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/* Translated into C++ by SciPy developers in 2024.
* Original header with Copyright information appears below.
*/
/* ellik.c
*
* Incomplete elliptic integral of the first kind
*
*
*
* SYNOPSIS:
*
* double phi, m, y, ellik();
*
* y = ellik( phi, m );
*
*
*
* DESCRIPTION:
*
* Approximates the integral
*
*
*
* phi
* -
* | |
* | dt
* F(phi | m) = | ------------------
* | 2
* | | sqrt( 1 - m sin t )
* -
* 0
*
* of amplitude phi and modulus m, using the arithmetic -
* geometric mean algorithm.
*
*
*
*
* ACCURACY:
*
* Tested at random points with m in [0, 1] and phi as indicated.
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -10,10 200000 7.4e-16 1.0e-16
*
*
*/
/*
* Cephes Math Library Release 2.0: April, 1987
* Copyright 1984, 1987 by Stephen L. Moshier
* Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
/* Copyright 2014, Eric W. Moore */
/* Incomplete elliptic integral of first kind */
#pragma once
#include "../config.h"
#include "../error.h"
#include "const.h"
#include "ellpk.h"
namespace xsf {
namespace cephes {
namespace detail {
/* To calculate legendre's incomplete elliptical integral of the first 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
* integral of the first kind.
*
* F(phi, m) = sin(phi) * R_F(cos(phi)^2, 1 - m * sin(phi)^2, 1.0)
* = R_F(c-1, c-m, c)
*
* 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 ellik_neg_m(double phi, double m) {
double x, y, z, x1, y1, z1, A0, A, Q, X, Y, Z, E2, E3, scale;
int n = 0;
double mpp = (m * phi) * phi;
if (-mpp < 1e-6 && phi < -m) {
return phi + (-mpp * phi * phi / 30.0 + 3.0 * mpp * mpp / 40.0 + mpp / 6.0) * phi;
}
if (-mpp > 4e7) {
double sm = std::sqrt(-m);
double sp = std::sin(phi);
double cp = std::cos(phi);
double a = std::log(4 * sp * sm / (1 + cp));
double b = -(1 + cp / sp / sp - a) / 4 / m;
return (a + b) / sm;
}
if (phi > 1e-153 && m > -1e305) {
double s = std::sin(phi);
double csc2 = 1.0 / (s * s);
scale = 1.0;
x = 1.0 / (std::tan(phi) * std::tan(phi));
y = csc2 - m;
z = csc2;
} else {
scale = phi;
x = 1.0;
y = 1 - m * scale * scale;
z = 1.0;
}
if (x == y && x == z) {
return scale / std::sqrt(x);
}
A0 = (x + y + z) / 3.0;
A = A0;
x1 = x;
y1 = y;
z1 = z;
/* Carlson gives 1/pow(3*r, 1.0/6.0) for this constant. if r == eps,
* it is ~338.38. */
Q = 400.0 * std::fmax(std::abs(A0 - x), std::fmax(std::abs(A0 - y), std::abs(A0 - z)));
while (Q > std::abs(A) && 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;
x1 = (x1 + lam) / 4.0;
y1 = (y1 + lam) / 4.0;
z1 = (z1 + lam) / 4.0;
A = (x1 + y1 + z1) / 3.0;
n += 1;
Q /= 4;
}
X = (A0 - x) / A / (1 << 2 * n);
Y = (A0 - y) / A / (1 << 2 * n);
Z = -(X + Y);
E2 = X * Y - Z * Z;
E3 = X * Y * Z;
return scale * (1.0 - E2 / 10.0 + E3 / 14.0 + E2 * E2 / 24.0 - 3.0 * E2 * E3 / 44.0) / sqrt(A);
}
} // namespace detail
XSF_HOST_DEVICE inline double ellik(double phi, double m) {
double a, b, c, e, temp, t, K, 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) || std::isinf(m)) {
if (std::isinf(m) && std::isfinite(phi))
return 0.0;
else if (std::isinf(phi) && std::isfinite(m))
return phi;
else
return std::numeric_limits<double>::quiet_NaN();
}
if (m == 0.0)
return (phi);
a = 1.0 - m;
if (a == 0.0) {
if (std::abs(phi) >= (double) M_PI_2) {
set_error("ellik", SF_ERROR_SINGULAR, NULL);
return (std::numeric_limits<double>::infinity());
}
/* DLMF 19.6.8, and 4.23.42 */
return std::asinh(std::tan(phi));
}
npio2 = floor(phi / M_PI_2);
if (std::fmod(std::abs(npio2), 2.0) == 1.0)
npio2 += 1;
if (npio2 != 0.0) {
K = ellpk(a);
phi = phi - npio2 * M_PI_2;
} else
K = 0.0;
if (phi < 0.0) {
phi = -phi;
sign = -1;
} else
sign = 0;
if (a > 1.0) {
temp = detail::ellik_neg_m(phi, m);
goto done;
}
b = std::sqrt(a);
t = std::tan(phi);
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);
if (npio2 == 0)
K = ellpk(a);
temp = K - ellik(e, m);
goto done;
}
}
a = 1.0;
c = std::sqrt(m);
d = 1;
mod = 0;
while (std::abs(c / a) > detail::MACHEP) {
temp = b / a;
phi = phi + atan(t * temp) + mod * M_PI;
denom = 1.0 - temp * t * t;
if (std::abs(denom) > 10 * detail::MACHEP) {
t = t * (1.0 + temp) / denom;
mod = (phi + M_PI_2) / M_PI;
} else {
t = std::tan(phi);
mod = static_cast<int>(std::floor((phi - std::atan(t)) / M_PI));
}
c = (a - b) / 2.0;
temp = std::sqrt(a * b);
a = (a + b) / 2.0;
b = temp;
d += d;
}
temp = (std::atan(t) + mod * M_PI) / (d * a);
done:
if (sign < 0)
temp = -temp;
temp += npio2 * K;
return (temp);
}
} // namespace cephes
} // namespace xsf
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