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/* Translated into C++ by SciPy developers in 2024.
* Original header with Copyright information appears below.
*/
/* ellie.c
*
* Incomplete elliptic integral of the second kind
*
*
*
* SYNOPSIS:
*
* double phi, m, y, ellie();
*
* y = ellie( phi, m );
*
*
*
* DESCRIPTION:
*
* Approximates the integral
*
*
* phi
* -
* | |
* | 2
* E(phi_\m) = | sqrt( 1 - m sin t ) dt
* |
* | |
* -
* 0
*
* of amplitude phi and modulus m, using the arithmetic -
* geometric mean algorithm.
*
*
*
* ACCURACY:
*
* Tested at random arguments with phi in [-10, 10] and m in
* [0, 1].
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -10,10 150000 3.3e-15 1.4e-16
*/
/*
* Cephes Math Library Release 2.0: April, 1987
* Copyright 1984, 1987, 1993 by Stephen L. Moshier
* Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
/* Copyright 2014, Eric W. Moore */
/* Incomplete elliptic integral of second kind */
#pragma once
#include "../config.h"
#include "const.h"
#include "ellpe.h"
#include "ellpk.h"
#include "unity.h"
namespace xsf {
namespace cephes {
namespace detail {
/* To calculate legendre's incomplete elliptical integral of the second kind for
* negative m, we use a power series in phi for small m*phi*phi, an asymptotic
* series in m for large m*phi*phi* and the relation to Carlson's symmetric
* integrals, R_F(x,y,z) and R_D(x,y,z).
*
* E(phi, m) = sin(phi) * R_F(cos(phi)^2, 1 - m * sin(phi)^2, 1.0)
* - m * sin(phi)^3 * R_D(cos(phi)^2, 1 - m * sin(phi)^2, 1.0) / 3
*
* = R_F(c-1, c-m, c) - m * R_D(c-1, c-m, c) / 3
*
* where c = csc(phi)^2. We use the second form of this for (approximately)
* phi > 1/(sqrt(DBL_MAX) ~ 1e-154, where csc(phi)^2 overflows. Elsewhere we
* use the first form, accounting for the smallness of phi.
*
* The algorithm used is described in Carlson, B. C. Numerical computation of
* real or complex elliptic integrals. (1994) https://arxiv.org/abs/math/9409227
* Most variable names reflect Carlson's usage.
*
* In this routine, we assume m < 0 and 0 > phi > pi/2.
*/
XSF_HOST_DEVICE inline double ellie_neg_m(double phi, double m) {
double x, y, z, x1, y1, z1, ret, Q;
double A0f, Af, Xf, Yf, Zf, E2f, E3f, scalef;
double A0d, Ad, seriesn, seriesd, Xd, Yd, Zd, E2d, E3d, E4d, E5d, scaled;
int n = 0;
double mpp = (m * phi) * phi;
if (-mpp < 1e-6 && phi < -m) {
return phi + (mpp * phi * phi / 30.0 - mpp * mpp / 40.0 - mpp / 6.0) * phi;
}
if (-mpp > 1e6) {
double sm = std::sqrt(-m);
double sp = std::sin(phi);
double cp = std::cos(phi);
double a = -cosm1(phi);
double b1 = std::log(4 * sp * sm / (1 + cp));
double b = -(0.5 + b1) / 2.0 / m;
double c = (0.75 + cp / sp / sp - b1) / 16.0 / m / m;
return (a + b + c) * sm;
}
if (phi > 1e-153 && m > -1e200) {
double s = std::sin(phi);
double csc2 = 1.0 / s / s;
scalef = 1.0;
scaled = m / 3.0;
x = 1.0 / std::tan(phi) / std::tan(phi);
y = csc2 - m;
z = csc2;
} else {
scalef = phi;
scaled = mpp * phi / 3.0;
x = 1.0;
y = 1 - mpp;
z = 1.0;
}
if (x == y && x == z) {
return (scalef + scaled / x) / std::sqrt(x);
}
A0f = (x + y + z) / 3.0;
Af = A0f;
A0d = (x + y + 3.0 * z) / 5.0;
Ad = A0d;
x1 = x;
y1 = y;
z1 = z;
seriesd = 0.0;
seriesn = 1.0;
/* Carlson gives 1/pow(3*r, 1.0/6.0) for this constant. if r == eps,
* it is ~338.38. */
/* N.B. This will evaluate its arguments multiple times. */
Q = 400.0 * std::fmax(std::abs(A0f - x), std::fmax(std::abs(A0f - y), std::abs(A0f - z)));
while (Q > std::abs(Af) && Q > std::abs(Ad) && n <= 100) {
double sx = std::sqrt(x1);
double sy = std::sqrt(y1);
double sz = std::sqrt(z1);
double lam = sx * sy + sx * sz + sy * sz;
seriesd += seriesn / (sz * (z1 + lam));
x1 = (x1 + lam) / 4.0;
y1 = (y1 + lam) / 4.0;
z1 = (z1 + lam) / 4.0;
Af = (x1 + y1 + z1) / 3.0;
Ad = (Ad + lam) / 4.0;
n += 1;
Q /= 4.0;
seriesn /= 4.0;
}
Xf = (A0f - x) / Af / (1 << 2 * n);
Yf = (A0f - y) / Af / (1 << 2 * n);
Zf = -(Xf + Yf);
E2f = Xf * Yf - Zf * Zf;
E3f = Xf * Yf * Zf;
ret = scalef * (1.0 - E2f / 10.0 + E3f / 14.0 + E2f * E2f / 24.0 - 3.0 * E2f * E3f / 44.0) / sqrt(Af);
Xd = (A0d - x) / Ad / (1 << 2 * n);
Yd = (A0d - y) / Ad / (1 << 2 * n);
Zd = -(Xd + Yd) / 3.0;
E2d = Xd * Yd - 6.0 * Zd * Zd;
E3d = (3 * Xd * Yd - 8.0 * Zd * Zd) * Zd;
E4d = 3.0 * (Xd * Yd - Zd * Zd) * Zd * Zd;
E5d = Xd * Yd * Zd * Zd * Zd;
ret -= scaled *
(1.0 - 3.0 * E2d / 14.0 + E3d / 6.0 + 9.0 * E2d * E2d / 88.0 - 3.0 * E4d / 22.0 -
9.0 * E2d * E3d / 52.0 + 3.0 * E5d / 26.0) /
(1 << 2 * n) / Ad / sqrt(Ad);
ret -= 3.0 * scaled * seriesd;
return ret;
}
} // namespace detail
XSF_HOST_DEVICE inline double ellie(double phi, double m) {
double a, b, c, e, temp;
double lphi, t, E, denom, npio2;
int d, mod, sign;
if (std::isnan(phi) || std::isnan(m))
return std::numeric_limits<double>::quiet_NaN();
if (m > 1.0)
return std::numeric_limits<double>::quiet_NaN();
;
if (std::isinf(phi))
return phi;
if (std::isinf(m))
return -m;
if (m == 0.0)
return (phi);
lphi = phi;
npio2 = std::floor(lphi / M_PI_2);
if (std::fmod(std::abs(npio2), 2.0) == 1.0)
npio2 += 1;
lphi = lphi - npio2 * M_PI_2;
if (lphi < 0.0) {
lphi = -lphi;
sign = -1;
} else {
sign = 1;
}
a = 1.0 - m;
E = ellpe(m);
if (a == 0.0) {
temp = std::sin(lphi);
goto done;
}
if (a > 1.0) {
temp = detail::ellie_neg_m(lphi, m);
goto done;
}
if (lphi < 0.135) {
double m11 = (((((-7.0 / 2816.0) * m + (5.0 / 1056.0)) * m - (7.0 / 2640.0)) * m + (17.0 / 41580.0)) * m -
(1.0 / 155925.0)) *
m;
double m9 = ((((-5.0 / 1152.0) * m + (1.0 / 144.0)) * m - (1.0 / 360.0)) * m + (1.0 / 5670.0)) * m;
double m7 = ((-m / 112.0 + (1.0 / 84.0)) * m - (1.0 / 315.0)) * m;
double m5 = (-m / 40.0 + (1.0 / 30)) * m;
double m3 = -m / 6.0;
double p2 = lphi * lphi;
temp = ((((m11 * p2 + m9) * p2 + m7) * p2 + m5) * p2 + m3) * p2 * lphi + lphi;
goto done;
}
t = std::tan(lphi);
b = std::sqrt(a);
/* Thanks to Brian Fitzgerald <[email protected]>
* for pointing out an instability near odd multiples of pi/2. */
if (std::abs(t) > 10.0) {
/* Transform the amplitude */
e = 1.0 / (b * t);
/* ... but avoid multiple recursions. */
if (std::abs(e) < 10.0) {
e = std::atan(e);
temp = E + m * std::sin(lphi) * std::sin(e) - ellie(e, m);
goto done;
}
}
c = std::sqrt(m);
a = 1.0;
d = 1;
e = 0.0;
mod = 0;
while (std::abs(c / a) > detail::MACHEP) {
temp = b / a;
lphi = lphi + atan(t * temp) + mod * M_PI;
denom = 1 - temp * t * t;
if (std::abs(denom) > 10 * detail::MACHEP) {
t = t * (1.0 + temp) / denom;
mod = (lphi + M_PI_2) / M_PI;
} else {
t = std::tan(lphi);
mod = static_cast<int>(std::floor((lphi - std::atan(t)) / M_PI));
}
c = (a - b) / 2.0;
temp = std::sqrt(a * b);
a = (a + b) / 2.0;
b = temp;
d += d;
e += c * std::sin(lphi);
}
temp = E / ellpk(1.0 - m);
temp *= (std::atan(t) + mod * M_PI) / (d * a);
temp += e;
done:
if (sign < 0)
temp = -temp;
temp += npio2 * E;
return (temp);
}
} // namespace cephes
} // namespace xsf
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