X-Git-Url: https://oss.titaniummirror.com/gitweb?a=blobdiff_plain;f=libstdc%2B%2B-v3%2Finclude%2Ftr1%2Fell_integral.tcc;fp=libstdc%2B%2B-v3%2Finclude%2Ftr1%2Fell_integral.tcc;h=09bda9aa90b94f801ee9bf31404d202d3c4cd3ea;hb=6fed43773c9b0ce596dca5686f37ac3fc0fa11c0;hp=0000000000000000000000000000000000000000;hpb=27b11d56b743098deb193d510b337ba22dc52e5c;p=msp430-gcc.git
diff --git a/libstdc++-v3/include/tr1/ell_integral.tcc b/libstdc++-v3/include/tr1/ell_integral.tcc
new file mode 100644
index 00000000..09bda9aa
--- /dev/null
+++ b/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
+// .
+
+/** @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
+ _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
+ _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
+ _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
+ _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
+ _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
+ _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
+ _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
+ _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
+ _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
+ _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
+ _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
+ _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
+