],b,z);

return 0;


Pulong Ma
Postdoctoral Fellow
Statistical and Applied Mathematical Sciences Institute
4051 Research Commons, Suite 300
79 T.W. Alexander Drive, P.O. Box 110207
Durham, NC 27709
Email: address@hidden<mailto:address@hidden>
Phone: (513) 652-2159
URL: https://pulongma.github.io

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