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/* Translated into C++ by SciPy developers in 2024.
* Original header with Copyright information appears below.
*/
/* igam.c
*
* Incomplete Gamma integral
*
*
*
* SYNOPSIS:
*
* double a, x, y, igam();
*
* y = igam( a, x );
*
* DESCRIPTION:
*
* The function is defined by
*
* x
* -
* 1 | | -t a-1
* igam(a,x) = ----- | e t dt.
* - | |
* | (a) -
* 0
*
*
* In this implementation both arguments must be positive.
* The integral is evaluated by either a power series or
* continued fraction expansion, depending on the relative
* values of a and x.
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0,30 200000 3.6e-14 2.9e-15
* IEEE 0,100 300000 9.9e-14 1.5e-14
*/
/* igamc()
*
* Complemented incomplete Gamma integral
*
*
*
* SYNOPSIS:
*
* double a, x, y, igamc();
*
* y = igamc( a, x );
*
* DESCRIPTION:
*
* The function is defined by
*
*
* igamc(a,x) = 1 - igam(a,x)
*
* inf.
* -
* 1 | | -t a-1
* = ----- | e t dt.
* - | |
* | (a) -
* x
*
*
* In this implementation both arguments must be positive.
* The integral is evaluated by either a power series or
* continued fraction expansion, depending on the relative
* values of a and x.
*
* ACCURACY:
*
* Tested at random a, x.
* a x Relative error:
* arithmetic domain domain # trials peak rms
* IEEE 0.5,100 0,100 200000 1.9e-14 1.7e-15
* IEEE 0.01,0.5 0,100 200000 1.4e-13 1.6e-15
*/
/*
* Cephes Math Library Release 2.0: April, 1987
* Copyright 1985, 1987 by Stephen L. Moshier
* Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
/* Sources
* [1] "The Digital Library of Mathematical Functions", dlmf.nist.gov
* [2] Maddock et. al., "Incomplete Gamma Functions",
* https://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html
*/
/* Scipy changes:
* - 05-01-2016: added asymptotic expansion for igam to improve the
* a ~ x regime.
* - 06-19-2016: additional series expansion added for igamc to
* improve accuracy at small arguments.
* - 06-24-2016: better choice of domain for the asymptotic series;
* improvements in accuracy for the asymptotic series when a and x
* are very close.
*/
#pragma once
#include "../config.h"
#include "../error.h"
#include "const.h"
#include "gamma.h"
#include "igam_asymp_coeff.h"
#include "lanczos.h"
#include "ndtr.h"
#include "unity.h"
namespace xsf {
namespace cephes {
namespace detail {
constexpr int igam_MAXITER = 2000;
constexpr int IGAM = 1;
constexpr int IGAMC = 0;
constexpr double igam_SMALL = 20;
constexpr double igam_LARGE = 200;
constexpr double igam_SMALLRATIO = 0.3;
constexpr double igam_LARGERATIO = 4.5;
constexpr double igam_big = 4.503599627370496e15;
constexpr double igam_biginv = 2.22044604925031308085e-16;
/* Compute
*
* x^a * exp(-x) / gamma(a)
*
* corrected from (15) and (16) in [2] by replacing exp(x - a) with
* exp(a - x).
*/
XSF_HOST_DEVICE inline double igam_fac(double a, double x) {
double ax, fac, res, num;
if (std::abs(a - x) > 0.4 * std::abs(a)) {
ax = a * std::log(x) - x - xsf::cephes::lgam(a);
if (ax < -MAXLOG) {
set_error("igam", SF_ERROR_UNDERFLOW, NULL);
return 0.0;
}
return std::exp(ax);
}
fac = a + xsf::cephes::lanczos_g - 0.5;
res = std::sqrt(fac / std::exp(1)) / xsf::cephes::lanczos_sum_expg_scaled(a);
if ((a < 200) && (x < 200)) {
res *= std::exp(a - x) * std::pow(x / fac, a);
} else {
num = x - a - xsf::cephes::lanczos_g + 0.5;
res *= std::exp(a * xsf::cephes::log1pmx(num / fac) + x * (0.5 - xsf::cephes::lanczos_g) / fac);
}
return res;
}
/* Compute igamc using DLMF 8.9.2. */
XSF_HOST_DEVICE inline double igamc_continued_fraction(double a, double x) {
int i;
double ans, ax, c, yc, r, t, y, z;
double pk, pkm1, pkm2, qk, qkm1, qkm2;
ax = igam_fac(a, x);
if (ax == 0.0) {
return 0.0;
}
/* continued fraction */
y = 1.0 - a;
z = x + y + 1.0;
c = 0.0;
pkm2 = 1.0;
qkm2 = x;
pkm1 = x + 1.0;
qkm1 = z * x;
ans = pkm1 / qkm1;
for (i = 0; i < igam_MAXITER; i++) {
c += 1.0;
y += 1.0;
z += 2.0;
yc = y * c;
pk = pkm1 * z - pkm2 * yc;
qk = qkm1 * z - qkm2 * yc;
if (qk != 0) {
r = pk / qk;
t = std::abs((ans - r) / r);
ans = r;
} else
t = 1.0;
pkm2 = pkm1;
pkm1 = pk;
qkm2 = qkm1;
qkm1 = qk;
if (std::abs(pk) > igam_big) {
pkm2 *= igam_biginv;
pkm1 *= igam_biginv;
qkm2 *= igam_biginv;
qkm1 *= igam_biginv;
}
if (t <= MACHEP) {
break;
}
}
return (ans * ax);
}
/* Compute igam using DLMF 8.11.4. */
XSF_HOST_DEVICE inline double igam_series(double a, double x) {
int i;
double ans, ax, c, r;
ax = igam_fac(a, x);
if (ax == 0.0) {
return 0.0;
}
/* power series */
r = a;
c = 1.0;
ans = 1.0;
for (i = 0; i < igam_MAXITER; i++) {
r += 1.0;
c *= x / r;
ans += c;
if (c <= MACHEP * ans) {
break;
}
}
return (ans * ax / a);
}
/* Compute igamc using DLMF 8.7.3. This is related to the series in
* igam_series but extra care is taken to avoid cancellation.
*/
XSF_HOST_DEVICE inline double igamc_series(double a, double x) {
int n;
double fac = 1;
double sum = 0;
double term, logx;
for (n = 1; n < igam_MAXITER; n++) {
fac *= -x / n;
term = fac / (a + n);
sum += term;
if (std::abs(term) <= MACHEP * std::abs(sum)) {
break;
}
}
logx = std::log(x);
term = -xsf::cephes::expm1(a * logx - xsf::cephes::lgam1p(a));
return term - std::exp(a * logx - xsf::cephes::lgam(a)) * sum;
}
/* Compute igam/igamc using DLMF 8.12.3/8.12.4. */
XSF_HOST_DEVICE inline double asymptotic_series(double a, double x, int func) {
int k, n, sgn;
int maxpow = 0;
double lambda = x / a;
double sigma = (x - a) / a;
double eta, res, ck, ckterm, term, absterm;
double absoldterm = std::numeric_limits<double>::infinity();
double etapow[detail::igam_asymp_coeff_N] = {1};
double sum = 0;
double afac = 1;
if (func == detail::IGAM) {
sgn = -1;
} else {
sgn = 1;
}
if (lambda > 1) {
eta = std::sqrt(-2 * xsf::cephes::log1pmx(sigma));
} else if (lambda < 1) {
eta = -std::sqrt(-2 * xsf::cephes::log1pmx(sigma));
} else {
eta = 0;
}
res = 0.5 * xsf::cephes::erfc(sgn * eta * std::sqrt(a / 2));
for (k = 0; k < igam_asymp_coeff_K; k++) {
ck = igam_asymp_coeff_d[k][0];
for (n = 1; n < igam_asymp_coeff_N; n++) {
if (n > maxpow) {
etapow[n] = eta * etapow[n - 1];
maxpow += 1;
}
ckterm = igam_asymp_coeff_d[k][n] * etapow[n];
ck += ckterm;
if (std::abs(ckterm) < MACHEP * std::abs(ck)) {
break;
}
}
term = ck * afac;
absterm = std::abs(term);
if (absterm > absoldterm) {
break;
}
sum += term;
if (absterm < MACHEP * std::abs(sum)) {
break;
}
absoldterm = absterm;
afac /= a;
}
res += sgn * std::exp(-0.5 * a * eta * eta) * sum / std::sqrt(2 * M_PI * a);
return res;
}
} // namespace detail
XSF_HOST_DEVICE inline double igamc(double a, double x);
XSF_HOST_DEVICE inline double igam(double a, double x) {
double absxma_a;
if (x < 0 || a < 0) {
set_error("gammainc", SF_ERROR_DOMAIN, NULL);
return std::numeric_limits<double>::quiet_NaN();
} else if (a == 0) {
if (x > 0) {
return 1;
} else {
return std::numeric_limits<double>::quiet_NaN();
}
} else if (x == 0) {
/* Zero integration limit */
return 0;
} else if (std::isinf(a)) {
if (std::isinf(x)) {
return std::numeric_limits<double>::quiet_NaN();
}
return 0;
} else if (std::isinf(x)) {
return 1;
}
/* Asymptotic regime where a ~ x; see [2]. */
absxma_a = std::abs(x - a) / a;
if ((a > detail::igam_SMALL) && (a < detail::igam_LARGE) && (absxma_a < detail::igam_SMALLRATIO)) {
return detail::asymptotic_series(a, x, detail::IGAM);
} else if ((a > detail::igam_LARGE) && (absxma_a < detail::igam_LARGERATIO / std::sqrt(a))) {
return detail::asymptotic_series(a, x, detail::IGAM);
}
if ((x > 1.0) && (x > a)) {
return (1.0 - igamc(a, x));
}
return detail::igam_series(a, x);
}
XSF_HOST_DEVICE double igamc(double a, double x) {
double absxma_a;
if (x < 0 || a < 0) {
set_error("gammaincc", SF_ERROR_DOMAIN, NULL);
return std::numeric_limits<double>::quiet_NaN();
} else if (a == 0) {
if (x > 0) {
return 0;
} else {
return std::numeric_limits<double>::quiet_NaN();
}
} else if (x == 0) {
return 1;
} else if (std::isinf(a)) {
if (std::isinf(x)) {
return std::numeric_limits<double>::quiet_NaN();
}
return 1;
} else if (std::isinf(x)) {
return 0;
}
/* Asymptotic regime where a ~ x; see [2]. */
absxma_a = std::abs(x - a) / a;
if ((a > detail::igam_SMALL) && (a < detail::igam_LARGE) && (absxma_a < detail::igam_SMALLRATIO)) {
return detail::asymptotic_series(a, x, detail::IGAMC);
} else if ((a > detail::igam_LARGE) && (absxma_a < detail::igam_LARGERATIO / std::sqrt(a))) {
return detail::asymptotic_series(a, x, detail::IGAMC);
}
/* Everywhere else; see [2]. */
if (x > 1.1) {
if (x < a) {
return 1.0 - detail::igam_series(a, x);
} else {
return detail::igamc_continued_fraction(a, x);
}
} else if (x <= 0.5) {
if (-0.4 / std::log(x) < a) {
return 1.0 - detail::igam_series(a, x);
} else {
return detail::igamc_series(a, x);
}
} else {
if (x * 1.1 < a) {
return 1.0 - detail::igam_series(a, x);
} else {
return detail::igamc_series(a, x);
}
}
}
} // namespace cephes
} // namespace xsf
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