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// 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<T>(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
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