Added boost header

test/external/boost/math/special_functions/ellint_rd.hpp

@@ -0,0 +1,130 @@
+//  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
+