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1 | // Copyright (c) 2006 Xiaogang Zhang |
2 | // Copyright (c) 2006 John Maddock | |
3 | // Use, modification and distribution are subject to the | |
4 | // Boost Software License, Version 1.0. (See accompanying file | |
5 | // LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) | |
6 | // | |
7 | // History: | |
8 | // XZ wrote the original of this file as part of the Google | |
9 | // Summer of Code 2006. JM modified it to fit into the | |
10 | // Boost.Math conceptual framework better, and to ensure | |
11 | // that the code continues to work no matter how many digits | |
12 | // type T has. | |
13 | ||
14 | #ifndef BOOST_MATH_ELLINT_2_HPP | |
15 | #define BOOST_MATH_ELLINT_2_HPP | |
16 | ||
17 | #ifdef _MSC_VER | |
18 | #pragma once | |
19 | #endif | |
20 | ||
21 | #include <boost/math/special_functions/math_fwd.hpp> | |
22 | #include <boost/math/special_functions/ellint_rf.hpp> | |
23 | #include <boost/math/special_functions/ellint_rd.hpp> | |
24 | #include <boost/math/special_functions/ellint_rg.hpp> | |
25 | #include <boost/math/constants/constants.hpp> | |
26 | #include <boost/math/policies/error_handling.hpp> | |
27 | #include <boost/math/tools/workaround.hpp> | |
28 | #include <boost/math/special_functions/round.hpp> | |
29 | ||
30 | // Elliptic integrals (complete and incomplete) of the second kind | |
31 | // Carlson, Numerische Mathematik, vol 33, 1 (1979) | |
32 | ||
33 | namespace boost { namespace math { | |
34 | ||
35 | template <class T1, class T2, class Policy> | |
36 | typename tools::promote_args<T1, T2>::type ellint_2(T1 k, T2 phi, const Policy& pol); | |
37 | ||
38 | namespace detail{ | |
39 | ||
40 | template <typename T, typename Policy> | |
41 | T ellint_e_imp(T k, const Policy& pol); | |
42 | ||
43 | // Elliptic integral (Legendre form) of the second kind | |
44 | template <typename T, typename Policy> | |
45 | T ellint_e_imp(T phi, T k, const Policy& pol) | |
46 | { | |
47 | BOOST_MATH_STD_USING | |
48 | using namespace boost::math::tools; | |
49 | using namespace boost::math::constants; | |
50 | ||
51 | bool invert = false; | |
52 | if(phi < 0) | |
53 | { | |
54 | phi = fabs(phi); | |
55 | invert = true; | |
56 | } | |
57 | ||
58 | T result; | |
59 | ||
60 | if(phi >= tools::max_value<T>()) | |
61 | { | |
62 | // Need to handle infinity as a special case: | |
63 | result = policies::raise_overflow_error<T>("boost::math::ellint_e<%1%>(%1%,%1%)", 0, pol); | |
64 | } | |
65 | else if(phi > 1 / tools::epsilon<T>()) | |
66 | { | |
67 | // Phi is so large that phi%pi is necessarily zero (or garbage), | |
68 | // just return the second part of the duplication formula: | |
69 | result = 2 * phi * ellint_e_imp(k, pol) / constants::pi<T>(); | |
70 | } | |
71 | else if(k == 0) | |
72 | { | |
73 | return invert ? T(-phi) : phi; | |
74 | } | |
75 | else if(fabs(k) == 1) | |
76 | { | |
77 | return invert ? T(-sin(phi)) : T(sin(phi)); | |
78 | } | |
79 | else | |
80 | { | |
81 | // Carlson's algorithm works only for |phi| <= pi/2, | |
82 | // use the integrand's periodicity to normalize phi | |
83 | // | |
84 | // Xiaogang's original code used a cast to long long here | |
85 | // but that fails if T has more digits than a long long, | |
86 | // so rewritten to use fmod instead: | |
87 | // | |
88 | T rphi = boost::math::tools::fmod_workaround(phi, T(constants::half_pi<T>())); | |
89 | T m = boost::math::round((phi - rphi) / constants::half_pi<T>()); | |
90 | int s = 1; | |
91 | if(boost::math::tools::fmod_workaround(m, T(2)) > 0.5) | |
92 | { | |
93 | m += 1; | |
94 | s = -1; | |
95 | rphi = constants::half_pi<T>() - rphi; | |
96 | } | |
97 | T k2 = k * k; | |
98 | if(k2 > 1) | |
99 | { | |
100 | return policies::raise_domain_error<T>("boost::math::ellint_2<%1%>(%1%, %1%)", "The parameter k is out of range, got k = %1%", k, pol); | |
101 | } | |
102 | else if(rphi < tools::root_epsilon<T>()) | |
103 | { | |
104 | // See http://functions.wolfram.com/EllipticIntegrals/EllipticE2/06/01/03/0001/ | |
105 | result = s * rphi; | |
106 | } | |
107 | else | |
108 | { | |
109 | // http://dlmf.nist.gov/19.25#E10 | |
110 | T sinp = sin(rphi); | |
111 | T cosp = cos(rphi); | |
112 | T c = 1 / (sinp * sinp); | |
113 | T cm1 = cosp * cosp / (sinp * sinp); // c - 1 | |
114 | result = s * ((1 - k2) * ellint_rf_imp(cm1, T(c - k2), c, pol) + k2 * (1 - k2) * ellint_rd(cm1, c, T(c - k2), pol) / 3 + k2 * sqrt(cm1 / (c * (c - k2)))); | |
115 | } | |
116 | if(m != 0) | |
117 | result += m * ellint_e_imp(k, pol); | |
118 | } | |
119 | return invert ? T(-result) : result; | |
120 | } | |
121 | ||
122 | // Complete elliptic integral (Legendre form) of the second kind | |
123 | template <typename T, typename Policy> | |
124 | T ellint_e_imp(T k, const Policy& pol) | |
125 | { | |
126 | BOOST_MATH_STD_USING | |
127 | using namespace boost::math::tools; | |
128 | ||
129 | if (abs(k) > 1) | |
130 | { | |
131 | return policies::raise_domain_error<T>("boost::math::ellint_e<%1%>(%1%)", | |
132 | "Got k = %1%, function requires |k| <= 1", k, pol); | |
133 | } | |
134 | if (abs(k) == 1) | |
135 | { | |
136 | return static_cast<T>(1); | |
137 | } | |
138 | ||
139 | T x = 0; | |
140 | T t = k * k; | |
141 | T y = 1 - t; | |
142 | T z = 1; | |
143 | T value = 2 * ellint_rg_imp(x, y, z, pol); | |
144 | ||
145 | return value; | |
146 | } | |
147 | ||
148 | template <typename T, typename Policy> | |
149 | inline typename tools::promote_args<T>::type ellint_2(T k, const Policy& pol, const mpl::true_&) | |
150 | { | |
151 | typedef typename tools::promote_args<T>::type result_type; | |
152 | typedef typename policies::evaluation<result_type, Policy>::type value_type; | |
153 | return policies::checked_narrowing_cast<result_type, Policy>(detail::ellint_e_imp(static_cast<value_type>(k), pol), "boost::math::ellint_2<%1%>(%1%)"); | |
154 | } | |
155 | ||
156 | // Elliptic integral (Legendre form) of the second kind | |
157 | template <class T1, class T2> | |
158 | inline typename tools::promote_args<T1, T2>::type ellint_2(T1 k, T2 phi, const mpl::false_&) | |
159 | { | |
160 | return boost::math::ellint_2(k, phi, policies::policy<>()); | |
161 | } | |
162 | ||
163 | } // detail | |
164 | ||
165 | // Complete elliptic integral (Legendre form) of the second kind | |
166 | template <typename T> | |
167 | inline typename tools::promote_args<T>::type ellint_2(T k) | |
168 | { | |
169 | return ellint_2(k, policies::policy<>()); | |
170 | } | |
171 | ||
172 | // Elliptic integral (Legendre form) of the second kind | |
173 | template <class T1, class T2> | |
174 | inline typename tools::promote_args<T1, T2>::type ellint_2(T1 k, T2 phi) | |
175 | { | |
176 | typedef typename policies::is_policy<T2>::type tag_type; | |
177 | return detail::ellint_2(k, phi, tag_type()); | |
178 | } | |
179 | ||
180 | template <class T1, class T2, class Policy> | |
181 | inline typename tools::promote_args<T1, T2>::type ellint_2(T1 k, T2 phi, const Policy& pol) | |
182 | { | |
183 | typedef typename tools::promote_args<T1, T2>::type result_type; | |
184 | typedef typename policies::evaluation<result_type, Policy>::type value_type; | |
185 | return policies::checked_narrowing_cast<result_type, Policy>(detail::ellint_e_imp(static_cast<value_type>(phi), static_cast<value_type>(k), pol), "boost::math::ellint_2<%1%>(%1%,%1%)"); | |
186 | } | |
187 | ||
188 | }} // namespaces | |
189 | ||
190 | #endif // BOOST_MATH_ELLINT_2_HPP | |
191 |