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/* Translated into C++ by SciPy developers in 2024.
* Original header with Copyright information appears below.
*/
/* 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
* 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.
*
*/
/*
* Cephes Math Library Release 2.8: June, 2000
* Copyright 1984, 1987, 1988, 2000 by Stephen L. Moshier
*/
#pragma once
#include "../config.h"
#include "../error.h"
#include "const.h"
#include "gamma.h"
#include "rgamma.h"
namespace xsf {
namespace cephes {
namespace detail {
/* the `type` parameter determines what converging factor to use */
XSF_HOST_DEVICE inline double hyp2f0(double a, double b, double x, int type, 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 = std::abs(u);
if ((temp > 1.0) && (maxt > (std::numeric_limits<double>::max() / temp)))
goto error;
a0 *= u;
t = std::abs(a0);
/* terminating condition for asymptotic series:
* the series is divergent (if a or b is not a negative integer),
* but its leading part can be used as an asymptotic expansion
*/
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 = std::abs(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) + std::abs(a0);
done:
sum += alast;
return (sum);
/* series blew up: */
error:
*err = std::numeric_limits<double>::infinity();
set_error("hyperg", SF_ERROR_NO_RESULT, NULL);
return (sum);
}
/* 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) )
*/
XSF_HOST_DEVICE inline double hy1f1a(double a, double b, double x, double *err) {
double h1, h2, t, u, temp, acanc, asum, err1, err2;
if (x == 0) {
acanc = 1.0;
asum = std::numeric_limits<double>::infinity();
goto adone;
}
temp = std::log(std::abs(x));
t = x + temp * (a - b);
u = -temp * a;
if (b > 0) {
temp = xsf::cephes::lgam(b);
t += temp;
u += temp;
}
h1 = hyp2f0(a, a - b + 1, -1.0 / x, 1, &err1);
temp = std::exp(u) * xsf::cephes::rgamma(b - a);
h1 *= temp;
err1 *= temp;
h2 = hyp2f0(b - a, 1.0 - a, 1.0 / x, 2, &err2);
if (a < 0)
temp = std::exp(t) * xsf::cephes::rgamma(a);
else
temp = std::exp(t - xsf::cephes::lgam(a));
h2 *= temp;
err2 *= temp;
if (x < 0.0)
asum = h1;
else
asum = h2;
acanc = std::abs(err1) + std::abs(err2);
if (b < 0) {
temp = xsf::cephes::Gamma(b);
asum *= temp;
acanc *= std::abs(temp);
}
if (asum != 0.0)
acanc /= std::abs(asum);
if (acanc != acanc)
/* nan */
acanc = 1.0;
if (std::isinf(asum))
/* infinity */
acanc = 0;
acanc *= 30.0; /* fudge factor, since error of asymptotic formula
* often seems this much larger than advertised */
adone:
*err = acanc;
return (asum);
}
/* Power series summation for confluent hypergeometric function */
XSF_HOST_DEVICE inline double hy1f1p(double a, double b, double x, double *err) {
double n, a0, sum, t, u, temp, maxn;
double an, bn, maxt;
double y, c, sumc;
/* set up for power series summation */
an = a;
bn = b;
a0 = 1.0;
sum = 1.0;
c = 0.0;
n = 1.0;
t = 1.0;
maxt = 0.0;
*err = 1.0;
maxn = 200.0 + 2 * fabs(a) + 2 * fabs(b);
while (t > MACHEP) {
if (bn == 0) { /* check bn first since if both */
sf_error("hyperg", SF_ERROR_SINGULAR, NULL);
return (std::numeric_limits<double>::infinity()); /* an and bn are zero it is */
}
if (an == 0) /* a singularity */
return (sum);
if (n > maxn) {
/* too many terms; take the last one as error estimate */
c = std::abs(c) + std::abs(t) * 50.0;
goto pdone;
}
u = x * (an / (bn * n));
/* check for blowup */
temp = std::abs(u);
if ((temp > 1.0) && (maxt > (std::numeric_limits<double>::max() / temp))) {
*err = 1.0; /* blowup: estimate 100% error */
return sum;
}
a0 *= u;
y = a0 - c;
sumc = sum + y;
c = (sumc - sum) - y;
sum = sumc;
t = std::abs(a0);
an += 1.0;
bn += 1.0;
n += 1.0;
}
pdone:
/* estimate error due to roundoff and cancellation */
if (sum != 0.0) {
*err = std::abs(c / sum);
} else {
*err = std::abs(c);
}
if (*err != *err) {
/* nan */
*err = 1.0;
}
return (sum);
}
} // namespace detail
XSF_HOST_DEVICE inline double hyperg(double a, double b, double x) {
double asum, psum, acanc, pcanc, temp;
/* See if a Kummer transformation will help */
temp = b - a;
if (std::abs(temp) < 0.001 * std::abs(a))
return (exp(x) * hyperg(temp, b, -x));
/* Try power & asymptotic series, starting from the one that is likely OK */
if (std::abs(x) < 10 + std::abs(a) + std::abs(b)) {
psum = detail::hy1f1p(a, b, x, &pcanc);
if (pcanc < 1.0e-15)
goto done;
asum = detail::hy1f1a(a, b, x, &acanc);
} else {
psum = detail::hy1f1a(a, b, x, &pcanc);
if (pcanc < 1.0e-15)
goto done;
asum = detail::hy1f1p(a, b, x, &acanc);
}
/* Pick the result with less estimated error */
if (acanc < pcanc) {
pcanc = acanc;
psum = asum;
}
done:
if (pcanc > 1.0e-12)
set_error("hyperg", SF_ERROR_LOSS, NULL);
return (psum);
}
} // namespace cephes
} // namespace xsf
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