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diff --git a/gcc-4.4.3/libstdc++-v3/include/tr1/ell_integral.tcc b/gcc-4.4.3/libstdc++-v3/include/tr1/ell_integral.tcc new file mode 100644 index 000000000..09bda9aa9 --- /dev/null +++ b/gcc-4.4.3/libstdc++-v3/include/tr1/ell_integral.tcc @@ -0,0 +1,750 @@ +// Special functions -*- C++ -*- + +// Copyright (C) 2006, 2007, 2008, 2009 +// 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. + * You should not attempt to use it directly. + */ + +// +// 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 +{ +namespace tr1 +{ + + // [5.2] Special functions + + // Implementation-space details. + namespace __detail + { + + /** + * @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(const _Tp __x, const _Tp __y, const _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(const _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(const _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(const _Tp __k, const _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(const _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(const _Tp __x, const _Tp __y, const _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(const _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(const _Tp __k, const _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(const _Tp __x, const _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(const _Tp __x, const _Tp __y, const _Tp __z, const _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(const _Tp __k, const _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(const _Tp __k, const _Tp __nu, const _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); + } + } + + } // namespace std::tr1::__detail +} +} + +#endif // _GLIBCXX_TR1_ELL_INTEGRAL_TCC + |