// Copyright (c) 2006 Xiaogang Zhang | |
// 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 correctly | |
// handle the y < 0 case. | |
// | |
#ifndef BOOST_MATH_ELLINT_RC_HPP | |
#define BOOST_MATH_ELLINT_RC_HPP | |
#ifdef _MSC_VER | |
#pragma once | |
#endif | |
#include <boost/math/policies/error_handling.hpp> | |
#include <boost/math/tools/config.hpp> | |
#include <boost/math/special_functions/math_fwd.hpp> | |
// Carlson's degenerate elliptic integral | |
// R_C(x, y) = R_F(x, y, y) = 0.5 * \int_{0}^{\infty} (t+x)^{-1/2} (t+y)^{-1} dt | |
// Carlson, Numerische Mathematik, vol 33, 1 (1979) | |
namespace boost { namespace math { namespace detail{ | |
template <typename T, typename Policy> | |
T ellint_rc_imp(T x, T y, const Policy& pol) | |
{ | |
T value, S, u, lambda, tolerance, prefix; | |
unsigned long k; | |
BOOST_MATH_STD_USING | |
using namespace boost::math::tools; | |
static const char* function = "boost::math::ellint_rc<%1%>(%1%,%1%)"; | |
if(x < 0) | |
{ | |
return policies::raise_domain_error<T>(function, | |
"Argument x must be non-negative but got %1%", x, pol); | |
} | |
if(y == 0) | |
{ | |
return policies::raise_domain_error<T>(function, | |
"Argument y must not be zero but got %1%", y, pol); | |
} | |
// error scales as the 6th power of tolerance | |
tolerance = pow(4 * tools::epsilon<T>(), T(1) / 6); | |
// for y < 0, the integral is singular, return Cauchy principal value | |
if (y < 0) | |
{ | |
prefix = sqrt(x / (x - y)); | |
x = x - y; | |
y = -y; | |
} | |
else | |
prefix = 1; | |
// duplication: | |
k = 1; | |
do | |
{ | |
u = (x + y + y) / 3; | |
S = y / u - 1; // 1 - x / u = 2 * S | |
if (2 * abs(S) < tolerance) | |
break; | |
T sx = sqrt(x); | |
T sy = sqrt(y); | |
lambda = 2 * sx * sy + y; | |
x = (x + lambda) / 4; | |
y = (y + lambda) / 4; | |
++k; | |
}while(k < policies::get_max_series_iterations<Policy>()); | |
// Check to see if we gave up too soon: | |
policies::check_series_iterations(function, k, pol); | |
// Taylor series expansion to the 5th order | |
value = (1 + S * S * (T(3) / 10 + S * (T(1) / 7 + S * (T(3) / 8 + S * T(9) / 22)))) / sqrt(u); | |
return value * prefix; | |
} | |
} // namespace detail | |
template <class T1, class T2, class Policy> | |
inline typename tools::promote_args<T1, T2>::type | |
ellint_rc(T1 x, T2 y, const Policy& pol) | |
{ | |
typedef typename tools::promote_args<T1, T2>::type result_type; | |
typedef typename policies::evaluation<result_type, Policy>::type value_type; | |
return policies::checked_narrowing_cast<result_type, Policy>( | |
detail::ellint_rc_imp( | |
static_cast<value_type>(x), | |
static_cast<value_type>(y), pol), "boost::math::ellint_rc<%1%>(%1%,%1%)"); | |
} | |
template <class T1, class T2> | |
inline typename tools::promote_args<T1, T2>::type | |
ellint_rc(T1 x, T2 y) | |
{ | |
return ellint_rc(x, y, policies::policy<>()); | |
} | |
}} // namespaces | |
#endif // BOOST_MATH_ELLINT_RC_HPP | |