// 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 slightly to fit into the | |
// Boost.Math conceptual framework better. | |
#ifndef BOOST_MATH_ELLINT_RD_HPP | |
#define BOOST_MATH_ELLINT_RD_HPP | |
#ifdef _MSC_VER | |
#pragma once | |
#endif | |
#include <boost/math/special_functions/math_fwd.hpp> | |
#include <boost/math/tools/config.hpp> | |
#include <boost/math/policies/error_handling.hpp> | |
// Carlson's elliptic integral of the second kind | |
// R_D(x, y, z) = R_J(x, y, z, z) = 1.5 * \int_{0}^{\infty} [(t+x)(t+y)]^{-1/2} (t+z)^{-3/2} dt | |
// Carlson, Numerische Mathematik, vol 33, 1 (1979) | |
namespace boost { namespace math { namespace detail{ | |
template <typename T, typename Policy> | |
T ellint_rd_imp(T x, T y, T z, const Policy& pol) | |
{ | |
T value, u, lambda, sigma, factor, tolerance; | |
T X, Y, Z, EA, EB, EC, ED, EE, S1, S2; | |
unsigned long k; | |
BOOST_MATH_STD_USING | |
using namespace boost::math::tools; | |
static const char* function = "boost::math::ellint_rd<%1%>(%1%,%1%,%1%)"; | |
if (x < 0) | |
{ | |
return policies::raise_domain_error<T>(function, | |
"Argument x must be >= 0, but got %1%", x, pol); | |
} | |
if (y < 0) | |
{ | |
return policies::raise_domain_error<T>(function, | |
"Argument y must be >= 0, but got %1%", y, pol); | |
} | |
if (z <= 0) | |
{ | |
return policies::raise_domain_error<T>(function, | |
"Argument z must be > 0, but got %1%", z, pol); | |
} | |
if (x + y == 0) | |
{ | |
return policies::raise_domain_error<T>(function, | |
"At most one argument can be zero, but got, x + y = %1%", x+y, pol); | |
} | |
// error scales as the 6th power of tolerance | |
tolerance = pow(tools::epsilon<T>() / 3, T(1)/6); | |
// duplication | |
sigma = 0; | |
factor = 1; | |
k = 1; | |
do | |
{ | |
u = (x + y + z + z + z) / 5; | |
X = (u - x) / u; | |
Y = (u - y) / u; | |
Z = (u - z) / u; | |
if ((tools::max)(abs(X), abs(Y), abs(Z)) < tolerance) | |
break; | |
T sx = sqrt(x); | |
T sy = sqrt(y); | |
T sz = sqrt(z); | |
lambda = sy * (sx + sz) + sz * sx; //sqrt(x * y) + sqrt(y * z) + sqrt(z * x); | |
sigma += factor / (sz * (z + lambda)); | |
factor /= 4; | |
x = (x + lambda) / 4; | |
y = (y + lambda) / 4; | |
z = (z + 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 | |
EA = X * Y; | |
EB = Z * Z; | |
EC = EA - EB; | |
ED = EA - 6 * EB; | |
EE = ED + EC + EC; | |
S1 = ED * (ED * T(9) / 88 - Z * EE * T(9) / 52 - T(3) / 14); | |
S2 = Z * (EE / 6 + Z * (-EC * T(9) / 22 + Z * EA * T(3) / 26)); | |
value = 3 * sigma + factor * (1 + S1 + S2) / (u * sqrt(u)); | |
return value; | |
} | |
} // namespace detail | |
template <class T1, class T2, class T3, class Policy> | |
inline typename tools::promote_args<T1, T2, T3>::type | |
ellint_rd(T1 x, T2 y, T3 z, const Policy& pol) | |
{ | |
typedef typename tools::promote_args<T1, T2, T3>::type result_type; | |
typedef typename policies::evaluation<result_type, Policy>::type value_type; | |
return policies::checked_narrowing_cast<result_type, Policy>( | |
detail::ellint_rd_imp( | |
static_cast<value_type>(x), | |
static_cast<value_type>(y), | |
static_cast<value_type>(z), pol), "boost::math::ellint_rd<%1%>(%1%,%1%,%1%)"); | |
} | |
template <class T1, class T2, class T3> | |
inline typename tools::promote_args<T1, T2, T3>::type | |
ellint_rd(T1 x, T2 y, T3 z) | |
{ | |
return ellint_rd(x, y, z, policies::policy<>()); | |
} | |
}} // namespaces | |
#endif // BOOST_MATH_ELLINT_RD_HPP | |