1 //  Copyright (c) 2006 Xiaogang Zhang, 2015 John Maddock
2 //  Use, modification and distribution are subject to the
3 //  Boost Software License, Version 1.0. (See accompanying file
4 //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
5 //
6 //  History:
7 //  XZ wrote the original of this file as part of the Google
8 //  Summer of Code 2006.  JM modified it to fit into the
9 //  Boost.Math conceptual framework better, and to handle
10 //  types longer than 80-bit reals.
11 //  Updated 2015 to use Carlson's latest methods.
12 //
13 #ifndef BOOST_MATH_ELLINT_RF_HPP
14 #define BOOST_MATH_ELLINT_RF_HPP
15 
16 #ifdef _MSC_VER
17 #pragma once
18 #endif
19 
20 #include <boost/math/special_functions/math_fwd.hpp>
21 #include <boost/math/tools/config.hpp>
22 #include <boost/math/constants/constants.hpp>
23 #include <boost/math/policies/error_handling.hpp>
24 #include <boost/math/special_functions/ellint_rc.hpp>
25 
26 // Carlson's elliptic integral of the first kind
27 // R_F(x, y, z) = 0.5 * \int_{0}^{\infty} [(t+x)(t+y)(t+z)]^{-1/2} dt
28 // Carlson, Numerische Mathematik, vol 33, 1 (1979)
29 
30 namespace boost { namespace math { namespace detail{
31 
32    template <typename T, typename Policy>
33    T ellint_rf_imp(T x, T y, T z, const Policy& pol)
34    {
35       BOOST_MATH_STD_USING
36       using namespace boost::math;
37       using std::swap;
38 
39       static const char* function = "boost::math::ellint_rf<%1%>(%1%,%1%,%1%)";
40 
41       if(x < 0 || y < 0 || z < 0)
42       {
43          return policies::raise_domain_error<T>(function,
44             "domain error, all arguments must be non-negative, "
45             "only sensible result is %1%.",
46             std::numeric_limits<T>::quiet_NaN(), pol);
47       }
48       if(x + y == 0 || y + z == 0 || z + x == 0)
49       {
50          return policies::raise_domain_error<T>(function,
51             "domain error, at most one argument can be zero, "
52             "only sensible result is %1%.",
53             std::numeric_limits<T>::quiet_NaN(), pol);
54       }
55       //
56       // Special cases from http://dlmf.nist.gov/19.20#i
57       //
58       if(x == y)
59       {
60          if(x == z)
61          {
62             // x, y, z equal:
63             return 1 / sqrt(x);
64          }
65          else
66          {
67             // 2 equal, x and y:
68             if(z == 0)
69                return constants::pi<T>() / (2 * sqrt(x));
70             else
71                return ellint_rc_imp(z, x, pol);
72          }
73       }
74       if(x == z)
75       {
76          if(y == 0)
77             return constants::pi<T>() / (2 * sqrt(x));
78          else
79             return ellint_rc_imp(y, x, pol);
80       }
81       if(y == z)
82       {
83          if(x == 0)
84             return constants::pi<T>() / (2 * sqrt(y));
85          else
86             return ellint_rc_imp(x, y, pol);
87       }
88       if(x == 0)
89          swap(x, z);
90       else if(y == 0)
91          swap(y, z);
92       if(z == 0)
93       {
94          //
95          // Special case for one value zero:
96          //
97          T xn = sqrt(x);
98          T yn = sqrt(y);
99 
100          while(fabs(xn - yn) >= 2.7 * tools::root_epsilon<T>() * fabs(xn))
101          {
102             T t = sqrt(xn * yn);
103             xn = (xn + yn) / 2;
104             yn = t;
105          }
106          return constants::pi<T>() / (xn + yn);
107       }
108 
109       T xn = x;
110       T yn = y;
111       T zn = z;
112       T An = (x + y + z) / 3;
113       T A0 = An;
114       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));
115       T fn = 1;
116 
117 
118       // duplication
119       unsigned k = 1;
120       for(; k < boost::math::policies::get_max_series_iterations<Policy>(); ++k)
121       {
122          T root_x = sqrt(xn);
123          T root_y = sqrt(yn);
124          T root_z = sqrt(zn);
125          T lambda = root_x * root_y + root_x * root_z + root_y * root_z;
126          An = (An + lambda) / 4;
127          xn = (xn + lambda) / 4;
128          yn = (yn + lambda) / 4;
129          zn = (zn + lambda) / 4;
130          Q /= 4;
131          fn *= 4;
132          if(Q < fabs(An))
133             break;
134       }
135       // Check to see if we gave up too soon:
136       policies::check_series_iterations<T>(function, k, pol);
137       BOOST_MATH_INSTRUMENT_VARIABLE(k);
138 
139       T X = (A0 - x) / (An * fn);
140       T Y = (A0 - y) / (An * fn);
141       T Z = -X - Y;
142 
143       // Taylor series expansion to the 7th order
144       T E2 = X * Y - Z * Z;
145       T E3 = X * Y * Z;
146       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);
147    }
148 
149 } // namespace detail
150 
151 template <class T1, class T2, class T3, class Policy>
152 inline typename tools::promote_args<T1, T2, T3>::type
ellint_rf(T1 x,T2 y,T3 z,const Policy & pol)153    ellint_rf(T1 x, T2 y, T3 z, const Policy& pol)
154 {
155    typedef typename tools::promote_args<T1, T2, T3>::type result_type;
156    typedef typename policies::evaluation<result_type, Policy>::type value_type;
157    return policies::checked_narrowing_cast<result_type, Policy>(
158       detail::ellint_rf_imp(
159          static_cast<value_type>(x),
160          static_cast<value_type>(y),
161          static_cast<value_type>(z), pol), "boost::math::ellint_rf<%1%>(%1%,%1%,%1%)");
162 }
163 
164 template <class T1, class T2, class T3>
165 inline typename tools::promote_args<T1, T2, T3>::type
ellint_rf(T1 x,T2 y,T3 z)166    ellint_rf(T1 x, T2 y, T3 z)
167 {
168    return ellint_rf(x, y, z, policies::policy<>());
169 }
170 
171 }} // namespaces
172 
173 #endif // BOOST_MATH_ELLINT_RF_HPP
174 
175