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/* Translated into C++ by SciPy developers in 2024.
* Original header with Copyright information appears below.
*/
/* expn.c
*
* Exponential integral En
*
*
*
* SYNOPSIS:
*
* int n;
* double x, y, expn();
*
* y = expn( n, x );
*
*
*
* DESCRIPTION:
*
* Evaluates the exponential integral
*
* inf.
* -
* | | -xt
* | e
* E (x) = | ---- dt.
* n | n
* | | t
* -
* 1
*
*
* Both n and x must be nonnegative.
*
* The routine employs either a power series, a continued
* fraction, or an asymptotic formula depending on the
* relative values of n and x.
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0, 30 10000 1.7e-15 3.6e-16
*
*/
/* expn.c */
/* Cephes Math Library Release 1.1: March, 1985
* Copyright 1985 by Stephen L. Moshier
* Direct inquiries to 30 Frost Street, Cambridge, MA 02140 */
/* Sources
* [1] NIST, "The Digital Library of Mathematical Functions", dlmf.nist.gov
*/
/* Scipy changes:
* - 09-10-2016: improved asymptotic expansion for large n
*/
#pragma once
#include "../config.h"
#include "../error.h"
#include "const.h"
#include "rgamma.h"
#include "polevl.h"
namespace xsf {
namespace cephes {
namespace detail {
constexpr int expn_nA = 13;
constexpr double expn_A0[] = {1.00000000000000000};
constexpr double expn_A1[] = {1.00000000000000000};
constexpr double expn_A2[] = {-2.00000000000000000, 1.00000000000000000};
constexpr double expn_A3[] = {6.00000000000000000, -8.00000000000000000, 1.00000000000000000};
constexpr double expn_A4[] = {-24.0000000000000000, 58.0000000000000000, -22.0000000000000000,
1.00000000000000000};
constexpr double expn_A5[] = {120.000000000000000, -444.000000000000000, 328.000000000000000,
-52.0000000000000000, 1.00000000000000000};
constexpr double expn_A6[] = {-720.000000000000000, 3708.00000000000000, -4400.00000000000000,
1452.00000000000000, -114.000000000000000, 1.00000000000000000};
constexpr double expn_A7[] = {5040.00000000000000, -33984.0000000000000, 58140.0000000000000,
-32120.0000000000000, 5610.00000000000000, -240.000000000000000,
1.00000000000000000};
constexpr double expn_A8[] = {-40320.0000000000000, 341136.000000000000, -785304.000000000000,
644020.000000000000, -195800.000000000000, 19950.0000000000000,
-494.000000000000000, 1.00000000000000000};
constexpr double expn_A9[] = {362880.000000000000, -3733920.00000000000, 11026296.0000000000,
-12440064.0000000000, 5765500.00000000000, -1062500.00000000000,
67260.0000000000000, -1004.00000000000000, 1.00000000000000000};
constexpr double expn_A10[] = {-3628800.00000000000, 44339040.0000000000, -162186912.000000000,
238904904.000000000, -155357384.000000000, 44765000.0000000000,
-5326160.00000000000, 218848.000000000000, -2026.00000000000000,
1.00000000000000000};
constexpr double expn_A11[] = {39916800.0000000000, -568356480.000000000, 2507481216.00000000,
-4642163952.00000000, 4002695088.00000000, -1648384304.00000000,
314369720.000000000, -25243904.0000000000, 695038.000000000000,
-4072.00000000000000, 1.00000000000000000};
constexpr double expn_A12[] = {-479001600.000000000, 7827719040.00000000, -40788301824.0000000,
92199790224.0000000, -101180433024.000000, 56041398784.0000000,
-15548960784.0000000, 2051482776.00000000, -114876376.000000000,
2170626.00000000000, -8166.00000000000000, 1.00000000000000000};
constexpr const double *expn_A[] = {expn_A0, expn_A1, expn_A2, expn_A3, expn_A4, expn_A5, expn_A6,
expn_A7, expn_A8, expn_A9, expn_A10, expn_A11, expn_A12};
constexpr int expn_Adegs[] = {0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11};
/* Asymptotic expansion for large n, DLMF 8.20(ii) */
XSF_HOST_DEVICE double expn_large_n(int n, double x) {
int k;
double p = n;
double lambda = x / p;
double multiplier = 1 / p / (lambda + 1) / (lambda + 1);
double fac = 1;
double res = 1; /* A[0] = 1 */
double expfac, term;
expfac = std::exp(-lambda * p) / (lambda + 1) / p;
if (expfac == 0) {
set_error("expn", SF_ERROR_UNDERFLOW, NULL);
return 0;
}
/* Do the k = 1 term outside the loop since A[1] = 1 */
fac *= multiplier;
res += fac;
for (k = 2; k < expn_nA; k++) {
fac *= multiplier;
term = fac * polevl(lambda, expn_A[k], expn_Adegs[k]);
res += term;
if (std::abs(term) < MACHEP * std::abs(res)) {
break;
}
}
return expfac * res;
}
} // namespace detail
XSF_HOST_DEVICE double expn(int n, double x) {
double ans, r, t, yk, xk;
double pk, pkm1, pkm2, qk, qkm1, qkm2;
double psi, z;
int i, k;
constexpr double big = 1.44115188075855872E+17;
if (std::isnan(x)) {
return std::numeric_limits<double>::quiet_NaN();
} else if (n < 0 || x < 0) {
set_error("expn", SF_ERROR_DOMAIN, NULL);
return std::numeric_limits<double>::quiet_NaN();
}
if (x > detail::MAXLOG) {
return (0.0);
}
if (x == 0.0) {
if (n < 2) {
set_error("expn", SF_ERROR_SINGULAR, NULL);
return std::numeric_limits<double>::infinity();
} else {
return (1.0 / (n - 1.0));
}
}
if (n == 0) {
return (std::exp(-x) / x);
}
/* Asymptotic expansion for large n, DLMF 8.20(ii) */
if (n > 50) {
ans = detail::expn_large_n(n, x);
return ans;
}
/* Continued fraction, DLMF 8.19.17 */
if (x > 1.0) {
k = 1;
pkm2 = 1.0;
qkm2 = x;
pkm1 = 1.0;
qkm1 = x + n;
ans = pkm1 / qkm1;
do {
k += 1;
if (k & 1) {
yk = 1.0;
xk = n + (k - 1) / 2;
} else {
yk = x;
xk = k / 2;
}
pk = pkm1 * yk + pkm2 * xk;
qk = qkm1 * yk + qkm2 * xk;
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) > big) {
pkm2 /= big;
pkm1 /= big;
qkm2 /= big;
qkm1 /= big;
}
} while (t > detail::MACHEP);
ans *= std::exp(-x);
return ans;
}
/* Power series expansion, DLMF 8.19.8 */
psi = -detail::SCIPY_EULER - std::log(x);
for (i = 1; i < n; i++) {
psi = psi + 1.0 / i;
}
z = -x;
xk = 0.0;
yk = 1.0;
pk = 1.0 - n;
if (n == 1) {
ans = 0.0;
} else {
ans = 1.0 / pk;
}
do {
xk += 1.0;
yk *= z / xk;
pk += 1.0;
if (pk != 0.0) {
ans += yk / pk;
}
if (ans != 0.0)
t = std::abs(yk / ans);
else
t = 1.0;
} while (t > detail::MACHEP);
k = xk;
t = n;
r = n - 1;
ans = (std::pow(z, r) * psi * rgamma(t)) - ans;
return ans;
}
} // namespace cephes
} // namespace xsf
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