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// Special functions -*- C++ -*- |
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// Copyright (C) 2006-2013 Free Software Foundation, Inc. |
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// |
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// This file is part of the GNU ISO C++ Library. This library is free |
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// software; you can redistribute it and/or modify it under the |
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// terms of the GNU General Public License as published by the |
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// Free Software Foundation; either version 3, or (at your option) |
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// any later version. |
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// |
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// This library is distributed in the hope that it will be useful, |
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// but WITHOUT ANY WARRANTY; without even the implied warranty of |
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// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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// GNU General Public License for more details. |
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// |
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// Under Section 7 of GPL version 3, you are granted additional |
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// permissions described in the GCC Runtime Library Exception, version |
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// 3.1, as published by the Free Software Foundation. |
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// You should have received a copy of the GNU General Public License and |
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// a copy of the GCC Runtime Library Exception along with this program; |
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// see the files COPYING3 and COPYING.RUNTIME respectively. If not, see |
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// . |
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/** @file tr1/ell_integral.tcc |
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* This is an internal header file, included by other library headers. |
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* Do not attempt to use it directly. @headername{tr1/cmath} |
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*/ |
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// |
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// ISO C++ 14882 TR1: 5.2 Special functions |
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// |
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// Written by Edward Smith-Rowland based on: |
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// (1) B. C. Carlson Numer. Math. 33, 1 (1979) |
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// (2) B. C. Carlson, Special Functions of Applied Mathematics (1977) |
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// (3) The Gnu Scientific Library, http://www.gnu.org/software/gsl |
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// (4) Numerical Recipes in C, 2nd ed, by W. H. Press, S. A. Teukolsky, |
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// W. T. Vetterling, B. P. Flannery, Cambridge University Press |
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// (1992), pp. 261-269 |
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#ifndef _GLIBCXX_TR1_ELL_INTEGRAL_TCC |
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#define _GLIBCXX_TR1_ELL_INTEGRAL_TCC 1 |
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namespace std _GLIBCXX_VISIBILITY(default) |
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{ |
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namespace tr1 |
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{ |
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// [5.2] Special functions |
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// Implementation-space details. |
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namespace __detail |
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{ |
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_GLIBCXX_BEGIN_NAMESPACE_VERSION |
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/** |
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* @brief Return the Carlson elliptic function @f$ R_F(x,y,z) @f$ |
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* of the first kind. |
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* |
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* The Carlson elliptic function of the first kind is defined by: |
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* @f[ |
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* R_F(x,y,z) = \frac{1}{2} \int_0^\infty |
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* \frac{dt}{(t + x)^{1/2}(t + y)^{1/2}(t + z)^{1/2}} |
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* @f] |
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* |
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* @param __x The first of three symmetric arguments. |
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* @param __y The second of three symmetric arguments. |
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* @param __z The third of three symmetric arguments. |
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* @return The Carlson elliptic function of the first kind. |
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*/ |
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template |
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_Tp |
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__ellint_rf(_Tp __x, _Tp __y, _Tp __z) |
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{ |
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const _Tp __min = std::numeric_limits<_Tp>::min(); |
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const _Tp __max = std::numeric_limits<_Tp>::max(); |
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const _Tp __lolim = _Tp(5) * __min; |
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const _Tp __uplim = __max / _Tp(5); |
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if (__x < _Tp(0) || __y < _Tp(0) || __z < _Tp(0)) |
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std::__throw_domain_error(__N("Argument less than zero " |
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"in __ellint_rf.")); |
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else if (__x + __y < __lolim || __x + __z < __lolim |
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|| __y + __z < __lolim) |
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std::__throw_domain_error(__N("Argument too small in __ellint_rf")); |
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else |
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{ |
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const _Tp __c0 = _Tp(1) / _Tp(4); |
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const _Tp __c1 = _Tp(1) / _Tp(24); |
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const _Tp __c2 = _Tp(1) / _Tp(10); |
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const _Tp __c3 = _Tp(3) / _Tp(44); |
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const _Tp __c4 = _Tp(1) / _Tp(14); |
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_Tp __xn = __x; |
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_Tp __yn = __y; |
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_Tp __zn = __z; |
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const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); |
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const _Tp __errtol = std::pow(__eps, _Tp(1) / _Tp(6)); |
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_Tp __mu; |
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_Tp __xndev, __yndev, __zndev; |
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const unsigned int __max_iter = 100; |
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for (unsigned int __iter = 0; __iter < __max_iter; ++__iter) |
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{ |
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__mu = (__xn + __yn + __zn) / _Tp(3); |
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__xndev = 2 - (__mu + __xn) / __mu; |
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__yndev = 2 - (__mu + __yn) / __mu; |
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__zndev = 2 - (__mu + __zn) / __mu; |
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_Tp __epsilon = std::max(std::abs(__xndev), std::abs(__yndev)); |
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__epsilon = std::max(__epsilon, std::abs(__zndev)); |
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if (__epsilon < __errtol) |
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break; |
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const _Tp __xnroot = std::sqrt(__xn); |
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const _Tp __ynroot = std::sqrt(__yn); |
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const _Tp __znroot = std::sqrt(__zn); |
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const _Tp __lambda = __xnroot * (__ynroot + __znroot) |
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+ __ynroot * __znroot; |
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__xn = __c0 * (__xn + __lambda); |
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__yn = __c0 * (__yn + __lambda); |
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__zn = __c0 * (__zn + __lambda); |
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} |
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const _Tp __e2 = __xndev * __yndev - __zndev * __zndev; |
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const _Tp __e3 = __xndev * __yndev * __zndev; |
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const _Tp __s = _Tp(1) + (__c1 * __e2 - __c2 - __c3 * __e3) * __e2 |
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+ __c4 * __e3; |
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return __s / std::sqrt(__mu); |
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} |
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} |
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/** |
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* @brief Return the complete elliptic integral of the first kind |
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* @f$ K(k) @f$ by series expansion. |
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* |
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* The complete elliptic integral of the first kind is defined as |
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* @f[ |
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* K(k) = F(k,\pi/2) = \int_0^{\pi/2}\frac{d\theta} |
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* {\sqrt{1 - k^2sin^2\theta}} |
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* @f] |
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* |
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* This routine is not bad as long as |k| is somewhat smaller than 1 |
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* but is not is good as the Carlson elliptic integral formulation. |
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* |
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* @param __k The argument of the complete elliptic function. |
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* @return The complete elliptic function of the first kind. |
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*/ |
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template |
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_Tp |
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__comp_ellint_1_series(_Tp __k) |
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{ |
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const _Tp __kk = __k * __k; |
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_Tp __term = __kk / _Tp(4); |
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_Tp __sum = _Tp(1) + __term; |
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const unsigned int __max_iter = 1000; |
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for (unsigned int __i = 2; __i < __max_iter; ++__i) |
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{ |
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__term *= (2 * __i - 1) * __kk / (2 * __i); |
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if (__term < std::numeric_limits<_Tp>::epsilon()) |
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break; |
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__sum += __term; |
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} |
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return __numeric_constants<_Tp>::__pi_2() * __sum; |
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} |
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/** |
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* @brief Return the complete elliptic integral of the first kind |
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* @f$ K(k) @f$ using the Carlson formulation. |
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* |
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* The complete elliptic integral of the first kind is defined as |
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* @f[ |
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* K(k) = F(k,\pi/2) = \int_0^{\pi/2}\frac{d\theta} |
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* {\sqrt{1 - k^2 sin^2\theta}} |
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* @f] |
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* where @f$ F(k,\phi) @f$ is the incomplete elliptic integral of the |
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* first kind. |
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* |
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* @param __k The argument of the complete elliptic function. |
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* @return The complete elliptic function of the first kind. |
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*/ |
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template |
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_Tp |
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__comp_ellint_1(_Tp __k) |
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{ |
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if (__isnan(__k)) |
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return std::numeric_limits<_Tp>::quiet_NaN(); |
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else if (std::abs(__k) >= _Tp(1)) |
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return std::numeric_limits<_Tp>::quiet_NaN(); |
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else |
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return __ellint_rf(_Tp(0), _Tp(1) - __k * __k, _Tp(1)); |
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} |
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/** |
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* @brief Return the incomplete elliptic integral of the first kind |
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* @f$ F(k,\phi) @f$ using the Carlson formulation. |
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* |
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* The incomplete elliptic integral of the first kind is defined as |
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* @f[ |
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* F(k,\phi) = \int_0^{\phi}\frac{d\theta} |
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* {\sqrt{1 - k^2 sin^2\theta}} |
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* @f] |
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* |
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* @param __k The argument of the elliptic function. |
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* @param __phi The integral limit argument of the elliptic function. |
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* @return The elliptic function of the first kind. |
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*/ |
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template |
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_Tp |
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__ellint_1(_Tp __k, _Tp __phi) |
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{ |
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if (__isnan(__k) || __isnan(__phi)) |
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return std::numeric_limits<_Tp>::quiet_NaN(); |
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else if (std::abs(__k) > _Tp(1)) |
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std::__throw_domain_error(__N("Bad argument in __ellint_1.")); |
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else |
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{ |
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// Reduce phi to -pi/2 < phi < +pi/2. |
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const int __n = std::floor(__phi / __numeric_constants<_Tp>::__pi() |
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+ _Tp(0.5L)); |
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const _Tp __phi_red = __phi |
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- __n * __numeric_constants<_Tp>::__pi(); |
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const _Tp __s = std::sin(__phi_red); |
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const _Tp __c = std::cos(__phi_red); |
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const _Tp __F = __s |
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* __ellint_rf(__c * __c, |
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_Tp(1) - __k * __k * __s * __s, _Tp(1)); |
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if (__n == 0) |
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return __F; |
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else |
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return __F + _Tp(2) * __n * __comp_ellint_1(__k); |
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} |
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} |
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/** |
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* @brief Return the complete elliptic integral of the second kind |
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* @f$ E(k) @f$ by series expansion. |
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* |
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* The complete elliptic integral of the second kind is defined as |
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* @f[ |
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* E(k,\pi/2) = \int_0^{\pi/2}\sqrt{1 - k^2 sin^2\theta} |
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* @f] |
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* |
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* This routine is not bad as long as |k| is somewhat smaller than 1 |
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* but is not is good as the Carlson elliptic integral formulation. |
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* |
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* @param __k The argument of the complete elliptic function. |
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* @return The complete elliptic function of the second kind. |
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*/ |
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template |
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_Tp |
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__comp_ellint_2_series(_Tp __k) |
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{ |
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const _Tp __kk = __k * __k; |
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_Tp __term = __kk; |
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_Tp __sum = __term; |
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const unsigned int __max_iter = 1000; |
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for (unsigned int __i = 2; __i < __max_iter; ++__i) |
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{ |
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const _Tp __i2m = 2 * __i - 1; |
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const _Tp __i2 = 2 * __i; |
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__term *= __i2m * __i2m * __kk / (__i2 * __i2); |
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if (__term < std::numeric_limits<_Tp>::epsilon()) |
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break; |
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__sum += __term / __i2m; |
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} |
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return __numeric_constants<_Tp>::__pi_2() * (_Tp(1) - __sum); |
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} |
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/** |
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* @brief Return the Carlson elliptic function of the second kind |
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* @f$ R_D(x,y,z) = R_J(x,y,z,z) @f$ where |
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* @f$ R_J(x,y,z,p) @f$ is the Carlson elliptic function |
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* of the third kind. |
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* |
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* The Carlson elliptic function of the second kind is defined by: |
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* @f[ |
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* R_D(x,y,z) = \frac{3}{2} \int_0^\infty |
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* \frac{dt}{(t + x)^{1/2}(t + y)^{1/2}(t + z)^{3/2}} |
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* @f] |
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* |
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* Based on Carlson's algorithms: |
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* - B. C. Carlson Numer. Math. 33, 1 (1979) |
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* - B. C. Carlson, Special Functions of Applied Mathematics (1977) |
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* - Numerical Recipes in C, 2nd ed, pp. 261-269, |
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* by Press, Teukolsky, Vetterling, Flannery (1992) |
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* |
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* @param __x The first of two symmetric arguments. |
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* @param __y The second of two symmetric arguments. |
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* @param __z The third argument. |
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* @return The Carlson elliptic function of the second kind. |
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*/ |
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template |
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_Tp |
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__ellint_rd(_Tp __x, _Tp __y, _Tp __z) |
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{ |
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const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); |
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const _Tp __errtol = std::pow(__eps / _Tp(8), _Tp(1) / _Tp(6)); |
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const _Tp __min = std::numeric_limits<_Tp>::min(); |
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const _Tp __max = std::numeric_limits<_Tp>::max(); |
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const _Tp __lolim = _Tp(2) / std::pow(__max, _Tp(2) / _Tp(3)); |
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const _Tp __uplim = std::pow(_Tp(0.1L) * __errtol / __min, _Tp(2) / _Tp(3)); |
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if (__x < _Tp(0) || __y < _Tp(0)) |
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std::__throw_domain_error(__N("Argument less than zero " |
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"in __ellint_rd.")); |
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else if (__x + __y < __lolim || __z < __lolim) |
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std::__throw_domain_error(__N("Argument too small " |
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"in __ellint_rd.")); |
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else |
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{ |
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const _Tp __c0 = _Tp(1) / _Tp(4); |
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const _Tp __c1 = _Tp(3) / _Tp(14); |
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const _Tp __c2 = _Tp(1) / _Tp(6); |
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const _Tp __c3 = _Tp(9) / _Tp(22); |
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const _Tp __c4 = _Tp(3) / _Tp(26); |
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_Tp __xn = __x; |
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_Tp __yn = __y; |
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_Tp __zn = __z; |
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_Tp __sigma = _Tp(0); |
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_Tp __power4 = _Tp(1); |
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_Tp __mu; |
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_Tp __xndev, __yndev, __zndev; |
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const unsigned int __max_iter = 100; |
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for (unsigned int __iter = 0; __iter < __max_iter; ++__iter) |
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{ |
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__mu = (__xn + __yn + _Tp(3) * __zn) / _Tp(5); |
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__xndev = (__mu - __xn) / __mu; |
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__yndev = (__mu - __yn) / __mu; |
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__zndev = (__mu - __zn) / __mu; |
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_Tp __epsilon = std::max(std::abs(__xndev), std::abs(__yndev)); |
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__epsilon = std::max(__epsilon, std::abs(__zndev)); |
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if (__epsilon < __errtol) |
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break; |
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_Tp __xnroot = std::sqrt(__xn); |
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357 |
_Tp __ynroot = std::sqrt(__yn); |
|
|
358 |
_Tp __znroot = std::sqrt(__zn); |
|
|
359 |
_Tp __lambda = __xnroot * (__ynroot + __znroot) |
|
|
360 |
+ __ynroot * __znroot; |
|
|
361 |
__sigma += __power4 / (__znroot * (__zn + __lambda)); |
|
|
362 |
__power4 *= __c0; |
|
|
363 |
__xn = __c0 * (__xn + __lambda); |
|
|
364 |
__yn = __c0 * (__yn + __lambda); |
|
|
365 |
__zn = __c0 * (__zn + __lambda); |
|
|
366 |
} |
|
|
367 |
|
|
|
368 |
// Note: __ea is an SPU badname. |
|
|
369 |
_Tp __eaa = __xndev * __yndev; |
|
|
370 |
_Tp __eb = __zndev * __zndev; |
|
|
371 |
_Tp __ec = __eaa - __eb; |
|
|
372 |
_Tp __ed = __eaa - _Tp(6) * __eb; |
|
|
373 |
_Tp __ef = __ed + __ec + __ec; |
|
|
374 |
_Tp __s1 = __ed * (-__c1 + __c3 * __ed |
|
|
375 |
/ _Tp(3) - _Tp(3) * __c4 * __zndev * __ef |
|
|
376 |
/ _Tp(2)); |
|
|
377 |
_Tp __s2 = __zndev |
|
|
378 |
* (__c2 * __ef |
|
|
379 |
+ __zndev * (-__c3 * __ec - __zndev * __c4 - __eaa)); |
|
|
380 |
|
|
|
381 |
return _Tp(3) * __sigma + __power4 * (_Tp(1) + __s1 + __s2) |
|
|
382 |
/ (__mu * std::sqrt(__mu)); |
|
|
383 |
} |
|
|
384 |
} |
|
|
385 |
|
|
|
386 |
|
|
|
387 |
/** |
|
|
388 |
* @brief Return the complete elliptic integral of the second kind |
|
|
389 |
* @f$ E(k) @f$ using the Carlson formulation. |
|
|
390 |
* |
|
|
391 |
* The complete elliptic integral of the second kind is defined as |
|
|
392 |
* @f[ |
|
|
393 |
* E(k,\pi/2) = \int_0^{\pi/2}\sqrt{1 - k^2 sin^2\theta} |
|
|
394 |
* @f] |
|
|
395 |
* |
|
|
396 |
* @param __k The argument of the complete elliptic function. |
|
|
397 |
* @return The complete elliptic function of the second kind. |
|
|
398 |
*/ |
|
|
399 |
template |
|
|
400 |
_Tp |
|
|
401 |
__comp_ellint_2(_Tp __k) |
|
|
402 |
{ |
|
|
403 |
|
|
|
404 |
if (__isnan(__k)) |
|
|
405 |
return std::numeric_limits<_Tp>::quiet_NaN(); |
|
|
406 |
else if (std::abs(__k) == 1) |
|
|
407 |
return _Tp(1); |
|
|
408 |
else if (std::abs(__k) > _Tp(1)) |
|
|
409 |
std::__throw_domain_error(__N("Bad argument in __comp_ellint_2.")); |
|
|
410 |
else |
|
|
411 |
{ |
|
|
412 |
const _Tp __kk = __k * __k; |
|
|
413 |
|
|
|
414 |
return __ellint_rf(_Tp(0), _Tp(1) - __kk, _Tp(1)) |
|
|
415 |
- __kk * __ellint_rd(_Tp(0), _Tp(1) - __kk, _Tp(1)) / _Tp(3); |
|
|
416 |
} |
|
|
417 |
} |
|
|
418 |
|
|
|
419 |
|
|
|
420 |
/** |
|
|
421 |
* @brief Return the incomplete elliptic integral of the second kind |
|
|
422 |
* @f$ E(k,\phi) @f$ using the Carlson formulation. |
|
|
423 |
* |
|
|
424 |
* The incomplete elliptic integral of the second kind is defined as |
|
|
425 |
* @f[ |
|
|
426 |
* E(k,\phi) = \int_0^{\phi} \sqrt{1 - k^2 sin^2\theta} |
|
|
427 |
* @f] |
|
|
428 |
* |
|
|
429 |
* @param __k The argument of the elliptic function. |
|
|
430 |
* @param __phi The integral limit argument of the elliptic function. |
|
|
431 |
* @return The elliptic function of the second kind. |
|
|
432 |
*/ |
|
|
433 |
template |
|
|
434 |
_Tp |
|
|
435 |
__ellint_2(_Tp __k, _Tp __phi) |
|
|
436 |
{ |
|
|
437 |
|
|
|
438 |
if (__isnan(__k) || __isnan(__phi)) |
|
|
439 |
return std::numeric_limits<_Tp>::quiet_NaN(); |
|
|
440 |
else if (std::abs(__k) > _Tp(1)) |
|
|
441 |
std::__throw_domain_error(__N("Bad argument in __ellint_2.")); |
|
|
442 |
else |
|
|
443 |
{ |
|
|
444 |
// Reduce phi to -pi/2 < phi < +pi/2. |
|
|
445 |
const int __n = std::floor(__phi / __numeric_constants<_Tp>::__pi() |
|
|
446 |
+ _Tp(0.5L)); |
|
|
447 |
const _Tp __phi_red = __phi |
|
|
448 |
- __n * __numeric_constants<_Tp>::__pi(); |
|
|
449 |
|
|
|
450 |
const _Tp __kk = __k * __k; |
|
|
451 |
const _Tp __s = std::sin(__phi_red); |
|
|
452 |
const _Tp __ss = __s * __s; |
|
|
453 |
const _Tp __sss = __ss * __s; |
|
|
454 |
const _Tp __c = std::cos(__phi_red); |
|
|
455 |
const _Tp __cc = __c * __c; |
|
|
456 |
|
|
|
457 |
const _Tp __E = __s |
|
|
458 |
* __ellint_rf(__cc, _Tp(1) - __kk * __ss, _Tp(1)) |
|
|
459 |
- __kk * __sss |
|
|
460 |
* __ellint_rd(__cc, _Tp(1) - __kk * __ss, _Tp(1)) |
|
|
461 |
/ _Tp(3); |
|
|
462 |
|
|
|
463 |
if (__n == 0) |
|
|
464 |
return __E; |
|
|
465 |
else |
|
|
466 |
return __E + _Tp(2) * __n * __comp_ellint_2(__k); |
|
|
467 |
} |
|
|
468 |
} |
|
|
469 |
|
|
|
470 |
|
|
|
471 |
/** |
|
|
472 |
* @brief Return the Carlson elliptic function |
|
|
473 |
* @f$ R_C(x,y) = R_F(x,y,y) @f$ where @f$ R_F(x,y,z) @f$ |
|
|
474 |
* is the Carlson elliptic function of the first kind. |
|
|
475 |
* |
|
|
476 |
* The Carlson elliptic function is defined by: |
|
|
477 |
* @f[ |
|
|
478 |
* R_C(x,y) = \frac{1}{2} \int_0^\infty |
|
|
479 |
* \frac{dt}{(t + x)^{1/2}(t + y)} |
|
|
480 |
* @f] |
|
|
481 |
* |
|
|
482 |
* Based on Carlson's algorithms: |
|
|
483 |
* - B. C. Carlson Numer. Math. 33, 1 (1979) |
|
|
484 |
* - B. C. Carlson, Special Functions of Applied Mathematics (1977) |
|
|
485 |
* - Numerical Recipes in C, 2nd ed, pp. 261-269, |
|
|
486 |
* by Press, Teukolsky, Vetterling, Flannery (1992) |
|
|
487 |
* |
|
|
488 |
* @param __x The first argument. |
|
|
489 |
* @param __y The second argument. |
|
|
490 |
* @return The Carlson elliptic function. |
|
|
491 |
*/ |
|
|
492 |
template |
|
|
493 |
_Tp |
|
|
494 |
__ellint_rc(_Tp __x, _Tp __y) |
|
|
495 |
{ |
|
|
496 |
const _Tp __min = std::numeric_limits<_Tp>::min(); |
|
|
497 |
const _Tp __max = std::numeric_limits<_Tp>::max(); |
|
|
498 |
const _Tp __lolim = _Tp(5) * __min; |
|
|
499 |
const _Tp __uplim = __max / _Tp(5); |
|
|
500 |
|
|
|
501 |
if (__x < _Tp(0) || __y < _Tp(0) || __x + __y < __lolim) |
|
|
502 |
std::__throw_domain_error(__N("Argument less than zero " |
|
|
503 |
"in __ellint_rc.")); |
|
|
504 |
else |
|
|
505 |
{ |
|
|
506 |
const _Tp __c0 = _Tp(1) / _Tp(4); |
|
|
507 |
const _Tp __c1 = _Tp(1) / _Tp(7); |
|
|
508 |
const _Tp __c2 = _Tp(9) / _Tp(22); |
|
|
509 |
const _Tp __c3 = _Tp(3) / _Tp(10); |
|
|
510 |
const _Tp __c4 = _Tp(3) / _Tp(8); |
|
|
511 |
|
|
|
512 |
_Tp __xn = __x; |
|
|
513 |
_Tp __yn = __y; |
|
|
514 |
|
|
|
515 |
const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); |
|
|
516 |
const _Tp __errtol = std::pow(__eps / _Tp(30), _Tp(1) / _Tp(6)); |
|
|
517 |
_Tp __mu; |
|
|
518 |
_Tp __sn; |
|
|
519 |
|
|
|
520 |
const unsigned int __max_iter = 100; |
|
|
521 |
for (unsigned int __iter = 0; __iter < __max_iter; ++__iter) |
|
|
522 |
{ |
|
|
523 |
__mu = (__xn + _Tp(2) * __yn) / _Tp(3); |
|
|
524 |
__sn = (__yn + __mu) / __mu - _Tp(2); |
|
|
525 |
if (std::abs(__sn) < __errtol) |
|
|
526 |
break; |
|
|
527 |
const _Tp __lambda = _Tp(2) * std::sqrt(__xn) * std::sqrt(__yn) |
|
|
528 |
+ __yn; |
|
|
529 |
__xn = __c0 * (__xn + __lambda); |
|
|
530 |
__yn = __c0 * (__yn + __lambda); |
|
|
531 |
} |
|
|
532 |
|
|
|
533 |
_Tp __s = __sn * __sn |
|
|
534 |
* (__c3 + __sn*(__c1 + __sn * (__c4 + __sn * __c2))); |
|
|
535 |
|
|
|
536 |
return (_Tp(1) + __s) / std::sqrt(__mu); |
|
|
537 |
} |
|
|
538 |
} |
|
|
539 |
|
|
|
540 |
|
|
|
541 |
/** |
|
|
542 |
* @brief Return the Carlson elliptic function @f$ R_J(x,y,z,p) @f$ |
|
|
543 |
* of the third kind. |
|
|
544 |
* |
|
|
545 |
* The Carlson elliptic function of the third kind is defined by: |
|
|
546 |
* @f[ |
|
|
547 |
* R_J(x,y,z,p) = \frac{3}{2} \int_0^\infty |
|
|
548 |
* \frac{dt}{(t + x)^{1/2}(t + y)^{1/2}(t + z)^{1/2}(t + p)} |
|
|
549 |
* @f] |
|
|
550 |
* |
|
|
551 |
* Based on Carlson's algorithms: |
|
|
552 |
* - B. C. Carlson Numer. Math. 33, 1 (1979) |
|
|
553 |
* - B. C. Carlson, Special Functions of Applied Mathematics (1977) |
|
|
554 |
* - Numerical Recipes in C, 2nd ed, pp. 261-269, |
|
|
555 |
* by Press, Teukolsky, Vetterling, Flannery (1992) |
|
|
556 |
* |
|
|
557 |
* @param __x The first of three symmetric arguments. |
|
|
558 |
* @param __y The second of three symmetric arguments. |
|
|
559 |
* @param __z The third of three symmetric arguments. |
|
|
560 |
* @param __p The fourth argument. |
|
|
561 |
* @return The Carlson elliptic function of the fourth kind. |
|
|
562 |
*/ |
|
|
563 |
template |
|
|
564 |
_Tp |
|
|
565 |
__ellint_rj(_Tp __x, _Tp __y, _Tp __z, _Tp __p) |
|
|
566 |
{ |
|
|
567 |
const _Tp __min = std::numeric_limits<_Tp>::min(); |
|
|
568 |
const _Tp __max = std::numeric_limits<_Tp>::max(); |
|
|
569 |
const _Tp __lolim = std::pow(_Tp(5) * __min, _Tp(1)/_Tp(3)); |
|
|
570 |
const _Tp __uplim = _Tp(0.3L) |
|
|
571 |
* std::pow(_Tp(0.2L) * __max, _Tp(1)/_Tp(3)); |
|
|
572 |
|
|
|
573 |
if (__x < _Tp(0) || __y < _Tp(0) || __z < _Tp(0)) |
|
|
574 |
std::__throw_domain_error(__N("Argument less than zero " |
|
|
575 |
"in __ellint_rj.")); |
|
|
576 |
else if (__x + __y < __lolim || __x + __z < __lolim |
|
|
577 |
|| __y + __z < __lolim || __p < __lolim) |
|
|
578 |
std::__throw_domain_error(__N("Argument too small " |
|
|
579 |
"in __ellint_rj")); |
|
|
580 |
else |
|
|
581 |
{ |
|
|
582 |
const _Tp __c0 = _Tp(1) / _Tp(4); |
|
|
583 |
const _Tp __c1 = _Tp(3) / _Tp(14); |
|
|
584 |
const _Tp __c2 = _Tp(1) / _Tp(3); |
|
|
585 |
const _Tp __c3 = _Tp(3) / _Tp(22); |
|
|
586 |
const _Tp __c4 = _Tp(3) / _Tp(26); |
|
|
587 |
|
|
|
588 |
_Tp __xn = __x; |
|
|
589 |
_Tp __yn = __y; |
|
|
590 |
_Tp __zn = __z; |
|
|
591 |
_Tp __pn = __p; |
|
|
592 |
_Tp __sigma = _Tp(0); |
|
|
593 |
_Tp __power4 = _Tp(1); |
|
|
594 |
|
|
|
595 |
const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); |
|
|
596 |
const _Tp __errtol = std::pow(__eps / _Tp(8), _Tp(1) / _Tp(6)); |
|
|
597 |
|
|
|
598 |
_Tp __lambda, __mu; |
|
|
599 |
_Tp __xndev, __yndev, __zndev, __pndev; |
|
|
600 |
|
|
|
601 |
const unsigned int __max_iter = 100; |
|
|
602 |
for (unsigned int __iter = 0; __iter < __max_iter; ++__iter) |
|
|
603 |
{ |
|
|
604 |
__mu = (__xn + __yn + __zn + _Tp(2) * __pn) / _Tp(5); |
|
|
605 |
__xndev = (__mu - __xn) / __mu; |
|
|
606 |
__yndev = (__mu - __yn) / __mu; |
|
|
607 |
__zndev = (__mu - __zn) / __mu; |
|
|
608 |
__pndev = (__mu - __pn) / __mu; |
|
|
609 |
_Tp __epsilon = std::max(std::abs(__xndev), std::abs(__yndev)); |
|
|
610 |
__epsilon = std::max(__epsilon, std::abs(__zndev)); |
|
|
611 |
__epsilon = std::max(__epsilon, std::abs(__pndev)); |
|
|
612 |
if (__epsilon < __errtol) |
|
|
613 |
break; |
|
|
614 |
const _Tp __xnroot = std::sqrt(__xn); |
|
|
615 |
const _Tp __ynroot = std::sqrt(__yn); |
|
|
616 |
const _Tp __znroot = std::sqrt(__zn); |
|
|
617 |
const _Tp __lambda = __xnroot * (__ynroot + __znroot) |
|
|
618 |
+ __ynroot * __znroot; |
|
|
619 |
const _Tp __alpha1 = __pn * (__xnroot + __ynroot + __znroot) |
|
|
620 |
+ __xnroot * __ynroot * __znroot; |
|
|
621 |
const _Tp __alpha2 = __alpha1 * __alpha1; |
|
|
622 |
const _Tp __beta = __pn * (__pn + __lambda) |
|
|
623 |
* (__pn + __lambda); |
|
|
624 |
__sigma += __power4 * __ellint_rc(__alpha2, __beta); |
|
|
625 |
__power4 *= __c0; |
|
|
626 |
__xn = __c0 * (__xn + __lambda); |
|
|
627 |
__yn = __c0 * (__yn + __lambda); |
|
|
628 |
__zn = __c0 * (__zn + __lambda); |
|
|
629 |
__pn = __c0 * (__pn + __lambda); |
|
|
630 |
} |
|
|
631 |
|
|
|
632 |
// Note: __ea is an SPU badname. |
|
|
633 |
_Tp __eaa = __xndev * (__yndev + __zndev) + __yndev * __zndev; |
|
|
634 |
_Tp __eb = __xndev * __yndev * __zndev; |
|
|
635 |
_Tp __ec = __pndev * __pndev; |
|
|
636 |
_Tp __e2 = __eaa - _Tp(3) * __ec; |
|
|
637 |
_Tp __e3 = __eb + _Tp(2) * __pndev * (__eaa - __ec); |
|
|
638 |
_Tp __s1 = _Tp(1) + __e2 * (-__c1 + _Tp(3) * __c3 * __e2 / _Tp(4) |
|
|
639 |
- _Tp(3) * __c4 * __e3 / _Tp(2)); |
|
|
640 |
_Tp __s2 = __eb * (__c2 / _Tp(2) |
|
|
641 |
+ __pndev * (-__c3 - __c3 + __pndev * __c4)); |
|
|
642 |
_Tp __s3 = __pndev * __eaa * (__c2 - __pndev * __c3) |
|
|
643 |
- __c2 * __pndev * __ec; |
|
|
644 |
|
|
|
645 |
return _Tp(3) * __sigma + __power4 * (__s1 + __s2 + __s3) |
|
|
646 |
/ (__mu * std::sqrt(__mu)); |
|
|
647 |
} |
|
|
648 |
} |
|
|
649 |
|
|
|
650 |
|
|
|
651 |
/** |
|
|
652 |
* @brief Return the complete elliptic integral of the third kind |
|
|
653 |
* @f$ \Pi(k,\nu) = \Pi(k,\nu,\pi/2) @f$ using the |
|
|
654 |
* Carlson formulation. |
|
|
655 |
* |
|
|
656 |
* The complete elliptic integral of the third kind is defined as |
|
|
657 |
* @f[ |
|
|
658 |
* \Pi(k,\nu) = \int_0^{\pi/2} |
|
|
659 |
* \frac{d\theta} |
|
|
660 |
* {(1 - \nu \sin^2\theta)\sqrt{1 - k^2 \sin^2\theta}} |
|
|
661 |
* @f] |
|
|
662 |
* |
|
|
663 |
* @param __k The argument of the elliptic function. |
|
|
664 |
* @param __nu The second argument of the elliptic function. |
|
|
665 |
* @return The complete elliptic function of the third kind. |
|
|
666 |
*/ |
|
|
667 |
template |
|
|
668 |
_Tp |
|
|
669 |
__comp_ellint_3(_Tp __k, _Tp __nu) |
|
|
670 |
{ |
|
|
671 |
|
|
|
672 |
if (__isnan(__k) || __isnan(__nu)) |
|
|
673 |
return std::numeric_limits<_Tp>::quiet_NaN(); |
|
|
674 |
else if (__nu == _Tp(1)) |
|
|
675 |
return std::numeric_limits<_Tp>::infinity(); |
|
|
676 |
else if (std::abs(__k) > _Tp(1)) |
|
|
677 |
std::__throw_domain_error(__N("Bad argument in __comp_ellint_3.")); |
|
|
678 |
else |
|
|
679 |
{ |
|
|
680 |
const _Tp __kk = __k * __k; |
|
|
681 |
|
|
|
682 |
return __ellint_rf(_Tp(0), _Tp(1) - __kk, _Tp(1)) |
|
|
683 |
- __nu |
|
|
684 |
* __ellint_rj(_Tp(0), _Tp(1) - __kk, _Tp(1), _Tp(1) + __nu) |
|
|
685 |
/ _Tp(3); |
|
|
686 |
} |
|
|
687 |
} |
|
|
688 |
|
|
|
689 |
|
|
|
690 |
/** |
|
|
691 |
* @brief Return the incomplete elliptic integral of the third kind |
|
|
692 |
* @f$ \Pi(k,\nu,\phi) @f$ using the Carlson formulation. |
|
|
693 |
* |
|
|
694 |
* The incomplete elliptic integral of the third kind is defined as |
|
|
695 |
* @f[ |
|
|
696 |
* \Pi(k,\nu,\phi) = \int_0^{\phi} |
|
|
697 |
* \frac{d\theta} |
|
|
698 |
* {(1 - \nu \sin^2\theta) |
|
|
699 |
* \sqrt{1 - k^2 \sin^2\theta}} |
|
|
700 |
* @f] |
|
|
701 |
* |
|
|
702 |
* @param __k The argument of the elliptic function. |
|
|
703 |
* @param __nu The second argument of the elliptic function. |
|
|
704 |
* @param __phi The integral limit argument of the elliptic function. |
|
|
705 |
* @return The elliptic function of the third kind. |
|
|
706 |
*/ |
|
|
707 |
template |
|
|
708 |
_Tp |
|
|
709 |
__ellint_3(_Tp __k, _Tp __nu, _Tp __phi) |
|
|
710 |
{ |
|
|
711 |
|
|
|
712 |
if (__isnan(__k) || __isnan(__nu) || __isnan(__phi)) |
|
|
713 |
return std::numeric_limits<_Tp>::quiet_NaN(); |
|
|
714 |
else if (std::abs(__k) > _Tp(1)) |
|
|
715 |
std::__throw_domain_error(__N("Bad argument in __ellint_3.")); |
|
|
716 |
else |
|
|
717 |
{ |
|
|
718 |
// Reduce phi to -pi/2 < phi < +pi/2. |
|
|
719 |
const int __n = std::floor(__phi / __numeric_constants<_Tp>::__pi() |
|
|
720 |
+ _Tp(0.5L)); |
|
|
721 |
const _Tp __phi_red = __phi |
|
|
722 |
- __n * __numeric_constants<_Tp>::__pi(); |
|
|
723 |
|
|
|
724 |
const _Tp __kk = __k * __k; |
|
|
725 |
const _Tp __s = std::sin(__phi_red); |
|
|
726 |
const _Tp __ss = __s * __s; |
|
|
727 |
const _Tp __sss = __ss * __s; |
|
|
728 |
const _Tp __c = std::cos(__phi_red); |
|
|
729 |
const _Tp __cc = __c * __c; |
|
|
730 |
|
|
|
731 |
const _Tp __Pi = __s |
|
|
732 |
* __ellint_rf(__cc, _Tp(1) - __kk * __ss, _Tp(1)) |
|
|
733 |
- __nu * __sss |
|
|
734 |
* __ellint_rj(__cc, _Tp(1) - __kk * __ss, _Tp(1), |
|
|
735 |
_Tp(1) + __nu * __ss) / _Tp(3); |
|
|
736 |
|
|
|
737 |
if (__n == 0) |
|
|
738 |
return __Pi; |
|
|
739 |
else |
|
|
740 |
return __Pi + _Tp(2) * __n * __comp_ellint_3(__k, __nu); |
|
|
741 |
} |
|
|
742 |
} |
|
|
743 |
|
|
|
744 |
_GLIBCXX_END_NAMESPACE_VERSION |
|
|
745 |
} // namespace std::tr1::__detail |
|
|
746 |
} |
|
|
747 |
} |
|
|
748 |
|
|
|
749 |
#endif // _GLIBCXX_TR1_ELL_INTEGRAL_TCC |
|
|
750 |
|