/* hyperg.c
*
* Confluent hypergeometric function
*
*
*
* SYNOPSIS:
*
* double a, b, x, y, hyperg();
*
* y = hyperg( a, b, x );
*
*
*
* DESCRIPTION:
*
* Computes the confluent hypergeometric function
*
* 1 2
* a x a(a+1) x
* F ( a,b;x ) = 1 + ---- + --------- + ...
* 1 1 b 1! b(b+1) 2!
*
* Many higher transcendental functions are special cases of
* this power series.
*
* As is evident from the formula, b must not be a negative
* integer or zero unless a is an integer with 0 >= a > b.
*
* The routine attempts both a direct summation of the series
* and an asymptotic expansion. In each case error due to
* roundoff, cancellation, and nonconvergence is estimated.
* The result with smaller estimated error is returned.
*
*
*
* ACCURACY:
*
* Tested at random points (a, b, x), all three variables
* ranging from 0 to 30.
* Relative error:
* arithmetic domain # trials peak rms
* DEC 0,30 2000 1.2e-15 1.3e-16
* IEEE 0,30 30000 1.8e-14 1.1e-15
*
* Larger errors can be observed when b is near a negative
* integer or zero. Certain combinations of arguments yield
* serious cancellation error in the power series summation
* and also are not in the region of near convergence of the
* asymptotic series. An error message is printed if the
* self-estimated relative error is greater than 1.0e-12.
*
*/
/* hyperg.c */
/*
Cephes Math Library Release 2.1: November, 1988
Copyright 1984, 1987, 1988 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
#include "mconf.h"
#include "cephes.h"
static double hy1f1p( double a, double b, double x, double *err );
static double hy1f1a( double a, double b, double x, double *err );
extern double MAXNUM, MACHEP;
double hyperg( a, b, x)
double a, b, x;
{
double asum, psum, acanc, pcanc=0;
/* See if a Kummer transformation will help */
/*
temp = b - a;
if( fabs(temp) < fabs(b) )
return( exp(x) * hyperg( temp, b, -x ) );
*/
psum = hy1f1p( a, b, x, &pcanc );
if( pcanc < 1.0e-15 )
goto done;
/* try asymptotic series */
asum = hy1f1a( a, b, x, &acanc );
/* Pick the result with less estimated error */
if( acanc < pcanc )
{
pcanc = acanc;
psum = asum;
}
done:
if( pcanc > 1.0e-12 )
mtherr( "hyperg", PLOSS );
return( psum );
}
/* Power series summation for confluent hypergeometric function */
static double hy1f1p( a, b, x, err )
double a, b, x;
double *err;
{
double n, a0, sum, t, u, temp;
double an, bn, maxt, pcanc;
/* set up for power series summation */
an = a;
bn = b;
a0 = 1.0;
sum = 1.0;
n = 1.0;
t = 1.0;
maxt = 0.0;
while( t > MACHEP )
{
if( bn == 0 ) /* check bn first since if both */
{
mtherr( "hyperg", SING );
return( MAXNUM ); /* an and bn are zero it is */
}
if( an == 0 ) /* a singularity */
return( sum );
if( n > 200 )
goto pdone;
u = x * ( an / (bn * n) );
/* check for blowup */
temp = fabs(u);
if( (temp > 1.0 ) && (maxt > (MAXNUM/temp)) )
{
pcanc = 1.0; /* estimate 100% error */
goto blowup;
}
a0 *= u;
sum += a0;
t = fabs(a0);
if( t > maxt )
maxt = t;
/*
if( (maxt/fabs(sum)) > 1.0e17 )
{
pcanc = 1.0;
goto blowup;
}
*/
an += 1.0;
bn += 1.0;
n += 1.0;
}
pdone:
/* estimate error due to roundoff and cancellation */
if( sum != 0.0 )
maxt /= fabs(sum);
maxt *= MACHEP; /* this way avoids multiply overflow */
pcanc = fabs( MACHEP * n + maxt );
blowup:
*err = pcanc;
return( sum );
}
/* hy1f1a() */
/* asymptotic formula for hypergeometric function:
*
* ( -a
* -- ( |z|
* | (b) ( -------- 2f0( a, 1+a-b, -1/x )
* ( --
* ( | (b-a)
*
*
* x a-b )
* e |x| )
* + -------- 2f0( b-a, 1-a, 1/x ) )
* -- )
* | (a) )
*/
static double hy1f1a( a, b, x, err )
double a, b, x;
double *err;
{
double h1, h2, t, u, temp, acanc, asum, err1, err2;
if( x == 0 )
{
acanc = 1.0;
asum = MAXNUM;
goto adone;
}
temp = log( fabs(x) );
t = x + temp * (a-b);
u = -temp * a;
if( b > 0 )
{
temp = lgam(b);
t += temp;
u += temp;
}
h1 = hyp2f0( a, a-b+1, -1.0/x, 1, &err1 );
temp = exp(u) / true_gamma(b-a);
h1 *= temp;
err1 *= temp;
h2 = hyp2f0( b-a, 1.0-a, 1.0/x, 2, &err2 );
if( a < 0 )
temp = exp(t) / true_gamma(a);
else
temp = exp( t - lgam(a) );
h2 *= temp;
err2 *= temp;
if( x < 0.0 )
asum = h1;
else
asum = h2;
acanc = fabs(err1) + fabs(err2);
if( b < 0 )
{
temp = true_gamma(b);
asum *= temp;
acanc *= fabs(temp);
}
if( asum != 0.0 )
acanc /= fabs(asum);
acanc *= 30.0; /* fudge factor, since error of asymptotic formula
* often seems this much larger than advertised */
adone:
*err = acanc;
return( asum );
}
/* hyp2f0() */
double hyp2f0( a, b, x, type, err )
double a, b, x;
int type; /* determines what converging factor to use */
double *err;
{
double a0, alast, t, tlast, maxt;
double n, an, bn, u, sum, temp;
an = a;
bn = b;
a0 = 1.0e0;
alast = 1.0e0;
sum = 0.0;
n = 1.0e0;
t = 1.0e0;
tlast = 1.0e9;
maxt = 0.0;
do
{
if( an == 0 )
goto pdone;
if( bn == 0 )
goto pdone;
u = an * (bn * x / n);
/* check for blowup */
temp = fabs(u);
if( (temp > 1.0 ) && (maxt > (MAXNUM/temp)) )
goto error;
a0 *= u;
t = fabs(a0);
/* terminating condition for asymptotic series */
if( t > tlast )
goto ndone;
tlast = t;
sum += alast; /* the sum is one term behind */
alast = a0;
if( n > 200 )
goto ndone;
an += 1.0e0;
bn += 1.0e0;
n += 1.0e0;
if( t > maxt )
maxt = t;
}
while( t > MACHEP );
pdone: /* series converged! */
/* estimate error due to roundoff and cancellation */
*err = fabs( MACHEP * (n + maxt) );
alast = a0;
goto done;
ndone: /* series did not converge */
/* The following "Converging factors" are supposed to improve accuracy,
* but do not actually seem to accomplish very much. */
n -= 1.0;
x = 1.0/x;
switch( type ) /* "type" given as subroutine argument */
{
case 1:
alast *= ( 0.5 + (0.125 + 0.25*b - 0.5*a + 0.25*x - 0.25*n)/x );
break;
case 2:
alast *= 2.0/3.0 - b + 2.0*a + x - n;
break;
default:
;
}
/* estimate error due to roundoff, cancellation, and nonconvergence */
*err = MACHEP * (n + maxt) + fabs ( a0 );
done:
sum += alast;
return( sum );
/* series blew up: */
error:
*err = MAXNUM;
mtherr( "hyperg", TLOSS );
return( sum );
}
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