/usr/include/boost/math/special_functions
Edit: /usr/include/boost/math/special_functions/ellint_rf.hpp (5255B)
// Copyright (c) 2006 Xiaogang Zhang
// Copyright (c) 2015 John Maddock
// Use, modification and distribution are subject to the
// Boost Software License, Version 1.0. (See accompanying file
// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
//
// History:
// XZ wrote the original of this file as part of the Google
// Summer of Code 2006. JM modified it to fit into the
// Boost.Math conceptual framework better, and to handle
// types longer than 80-bit reals.
// Updated 2015 to use Carlson's latest methods.
//
#ifndef BOOST_MATH_ELLINT_RF_HPP
#define BOOST_MATH_ELLINT_RF_HPP
#ifdef _MSC_VER
#pragma once
#endif
#include
#include
#include
#include
#include
// Carlson's elliptic integral of the first kind
// R_F(x, y, z) = 0.5 * \int_{0}^{\infty} [(t+x)(t+y)(t+z)]^{-1/2} dt
// Carlson, Numerische Mathematik, vol 33, 1 (1979)
namespace boost { namespace math { namespace detail{
template
T ellint_rf_imp(T x, T y, T z, const Policy& pol)
{
BOOST_MATH_STD_USING
using namespace boost::math;
using std::swap;
static const char* function = "boost::math::ellint_rf<%1%>(%1%,%1%,%1%)";
if(x < 0 || y < 0 || z < 0)
{
return policies::raise_domain_error(function,
"domain error, all arguments must be non-negative, "
"only sensible result is %1%.",
std::numeric_limits::quiet_NaN(), pol);
}
if(x + y == 0 || y + z == 0 || z + x == 0)
{
return policies::raise_domain_error(function,
"domain error, at most one argument can be zero, "
"only sensible result is %1%.",
std::numeric_limits::quiet_NaN(), pol);
}
//
// Special cases from http://dlmf.nist.gov/19.20#i
//
if(x == y)
{
if(x == z)
{
// x, y, z equal:
return 1 / sqrt(x);
}
else
{
// 2 equal, x and y:
if(z == 0)
return constants::pi() / (2 * sqrt(x));
else
return ellint_rc_imp(z, x, pol);
}
}
if(x == z)
{
if(y == 0)
return constants::pi() / (2 * sqrt(x));
else
return ellint_rc_imp(y, x, pol);
}
if(y == z)
{
if(x == 0)
return constants::pi() / (2 * sqrt(y));
else
return ellint_rc_imp(x, y, pol);
}
if(x == 0)
swap(x, z);
else if(y == 0)
swap(y, z);
if(z == 0)
{
//
// Special case for one value zero:
//
T xn = sqrt(x);
T yn = sqrt(y);
while(fabs(xn - yn) >= 2.7 * tools::root_epsilon() * fabs(xn))
{
T t = sqrt(xn * yn);
xn = (xn + yn) / 2;
yn = t;
}
return constants::pi() / (xn + yn);
}
T xn = x;
T yn = y;
T zn = z;
T An = (x + y + z) / 3;
T A0 = An;
T Q = pow(3 * boost::math::tools::epsilon(), T(-1) / 8) * (std::max)((std::max)(fabs(An - xn), fabs(An - yn)), fabs(An - zn));
T fn = 1;
// duplication
unsigned k = 1;
for(; k < boost::math::policies::get_max_series_iterations(); ++k)
{
T root_x = sqrt(xn);
T root_y = sqrt(yn);
T root_z = sqrt(zn);
T lambda = root_x * root_y + root_x * root_z + root_y * root_z;
An = (An + lambda) / 4;
xn = (xn + lambda) / 4;
yn = (yn + lambda) / 4;
zn = (zn + lambda) / 4;
Q /= 4;
fn *= 4;
if(Q < fabs(An))
break;
}
// Check to see if we gave up too soon:
policies::check_series_iterations(function, k, pol);
BOOST_MATH_INSTRUMENT_VARIABLE(k);
T X = (A0 - x) / (An * fn);
T Y = (A0 - y) / (An * fn);
T Z = -X - Y;
// Taylor series expansion to the 7th order
T E2 = X * Y - Z * Z;
T E3 = X * Y * Z;
return (1 + E3 * (T(1) / 14 + 3 * E3 / 104) + E2 * (T(-1) / 10 + E2 / 24 - (3 * E3) / 44 - 5 * E2 * E2 / 208 + E2 * E3 / 16)) / sqrt(An);
}
} // namespace detail
template
inline typename tools::promote_args::type
ellint_rf(T1 x, T2 y, T3 z, const Policy& pol)
{
typedef typename tools::promote_args::type result_type;
typedef typename policies::evaluation::type value_type;
return policies::checked_narrowing_cast(
detail::ellint_rf_imp(
static_cast(x),
static_cast(y),
static_cast(z), pol), "boost::math::ellint_rf<%1%>(%1%,%1%,%1%)");
}
template
inline typename tools::promote_args::type
ellint_rf(T1 x, T2 y, T3 z)
{
return ellint_rf(x, y, z, policies::policy<>());
}
}} // namespaces
#endif // BOOST_MATH_ELLINT_RF_HPP