// Special functions -*- C++ -*-
// Copyright (C) 2006-2013 Free Software Foundation, Inc.
//
// This file is part of the GNU ISO C++ Library. This library is free
// software; you can redistribute it and/or modify it under the
// terms of the GNU General Public License as published by the
// Free Software Foundation; either version 3, or (at your option)
// any later version.
//
// This library is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
// GNU General Public License for more details.
//
// Under Section 7 of GPL version 3, you are granted additional
// permissions described in the GCC Runtime Library Exception, version
// 3.1, as published by the Free Software Foundation.
// You should have received a copy of the GNU General Public License and
// a copy of the GCC Runtime Library Exception along with this program;
// see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
// <http://www.gnu.org/licenses/>.
/** @file tr1/ell_integral.tcc
* This is an internal header file, included by other library headers.
* Do not attempt to use it directly. @headername{tr1/cmath}
*/
//
// ISO C++ 14882 TR1: 5.2 Special functions
//
// Written by Edward Smith-Rowland based on:
// (1) B. C. Carlson Numer. Math. 33, 1 (1979)
// (2) B. C. Carlson, Special Functions of Applied Mathematics (1977)
// (3) The Gnu Scientific Library, http://www.gnu.org/software/gsl
// (4) Numerical Recipes in C, 2nd ed, by W. H. Press, S. A. Teukolsky,
// W. T. Vetterling, B. P. Flannery, Cambridge University Press
// (1992), pp. 261-269
#ifndef _GLIBCXX_TR1_ELL_INTEGRAL_TCC
#define _GLIBCXX_TR1_ELL_INTEGRAL_TCC 1
namespace std _GLIBCXX_VISIBILITY(default)
{
namespace tr1
{
// [5.2] Special functions
// Implementation-space details.
namespace __detail
{
_GLIBCXX_BEGIN_NAMESPACE_VERSION
/**
* @brief Return the Carlson elliptic function @f$ R_F(x,y,z) @f$
* of the first kind.
*
* The Carlson elliptic function of the first kind is defined by:
* @f[
* R_F(x,y,z) = \frac{1}{2} \int_0^\infty
* \frac{dt}{(t + x)^{1/2}(t + y)^{1/2}(t + z)^{1/2}}
* @f]
*
* @param __x The first of three symmetric arguments.
* @param __y The second of three symmetric arguments.
* @param __z The third of three symmetric arguments.
* @return The Carlson elliptic function of the first kind.
*/
template<typename _Tp>
_Tp
__ellint_rf(_Tp __x, _Tp __y, _Tp __z)
{
const _Tp __min = std::numeric_limits<_Tp>::min();
const _Tp __max = std::numeric_limits<_Tp>::max();
const _Tp __lolim = _Tp(5) * __min;
const _Tp __uplim = __max / _Tp(5);
if (__x < _Tp(0) || __y < _Tp(0) || __z < _Tp(0))
std::__throw_domain_error(__N("Argument less than zero "
"in __ellint_rf."));
else if (__x + __y < __lolim || __x + __z < __lolim
|| __y + __z < __lolim)
std::__throw_domain_error(__N("Argument too small in __ellint_rf"));
else
{
const _Tp __c0 = _Tp(1) / _Tp(4);
const _Tp __c1 = _Tp(1) / _Tp(24);
const _Tp __c2 = _Tp(1) / _Tp(10);
const _Tp __c3 = _Tp(3) / _Tp(44);
const _Tp __c4 = _Tp(1) / _Tp(14);
_Tp __xn = __x;
_Tp __yn = __y;
_Tp __zn = __z;
const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
const _Tp __errtol = std::pow(__eps, _Tp(1) / _Tp(6));
_Tp __mu;
_Tp __xndev, __yndev, __zndev;
const unsigned int __max_iter = 100;
for (unsigned int __iter = 0; __iter < __max_iter; ++__iter)
{
__mu = (__xn + __yn + __zn) / _Tp(3);
__xndev = 2 - (__mu + __xn) / __mu;
__yndev = 2 - (__mu + __yn) / __mu;
__zndev = 2 - (__mu + __zn) / __mu;
_Tp __epsilon = std::max(std::abs(__xndev), std::abs(__yndev));
__epsilon = std::max(__epsilon, std::abs(__zndev));
if (__epsilon < __errtol)
break;
const _Tp __xnroot = std::sqrt(__xn);
const _Tp __ynroot = std::sqrt(__yn);
const _Tp __znroot = std::sqrt(__zn);
const _Tp __lambda = __xnroot * (__ynroot + __znroot)
+ __ynroot * __znroot;
__xn = __c0 * (__xn + __lambda);
__yn = __c0 * (__yn + __lambda);
__zn = __c0 * (__zn + __lambda);
}
const _Tp __e2 = __xndev * __yndev - __zndev * __zndev;
const _Tp __e3 = __xndev * __yndev * __zndev;
const _Tp __s = _Tp(1) + (__c1 * __e2 - __c2 - __c3 * __e3) * __e2
+ __c4 * __e3;
return __s / std::sqrt(__mu);
}
}
/**
* @brief Return the complete elliptic integral of the first kind
* @f$ K(k) @f$ by series expansion.
*
* The complete elliptic integral of the first kind is defined as
* @f[
* K(k) = F(k,\pi/2) = \int_0^{\pi/2}\frac{d\theta}
* {\sqrt{1 - k^2sin^2\theta}}
* @f]
*
* This routine is not bad as long as |k| is somewhat smaller than 1
* but is not is good as the Carlson elliptic integral formulation.
*
* @param __k The argument of the complete elliptic function.
* @return The complete elliptic function of the first kind.
*/
template<typename _Tp>
_Tp
__comp_ellint_1_series(_Tp __k)
{
const _Tp __kk = __k * __k;
_Tp __term = __kk / _Tp(4);
_Tp __sum = _Tp(1) + __term;
const unsigned int __max_iter = 1000;
for (unsigned int __i = 2; __i < __max_iter; ++__i)
{
__term *= (2 * __i - 1) * __kk / (2 * __i);
if (__term < std::numeric_limits<_Tp>::epsilon())
break;
__sum += __term;
}
return __numeric_constants<_Tp>::__pi_2() * __sum;
}
/**
* @brief Return the complete elliptic integral of the first kind
* @f$ K(k) @f$ using the Carlson formulation.
*
* The complete elliptic integral of the first kind is defined as
* @f[
* K(k) = F(k,\pi/2) = \int_0^{\pi/2}\frac{d\theta}
* {\sqrt{1 - k^2 sin^2\theta}}
* @f]
* where @f$ F(k,\phi) @f$ is the incomplete elliptic integral of the
* first kind.
*
* @param __k The argument of the complete elliptic function.
* @return The complete elliptic function of the first kind.
*/
template<typename _Tp>
_Tp
__comp_ellint_1(_Tp __k)
{
if (__isnan(__k))
return std::numeric_limits<_Tp>::quiet_NaN();
else if (std::abs(__k) >= _Tp(1))
return std::numeric_limits<_Tp>::quiet_NaN();
else
return __ellint_rf(_Tp(0), _Tp(1) - __k * __k, _Tp(1));
}
/**
* @brief Return the incomplete elliptic integral of the first kind
* @f$ F(k,\phi) @f$ using the Carlson formulation.
*
* The incomplete elliptic integral of the first kind is defined as
* @f[
* F(k,\phi) = \int_0^{\phi}\frac{d\theta}
* {\sqrt{1 - k^2 sin^2\theta}}
* @f]
*
* @param __k The argument of the elliptic function.
* @param __phi The integral limit argument of the elliptic function.
* @return The elliptic function of the first kind.
*/
template<typename _Tp>
_Tp
__ellint_1(_Tp __k, _Tp __phi)
{
if (__isnan(__k) || __isnan(__phi))
return std::numeric_limits<_Tp>::quiet_NaN();
else if (std::abs(__k) > _Tp(1))
std::__throw_domain_error(__N("Bad argument in __ellint_1."));
else
{
// Reduce phi to -pi/2 < phi < +pi/2.
const int __n = std::floor(__phi / __numeric_constants<_Tp>::__pi()
+ _Tp(0.5L));
const _Tp __phi_red = __phi
- __n * __numeric_constants<_Tp>::__pi();
const _Tp __s = std::sin(__phi_red);
const _Tp __c = std::cos(__phi_red);
const _Tp __F = __s
* __ellint_rf(__c * __c,
_Tp(1) - __k * __k * __s * __s, _Tp(1));
if (__n == 0)
return __F;
else
return __F + _Tp(2) * __n * __comp_ellint_1(__k);
}
}
/**
* @brief Return the complete elliptic integral of the second kind
* @f$ E(k) @f$ by series expansion.
*
* The complete elliptic integral of the second kind is defined as
* @f[
* E(k,\pi/2) = \int_0^{\pi/2}\sqrt{1 - k^2 sin^2\theta}
* @f]
*
* This routine is not bad as long as |k| is somewhat smaller than 1
* but is not is good as the Carlson elliptic integral formulation.
*
* @param __k The argument of the complete elliptic function.
* @return The complete elliptic function of the second kind.
*/
template<typename _Tp>
_Tp
__comp_ellint_2_series(_Tp __k)
{
const _Tp __kk = __k * __k;
_Tp __term = __kk;
_Tp __sum = __term;
const unsigned int __max_iter = 1000;
for (unsigned int __i = 2; __i < __max_iter; ++__i)
{
const _Tp __i2m = 2 * __i - 1;
const _Tp __i2 = 2 * __i;
__term *= __i2m * __i2m * __kk / (__i2 * __i2);
if (__term < std::numeric_limits<_Tp>::epsilon())
break;
__sum += __term / __i2m;
}
return __numeric_constants<_Tp>::__pi_2() * (_Tp(1) - __sum);
}
/**
* @brief Return the Carlson elliptic function of the second kind
* @f$ R_D(x,y,z) = R_J(x,y,z,z) @f$ where
* @f$ R_J(x,y,z,p) @f$ is the Carlson elliptic function
* of the third kind.
*
* The Carlson elliptic function of the second kind is defined by:
* @f[
* R_D(x,y,z) = \frac{3}{2} \int_0^\infty
* \frac{dt}{(t + x)^{1/2}(t + y)^{1/2}(t + z)^{3/2}}
* @f]
*
* Based on Carlson's algorithms:
* - B. C. Carlson Numer. Math. 33, 1 (1979)
* - B. C. Carlson, Special Functions of Applied Mathematics (1977)
* - Numerical Recipes in C, 2nd ed, pp. 261-269,
* by Press, Teukolsky, Vetterling, Flannery (1992)
*
* @param __x The first of two symmetric arguments.
* @param __y The second of two symmetric arguments.
* @param __z The third argument.
* @return The Carlson elliptic function of the second kind.
*/
template<typename _Tp>
_Tp
__ellint_rd(_Tp __x, _Tp __y, _Tp __z)
{
const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
const _Tp __errtol = std::pow(__eps / _Tp(8), _Tp(1) / _Tp(6));
const _Tp __min = std::numeric_limits<_Tp>::min();
const _Tp __max = std::numeric_limits<_Tp>::max();
const _Tp __lolim = _Tp(2) / std::pow(__max, _Tp(2) / _Tp(3));
const _Tp __uplim = std::pow(_Tp(0.1L) * __errtol / __min, _Tp(2) / _Tp(3));
if (__x < _Tp(0) || __y < _Tp(0))
std::__throw_domain_error(__N("Argument less than zero "
"in __ellint_rd."));
else if (__x + __y < __lolim || __z < __lolim)
std::__throw_domain_error(__N("Argument too small "
"in __ellint_rd."));
else
{
const _Tp __c0 = _Tp(1) / _Tp(4);
const _Tp __c1 = _Tp(3) / _Tp(14);
const _Tp __c2 = _Tp(1) / _Tp(6);
const _Tp __c3 = _Tp(9) / _Tp(22);
const _Tp __c4 = _Tp(3) / _Tp(26);
_Tp __xn = __x;
_Tp __yn = __y;
_Tp __zn = __z;
_Tp __sigma = _Tp(0);
_Tp __power4 = _Tp(1);
_Tp __mu;
_Tp __xndev, __yndev, __zndev;
const unsigned int __max_iter = 100;
for (unsigned int __iter = 0; __iter < __max_iter; ++__iter)
{
__mu = (__xn + __yn + _Tp(3) * __zn) / _Tp(5);
__xndev = (__mu - __xn) / __mu;
__yndev = (__mu - __yn) / __mu;
__zndev = (__mu - __zn) / __mu;
_Tp __epsilon = std::max(std::abs(__xndev), std::abs(__yndev));
__epsilon = std::max(__epsilon, std::abs(__zndev));
if (__epsilon < __errtol)
break;
_Tp __xnroot = std::sqrt(__xn);
_Tp __ynroot = std::sqrt(__yn);
_Tp __znroot = std::sqrt(__zn);
_Tp __lambda = __xnroot * (__ynroot + __znroot)
+ __ynroot * __znroot;
__sigma += __power4 / (__znroot * (__zn + __lambda));
__power4 *= __c0;
__xn = __c0 * (__xn + __lambda);
__yn = __c0 * (__yn + __lambda);
__zn = __c0 * (__zn + __lambda);
}
// Note: __ea is an SPU badname.
_Tp __eaa = __xndev * __yndev;
_Tp __eb = __zndev * __zndev;
_Tp __ec = __eaa - __eb;
_Tp __ed = __eaa - _Tp(6) * __eb;
_Tp __ef = __ed + __ec + __ec;
_Tp __s1 = __ed * (-__c1 + __c3 * __ed
/ _Tp(3) - _Tp(3) * __c4 * __zndev * __ef
/ _Tp(2));
_Tp __s2 = __zndev
* (__c2 * __ef
+ __zndev * (-__c3 * __ec - __zndev * __c4 - __eaa));
return _Tp(3) * __sigma + __power4 * (_Tp(1) + __s1 + __s2)
/ (__mu * std::sqrt(__mu));
}
}
/**
* @brief Return the complete elliptic integral of the second kind
* @f$ E(k) @f$ using the Carlson formulation.
*
* The complete elliptic integral of the second kind is defined as
* @f[
* E(k,\pi/2) = \int_0^{\pi/2}\sqrt{1 - k^2 sin^2\theta}
* @f]
*
* @param __k The argument of the complete elliptic function.
* @return The complete elliptic function of the second kind.
*/
template<typename _Tp>
_Tp
__comp_ellint_2(_Tp __k)
{
if (__isnan(__k))
return std::numeric_limits<_Tp>::quiet_NaN();
else if (std::abs(__k) == 1)
return _Tp(1);
else if (std::abs(__k) > _Tp(1))
std::__throw_domain_error(__N("Bad argument in __comp_ellint_2."));
else
{
const _Tp __kk = __k * __k;
return __ellint_rf(_Tp(0), _Tp(1) - __kk, _Tp(1))
- __kk * __ellint_rd(_Tp(0), _Tp(1) - __kk, _Tp(1)) / _Tp(3);
}
}
/**
* @brief Return the incomplete elliptic integral of the second kind
* @f$ E(k,\phi) @f$ using the Carlson formulation.
*
* The incomplete elliptic integral of the second kind is defined as
* @f[
* E(k,\phi) = \int_0^{\phi} \sqrt{1 - k^2 sin^2\theta}
* @f]
*
* @param __k The argument of the elliptic function.
* @param __phi The integral limit argument of the elliptic function.
* @return The elliptic function of the second kind.
*/
template<typename _Tp>
_Tp
__ellint_2(_Tp __k, _Tp __phi)
{
if (__isnan(__k) || __isnan(__phi))
return std::numeric_limits<_Tp>::quiet_NaN();
else if (std::abs(__k) > _Tp(1))
std::__throw_domain_error(__N("Bad argument in __ellint_2."));
else
{
// Reduce phi to -pi/2 < phi < +pi/2.
const int __n = std::floor(__phi / __numeric_constants<_Tp>::__pi()
+ _Tp(0.5L));
const _Tp __phi_red = __phi
- __n * __numeric_constants<_Tp>::__pi();
const _Tp __kk = __k * __k;
const _Tp __s = std::sin(__phi_red);
const _Tp __ss = __s * __s;
const _Tp __sss = __ss * __s;
const _Tp __c = std::cos(__phi_red);
const _Tp __cc = __c * __c;
const _Tp __E = __s
* __ellint_rf(__cc, _Tp(1) - __kk * __ss, _Tp(1))
- __kk * __sss
* __ellint_rd(__cc, _Tp(1) - __kk * __ss, _Tp(1))
/ _Tp(3);
if (__n == 0)
return __E;
else
return __E + _Tp(2) * __n * __comp_ellint_2(__k);
}
}
/**
* @brief Return the Carlson elliptic function
* @f$ R_C(x,y) = R_F(x,y,y) @f$ where @f$ R_F(x,y,z) @f$
* is the Carlson elliptic function of the first kind.
*
* The Carlson elliptic function is defined by:
* @f[
* R_C(x,y) = \frac{1}{2} \int_0^\infty
* \frac{dt}{(t + x)^{1/2}(t + y)}
* @f]
*
* Based on Carlson's algorithms:
* - B. C. Carlson Numer. Math. 33, 1 (1979)
* - B. C. Carlson, Special Functions of Applied Mathematics (1977)
* - Numerical Recipes in C, 2nd ed, pp. 261-269,
* by Press, Teukolsky, Vetterling, Flannery (1992)
*
* @param __x The first argument.
* @param __y The second argument.
* @return The Carlson elliptic function.
*/
template<typename _Tp>
_Tp
__ellint_rc(_Tp __x, _Tp __y)
{
const _Tp __min = std::numeric_limits<_Tp>::min();
const _Tp __max = std::numeric_limits<_Tp>::max();
const _Tp __lolim = _Tp(5) * __min;
const _Tp __uplim = __max / _Tp(5);
if (__x < _Tp(0) || __y < _Tp(0) || __x + __y < __lolim)
std::__throw_domain_error(__N("Argument less than zero "
"in __ellint_rc."));
else
{
const _Tp __c0 = _Tp(1) / _Tp(4);
const _Tp __c1 = _Tp(1) / _Tp(7);
const _Tp __c2 = _Tp(9) / _Tp(22);
const _Tp __c3 = _Tp(3) / _Tp(10);
const _Tp __c4 = _Tp(3) / _Tp(8);
_Tp __xn = __x;
_Tp __yn = __y;
const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
const _Tp __errtol = std::pow(__eps / _Tp(30), _Tp(1) / _Tp(6));
_Tp __mu;
_Tp __sn;
const unsigned int __max_iter = 100;
for (unsigned int __iter = 0; __iter < __max_iter; ++__iter)
{
__mu = (__xn + _Tp(2) * __yn) / _Tp(3);
__sn = (__yn + __mu) / __mu - _Tp(2);
if (std::abs(__sn) < __errtol)
break;
const _Tp __lambda = _Tp(2) * std::sqrt(__xn) * std::sqrt(__yn)
+ __yn;
__xn = __c0 * (__xn + __lambda);
__yn = __c0 * (__yn + __lambda);
}
_Tp __s = __sn * __sn
* (__c3 + __sn*(__c1 + __sn * (__c4 + __sn * __c2)));
return (_Tp(1) + __s) / std::sqrt(__mu);
}
}
/**
* @brief Return the Carlson elliptic function @f$ R_J(x,y,z,p) @f$
* of the third kind.
*
* The Carlson elliptic function of the third kind is defined by:
* @f[
* R_J(x,y,z,p) = \frac{3}{2} \int_0^\infty
* \frac{dt}{(t + x)^{1/2}(t + y)^{1/2}(t + z)^{1/2}(t + p)}
* @f]
*
* Based on Carlson's algorithms:
* - B. C. Carlson Numer. Math. 33, 1 (1979)
* - B. C. Carlson, Special Functions of Applied Mathematics (1977)
* - Numerical Recipes in C, 2nd ed, pp. 261-269,
* by Press, Teukolsky, Vetterling, Flannery (1992)
*
* @param __x The first of three symmetric arguments.
* @param __y The second of three symmetric arguments.
* @param __z The third of three symmetric arguments.
* @param __p The fourth argument.
* @return The Carlson elliptic function of the fourth kind.
*/
template<typename _Tp>
_Tp
__ellint_rj(_Tp __x, _Tp __y, _Tp __z, _Tp __p)
{
const _Tp __min = std::numeric_limits<_Tp>::min();
const _Tp __max = std::numeric_limits<_Tp>::max();
const _Tp __lolim = std::pow(_Tp(5) * __min, _Tp(1)/_Tp(3));
const _Tp __uplim = _Tp(0.3L)
* std::pow(_Tp(0.2L) * __max, _Tp(1)/_Tp(3));
if (__x < _Tp(0) || __y < _Tp(0) || __z < _Tp(0))
std::__throw_domain_error(__N("Argument less than zero "
"in __ellint_rj."));
else if (__x + __y < __lolim || __x + __z < __lolim
|| __y + __z < __lolim || __p < __lolim)
std::__throw_domain_error(__N("Argument too small "
"in __ellint_rj"));
else
{
const _Tp __c0 = _Tp(1) / _Tp(4);
const _Tp __c1 = _Tp(3) / _Tp(14);
const _Tp __c2 = _Tp(1) / _Tp(3);
const _Tp __c3 = _Tp(3) / _Tp(22);
const _Tp __c4 = _Tp(3) / _Tp(26);
_Tp __xn = __x;
_Tp __yn = __y;
_Tp __zn = __z;
_Tp __pn = __p;
_Tp __sigma = _Tp(0);
_Tp __power4 = _Tp(1);
const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
const _Tp __errtol = std::pow(__eps / _Tp(8), _Tp(1) / _Tp(6));
_Tp __lambda, __mu;
_Tp __xndev, __yndev, __zndev, __pndev;
const unsigned int __max_iter = 100;
for (unsigned int __iter = 0; __iter < __max_iter; ++__iter)
{
__mu = (__xn + __yn + __zn + _Tp(2) * __pn) / _Tp(5);
__xndev = (__mu - __xn) / __mu;
__yndev = (__mu - __yn) / __mu;
__zndev = (__mu - __zn) / __mu;
__pndev = (__mu - __pn) / __mu;
_Tp __epsilon = std::max(std::abs(__xndev), std::abs(__yndev));
__epsilon = std::max(__epsilon, std::abs(__zndev));
__epsilon = std::max(__epsilon, std::abs(__pndev));
if (__epsilon < __errtol)
break;
const _Tp __xnroot = std::sqrt(__xn);
const _Tp __ynroot = std::sqrt(__yn);
const _Tp __znroot = std::sqrt(__zn);
const _Tp __lambda = __xnroot * (__ynroot + __znroot)
+ __ynroot * __znroot;
const _Tp __alpha1 = __pn * (__xnroot + __ynroot + __znroot)
+ __xnroot * __ynroot * __znroot;
const _Tp __alpha2 = __alpha1 * __alpha1;
const _Tp __beta = __pn * (__pn + __lambda)
* (__pn + __lambda);
__sigma += __power4 * __ellint_rc(__alpha2, __beta);
__power4 *= __c0;
__xn = __c0 * (__xn + __lambda);
__yn = __c0 * (__yn + __lambda);
__zn = __c0 * (__zn + __lambda);
__pn = __c0 * (__pn + __lambda);
}
// Note: __ea is an SPU badname.
_Tp __eaa = __xndev * (__yndev + __zndev) + __yndev * __zndev;
_Tp __eb = __xndev * __yndev * __zndev;
_Tp __ec = __pndev * __pndev;
_Tp __e2 = __eaa - _Tp(3) * __ec;
_Tp __e3 = __eb + _Tp(2) * __pndev * (__eaa - __ec);
_Tp __s1 = _Tp(1) + __e2 * (-__c1 + _Tp(3) * __c3 * __e2 / _Tp(4)
- _Tp(3) * __c4 * __e3 / _Tp(2));
_Tp __s2 = __eb * (__c2 / _Tp(2)
+ __pndev * (-__c3 - __c3 + __pndev * __c4));
_Tp __s3 = __pndev * __eaa * (__c2 - __pndev * __c3)
- __c2 * __pndev * __ec;
return _Tp(3) * __sigma + __power4 * (__s1 + __s2 + __s3)
/ (__mu * std::sqrt(__mu));
}
}
/**
* @brief Return the complete elliptic integral of the third kind
* @f$ \Pi(k,\nu) = \Pi(k,\nu,\pi/2) @f$ using the
* Carlson formulation.
*
* The complete elliptic integral of the third kind is defined as
* @f[
* \Pi(k,\nu) = \int_0^{\pi/2}
* \frac{d\theta}
* {(1 - \nu \sin^2\theta)\sqrt{1 - k^2 \sin^2\theta}}
* @f]
*
* @param __k The argument of the elliptic function.
* @param __nu The second argument of the elliptic function.
* @return The complete elliptic function of the third kind.
*/
template<typename _Tp>
_Tp
__comp_ellint_3(_Tp __k, _Tp __nu)
{
if (__isnan(__k) || __isnan(__nu))
return std::numeric_limits<_Tp>::quiet_NaN();
else if (__nu == _Tp(1))
return std::numeric_limits<_Tp>::infinity();
else if (std::abs(__k) > _Tp(1))
std::__throw_domain_error(__N("Bad argument in __comp_ellint_3."));
else
{
const _Tp __kk = __k * __k;
return __ellint_rf(_Tp(0), _Tp(1) - __kk, _Tp(1))
- __nu
* __ellint_rj(_Tp(0), _Tp(1) - __kk, _Tp(1), _Tp(1) + __nu)
/ _Tp(3);
}
}
/**
* @brief Return the incomplete elliptic integral of the third kind
* @f$ \Pi(k,\nu,\phi) @f$ using the Carlson formulation.
*
* The incomplete elliptic integral of the third kind is defined as
* @f[
* \Pi(k,\nu,\phi) = \int_0^{\phi}
* \frac{d\theta}
* {(1 - \nu \sin^2\theta)
* \sqrt{1 - k^2 \sin^2\theta}}
* @f]
*
* @param __k The argument of the elliptic function.
* @param __nu The second argument of the elliptic function.
* @param __phi The integral limit argument of the elliptic function.
* @return The elliptic function of the third kind.
*/
template<typename _Tp>
_Tp
__ellint_3(_Tp __k, _Tp __nu, _Tp __phi)
{
if (__isnan(__k) || __isnan(__nu) || __isnan(__phi))
return std::numeric_limits<_Tp>::quiet_NaN();
else if (std::abs(__k) > _Tp(1))
std::__throw_domain_error(__N("Bad argument in __ellint_3."));
else
{
// Reduce phi to -pi/2 < phi < +pi/2.
const int __n = std::floor(__phi / __numeric_constants<_Tp>::__pi()
+ _Tp(0.5L));
const _Tp __phi_red = __phi
- __n * __numeric_constants<_Tp>::__pi();
const _Tp __kk = __k * __k;
const _Tp __s = std::sin(__phi_red);
const _Tp __ss = __s * __s;
const _Tp __sss = __ss * __s;
const _Tp __c = std::cos(__phi_red);
const _Tp __cc = __c * __c;
const _Tp __Pi = __s
* __ellint_rf(__cc, _Tp(1) - __kk * __ss, _Tp(1))
- __nu * __sss
* __ellint_rj(__cc, _Tp(1) - __kk * __ss, _Tp(1),
_Tp(1) + __nu * __ss) / _Tp(3);
if (__n == 0)
return __Pi;
else
return __Pi + _Tp(2) * __n * __comp_ellint_3(__k, __nu);
}
}
_GLIBCXX_END_NAMESPACE_VERSION
} // namespace std::tr1::__detail
}
}
#endif // _GLIBCXX_TR1_ELL_INTEGRAL_TCC