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 slightly to fit into the
9 // Boost.Math conceptual framework better.
10 // Updated 2015 to use Carlson's latest methods.
11
12 #ifndef BOOST_MATH_ELLINT_RD_HPP
13 #define BOOST_MATH_ELLINT_RD_HPP
14
15 #ifdef _MSC_VER
16 #pragma once
17 #endif
18
19 #include <boost/math/special_functions/math_fwd.hpp>
20 #include <boost/math/special_functions/ellint_rc.hpp>
21 #include <boost/math/special_functions/pow.hpp>
22 #include <boost/math/tools/config.hpp>
23 #include <boost/math/policies/error_handling.hpp>
24
25 // Carlson's elliptic integral of the second kind
26 // 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
27 // Carlson, Numerische Mathematik, vol 33, 1 (1979)
28
29 namespace boost { namespace math { namespace detail{
30
31 template <typename T, typename Policy>
ellint_rd_imp(T x,T y,T z,const Policy & pol)32 T ellint_rd_imp(T x, T y, T z, const Policy& pol)
33 {
34 BOOST_MATH_STD_USING
35 using std::swap;
36
37 static const char* function = "boost::math::ellint_rd<%1%>(%1%,%1%,%1%)";
38
39 if(x < 0)
40 {
41 return policies::raise_domain_error<T>(function,
42 "Argument x must be >= 0, but got %1%", x, pol);
43 }
44 if(y < 0)
45 {
46 return policies::raise_domain_error<T>(function,
47 "Argument y must be >= 0, but got %1%", y, pol);
48 }
49 if(z <= 0)
50 {
51 return policies::raise_domain_error<T>(function,
52 "Argument z must be > 0, but got %1%", z, pol);
53 }
54 if(x + y == 0)
55 {
56 return policies::raise_domain_error<T>(function,
57 "At most one argument can be zero, but got, x + y = %1%", x + y, pol);
58 }
59 //
60 // Special cases from http://dlmf.nist.gov/19.20#iv
61 //
62 using std::swap;
63 if(x == z)
64 swap(x, y);
65 if(y == z)
66 {
67 if(x == y)
68 {
69 return 1 / (x * sqrt(x));
70 }
71 else if(x == 0)
72 {
73 return 3 * constants::pi<T>() / (4 * y * sqrt(y));
74 }
75 else
76 {
77 if((std::min)(x, y) / (std::max)(x, y) > 1.3)
78 return 3 * (ellint_rc_imp(x, y, pol) - sqrt(x) / y) / (2 * (y - x));
79 // Otherwise fall through to avoid cancellation in the above (RC(x,y) -> 1/x^0.5 as x -> y)
80 }
81 }
82 if(x == y)
83 {
84 if((std::min)(x, z) / (std::max)(x, z) > 1.3)
85 return 3 * (ellint_rc_imp(z, x, pol) - 1 / sqrt(z)) / (z - x);
86 // Otherwise fall through to avoid cancellation in the above (RC(x,y) -> 1/x^0.5 as x -> y)
87 }
88 if(y == 0)
89 swap(x, y);
90 if(x == 0)
91 {
92 //
93 // Special handling for common case, from
94 // Numerical Computation of Real or Complex Elliptic Integrals, eq.47
95 //
96 T xn = sqrt(y);
97 T yn = sqrt(z);
98 T x0 = xn;
99 T y0 = yn;
100 T sum = 0;
101 T sum_pow = 0.25f;
102
103 while(fabs(xn - yn) >= 2.7 * tools::root_epsilon<T>() * fabs(xn))
104 {
105 T t = sqrt(xn * yn);
106 xn = (xn + yn) / 2;
107 yn = t;
108 sum_pow *= 2;
109 sum += sum_pow * boost::math::pow<2>(xn - yn);
110 }
111 T RF = constants::pi<T>() / (xn + yn);
112 //
113 // This following calculation suffers from serious cancellation when y ~ z
114 // unless we combine terms. We have:
115 //
116 // ( ((x0 + y0)/2)^2 - z ) / (z(y-z))
117 //
118 // Substituting y = x0^2 and z = y0^2 and simplifying we get the following:
119 //
120 T pt = (x0 + 3 * y0) / (4 * z * (x0 + y0));
121 //
122 // Since we've moved the denominator from eq.47 inside the expression, we
123 // need to also scale "sum" by the same value:
124 //
125 pt -= sum / (z * (y - z));
126 return pt * RF * 3;
127 }
128
129 T xn = x;
130 T yn = y;
131 T zn = z;
132 T An = (x + y + 3 * z) / 5;
133 T A0 = An;
134 // This has an extra 1.2 fudge factor which is really only needed when x, y and z are close in magnitude:
135 T Q = pow(tools::epsilon<T>() / 4, -T(1) / 8) * (std::max)((std::max)(An - x, An - y), An - z) * 1.2f;
136 BOOST_MATH_INSTRUMENT_VARIABLE(Q);
137 T lambda, rx, ry, rz;
138 unsigned k = 0;
139 T fn = 1;
140 T RD_sum = 0;
141
142 for(; k < policies::get_max_series_iterations<Policy>(); ++k)
143 {
144 rx = sqrt(xn);
145 ry = sqrt(yn);
146 rz = sqrt(zn);
147 lambda = rx * ry + rx * rz + ry * rz;
148 RD_sum += fn / (rz * (zn + lambda));
149 An = (An + lambda) / 4;
150 xn = (xn + lambda) / 4;
151 yn = (yn + lambda) / 4;
152 zn = (zn + lambda) / 4;
153 fn /= 4;
154 Q /= 4;
155 BOOST_MATH_INSTRUMENT_VARIABLE(k);
156 BOOST_MATH_INSTRUMENT_VARIABLE(RD_sum);
157 BOOST_MATH_INSTRUMENT_VARIABLE(Q);
158 if(Q < An)
159 break;
160 }
161
162 policies::check_series_iterations<T, Policy>(function, k, pol);
163
164 T X = fn * (A0 - x) / An;
165 T Y = fn * (A0 - y) / An;
166 T Z = -(X + Y) / 3;
167 T E2 = X * Y - 6 * Z * Z;
168 T E3 = (3 * X * Y - 8 * Z * Z) * Z;
169 T E4 = 3 * (X * Y - Z * Z) * Z * Z;
170 T E5 = X * Y * Z * Z * Z;
171
172 T result = fn * pow(An, T(-3) / 2) *
173 (1 - 3 * E2 / 14 + E3 / 6 + 9 * E2 * E2 / 88 - 3 * E4 / 22 - 9 * E2 * E3 / 52 + 3 * E5 / 26 - E2 * E2 * E2 / 16
174 + 3 * E3 * E3 / 40 + 3 * E2 * E4 / 20 + 45 * E2 * E2 * E3 / 272 - 9 * (E3 * E4 + E2 * E5) / 68);
175 BOOST_MATH_INSTRUMENT_VARIABLE(result);
176 result += 3 * RD_sum;
177
178 return result;
179 }
180
181 } // namespace detail
182
183 template <class T1, class T2, class T3, class Policy>
184 inline typename tools::promote_args<T1, T2, T3>::type
ellint_rd(T1 x,T2 y,T3 z,const Policy & pol)185 ellint_rd(T1 x, T2 y, T3 z, const Policy& pol)
186 {
187 typedef typename tools::promote_args<T1, T2, T3>::type result_type;
188 typedef typename policies::evaluation<result_type, Policy>::type value_type;
189 return policies::checked_narrowing_cast<result_type, Policy>(
190 detail::ellint_rd_imp(
191 static_cast<value_type>(x),
192 static_cast<value_type>(y),
193 static_cast<value_type>(z), pol), "boost::math::ellint_rd<%1%>(%1%,%1%,%1%)");
194 }
195
196 template <class T1, class T2, class T3>
197 inline typename tools::promote_args<T1, T2, T3>::type
ellint_rd(T1 x,T2 y,T3 z)198 ellint_rd(T1 x, T2 y, T3 z)
199 {
200 return ellint_rd(x, y, z, policies::policy<>());
201 }
202
203 }} // namespaces
204
205 #endif // BOOST_MATH_ELLINT_RD_HPP
206
207