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-rw-r--r--boost/math/special_functions/ellint_rf.hpp182
1 files changed, 112 insertions, 70 deletions
diff --git a/boost/math/special_functions/ellint_rf.hpp b/boost/math/special_functions/ellint_rf.hpp
index ac57257..a8a7b4b 100644
--- a/boost/math/special_functions/ellint_rf.hpp
+++ b/boost/math/special_functions/ellint_rf.hpp
@@ -1,4 +1,4 @@
-// Copyright (c) 2006 Xiaogang Zhang
+// Copyright (c) 2006 Xiaogang Zhang, 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)
@@ -8,6 +8,7 @@
// 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
@@ -18,8 +19,9 @@
#include <boost/math/special_functions/math_fwd.hpp>
#include <boost/math/tools/config.hpp>
-
+#include <boost/math/constants/constants.hpp>
#include <boost/math/policies/error_handling.hpp>
+#include <boost/math/special_functions/ellint_rc.hpp>
// 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
@@ -27,82 +29,122 @@
namespace boost { namespace math { namespace detail{
-template <typename T, typename Policy>
-T ellint_rf_imp(T x, T y, T z, const Policy& pol)
-{
- T value, X, Y, Z, E2, E3, u, lambda, tolerance;
- unsigned long k;
-
- BOOST_MATH_STD_USING
- using namespace boost::math::tools;
+ template <typename T, typename Policy>
+ 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%)";
+ static const char* function = "boost::math::ellint_rf<%1%>(%1%,%1%,%1%)";
- if (x < 0 || y < 0 || z < 0)
- {
- return policies::raise_domain_error<T>(function,
+ if(x < 0 || y < 0 || z < 0)
+ {
+ return policies::raise_domain_error<T>(function,
"domain error, all arguments must be non-negative, "
"only sensible result is %1%.",
std::numeric_limits<T>::quiet_NaN(), pol);
- }
- if (x + y == 0 || y + z == 0 || z + x == 0)
- {
- return policies::raise_domain_error<T>(function,
+ }
+ if(x + y == 0 || y + z == 0 || z + x == 0)
+ {
+ return policies::raise_domain_error<T>(function,
"domain error, at most one argument can be zero, "
"only sensible result is %1%.",
std::numeric_limits<T>::quiet_NaN(), pol);
- }
-
- // Carlson scales error as the 6th power of tolerance,
- // but this seems not to work for types larger than
- // 80-bit reals, this heuristic seems to work OK:
- if(policies::digits<T, Policy>() > 64)
- {
- tolerance = pow(tools::epsilon<T>(), T(1)/4.25f);
- BOOST_MATH_INSTRUMENT_VARIABLE(tolerance);
- }
- else
- {
- tolerance = pow(4*tools::epsilon<T>(), T(1)/6);
- BOOST_MATH_INSTRUMENT_VARIABLE(tolerance);
- }
-
- // duplication
- k = 1;
- do
- {
- u = (x + y + z) / 3;
- X = (u - x) / u;
- Y = (u - y) / u;
- Z = (u - z) / u;
-
- // Termination condition:
- 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;
- 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);
- BOOST_MATH_INSTRUMENT_VARIABLE(k);
-
- // Taylor series expansion to the 5th order
- E2 = X * Y - Z * Z;
- E3 = X * Y * Z;
- value = (1 + E2*(E2/24 - E3*T(3)/44 - T(0.1)) + E3/14) / sqrt(u);
- BOOST_MATH_INSTRUMENT_VARIABLE(value);
-
- return value;
-}
+ }
+ //
+ // 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<T>() / (2 * sqrt(x));
+ else
+ return ellint_rc_imp(z, x, pol);
+ }
+ }
+ if(x == z)
+ {
+ if(y == 0)
+ return constants::pi<T>() / (2 * sqrt(x));
+ else
+ return ellint_rc_imp(y, x, pol);
+ }
+ if(y == z)
+ {
+ if(x == 0)
+ return constants::pi<T>() / (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<T>() * fabs(xn))
+ {
+ T t = sqrt(xn * yn);
+ xn = (xn + yn) / 2;
+ yn = t;
+ }
+ return constants::pi<T>() / (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>(), 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<Policy>(); ++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<T>(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