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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> | |
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 demoninator 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 | T lambda, rx, ry, rz; | |
137 | unsigned k = 0; | |
138 | T fn = 1; | |
139 | T RD_sum = 0; | |
140 | ||
141 | for(; k < policies::get_max_series_iterations<Policy>(); ++k) | |
142 | { | |
143 | rx = sqrt(xn); | |
144 | ry = sqrt(yn); | |
145 | rz = sqrt(zn); | |
146 | lambda = rx * ry + rx * rz + ry * rz; | |
147 | RD_sum += fn / (rz * (zn + lambda)); | |
148 | An = (An + lambda) / 4; | |
149 | xn = (xn + lambda) / 4; | |
150 | yn = (yn + lambda) / 4; | |
151 | zn = (zn + lambda) / 4; | |
152 | fn /= 4; | |
153 | Q /= 4; | |
154 | if(Q < An) | |
155 | break; | |
156 | } | |
157 | ||
158 | policies::check_series_iterations<T, Policy>(function, k, pol); | |
159 | ||
160 | T X = fn * (A0 - x) / An; | |
161 | T Y = fn * (A0 - y) / An; | |
162 | T Z = -(X + Y) / 3; | |
163 | T E2 = X * Y - 6 * Z * Z; | |
164 | T E3 = (3 * X * Y - 8 * Z * Z) * Z; | |
165 | T E4 = 3 * (X * Y - Z * Z) * Z * Z; | |
166 | T E5 = X * Y * Z * Z * Z; | |
167 | ||
168 | T result = fn * pow(An, T(-3) / 2) * | |
169 | (1 - 3 * E2 / 14 + E3 / 6 + 9 * E2 * E2 / 88 - 3 * E4 / 22 - 9 * E2 * E3 / 52 + 3 * E5 / 26 - E2 * E2 * E2 / 16 | |
170 | + 3 * E3 * E3 / 40 + 3 * E2 * E4 / 20 + 45 * E2 * E2 * E3 / 272 - 9 * (E3 * E4 + E2 * E5) / 68); | |
171 | result += 3 * RD_sum; | |
172 | ||
173 | return result; | |
174 | } | |
175 | ||
176 | } // namespace detail | |
177 | ||
178 | template <class T1, class T2, class T3, class Policy> | |
179 | inline typename tools::promote_args<T1, T2, T3>::type | |
180 | ellint_rd(T1 x, T2 y, T3 z, const Policy& pol) | |
181 | { | |
182 | typedef typename tools::promote_args<T1, T2, T3>::type result_type; | |
183 | typedef typename policies::evaluation<result_type, Policy>::type value_type; | |
184 | return policies::checked_narrowing_cast<result_type, Policy>( | |
185 | detail::ellint_rd_imp( | |
186 | static_cast<value_type>(x), | |
187 | static_cast<value_type>(y), | |
188 | static_cast<value_type>(z), pol), "boost::math::ellint_rd<%1%>(%1%,%1%,%1%)"); | |
189 | } | |
190 | ||
191 | template <class T1, class T2, class T3> | |
192 | inline typename tools::promote_args<T1, T2, T3>::type | |
193 | ellint_rd(T1 x, T2 y, T3 z) | |
194 | { | |
195 | return ellint_rd(x, y, z, policies::policy<>()); | |
196 | } | |
197 | ||
198 | }} // namespaces | |
199 | ||
200 | #endif // BOOST_MATH_ELLINT_RD_HPP | |
201 |