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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_1_HPP | |
15 | #define BOOST_MATH_ELLINT_1_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/constants/constants.hpp> | |
24 | #include <boost/math/policies/error_handling.hpp> | |
25 | #include <boost/math/tools/workaround.hpp> | |
26 | #include <boost/math/special_functions/round.hpp> | |
27 | ||
28 | // Elliptic integrals (complete and incomplete) of the first kind | |
29 | // Carlson, Numerische Mathematik, vol 33, 1 (1979) | |
30 | ||
31 | namespace boost { namespace math { | |
32 | ||
33 | template <class T1, class T2, class Policy> | |
34 | typename tools::promote_args<T1, T2>::type ellint_1(T1 k, T2 phi, const Policy& pol); | |
35 | ||
36 | namespace detail{ | |
37 | ||
38 | template <typename T, typename Policy> | |
39 | T ellint_k_imp(T k, const Policy& pol); | |
40 | ||
41 | // Elliptic integral (Legendre form) of the first kind | |
42 | template <typename T, typename Policy> | |
43 | T ellint_f_imp(T phi, T k, const Policy& pol) | |
44 | { | |
45 | BOOST_MATH_STD_USING | |
46 | using namespace boost::math::tools; | |
47 | using namespace boost::math::constants; | |
48 | ||
49 | static const char* function = "boost::math::ellint_f<%1%>(%1%,%1%)"; | |
50 | BOOST_MATH_INSTRUMENT_VARIABLE(phi); | |
51 | BOOST_MATH_INSTRUMENT_VARIABLE(k); | |
52 | BOOST_MATH_INSTRUMENT_VARIABLE(function); | |
53 | ||
54 | if (abs(k) > 1) | |
55 | { | |
56 | return policies::raise_domain_error<T>(function, | |
57 | "Got k = %1%, function requires |k| <= 1", k, pol); | |
58 | } | |
59 | ||
60 | bool invert = false; | |
61 | if(phi < 0) | |
62 | { | |
63 | BOOST_MATH_INSTRUMENT_VARIABLE(phi); | |
64 | phi = fabs(phi); | |
65 | invert = true; | |
66 | } | |
67 | ||
68 | T result; | |
69 | ||
70 | if(phi >= tools::max_value<T>()) | |
71 | { | |
72 | // Need to handle infinity as a special case: | |
73 | result = policies::raise_overflow_error<T>(function, 0, pol); | |
74 | BOOST_MATH_INSTRUMENT_VARIABLE(result); | |
75 | } | |
76 | else if(phi > 1 / tools::epsilon<T>()) | |
77 | { | |
78 | // Phi is so large that phi%pi is necessarily zero (or garbage), | |
79 | // just return the second part of the duplication formula: | |
80 | result = 2 * phi * ellint_k_imp(k, pol) / constants::pi<T>(); | |
81 | BOOST_MATH_INSTRUMENT_VARIABLE(result); | |
82 | } | |
83 | else | |
84 | { | |
85 | // Carlson's algorithm works only for |phi| <= pi/2, | |
86 | // use the integrand's periodicity to normalize phi | |
87 | // | |
88 | // Xiaogang's original code used a cast to long long here | |
89 | // but that fails if T has more digits than a long long, | |
90 | // so rewritten to use fmod instead: | |
91 | // | |
92 | BOOST_MATH_INSTRUMENT_CODE("pi/2 = " << constants::pi<T>() / 2); | |
93 | T rphi = boost::math::tools::fmod_workaround(phi, T(constants::half_pi<T>())); | |
94 | BOOST_MATH_INSTRUMENT_VARIABLE(rphi); | |
95 | T m = boost::math::round((phi - rphi) / constants::half_pi<T>()); | |
96 | BOOST_MATH_INSTRUMENT_VARIABLE(m); | |
97 | int s = 1; | |
98 | if(boost::math::tools::fmod_workaround(m, T(2)) > 0.5) | |
99 | { | |
100 | m += 1; | |
101 | s = -1; | |
102 | rphi = constants::half_pi<T>() - rphi; | |
103 | BOOST_MATH_INSTRUMENT_VARIABLE(rphi); | |
104 | } | |
105 | T sinp = sin(rphi); | |
106 | sinp *= sinp; | |
107 | T cosp = cos(rphi); | |
108 | cosp *= cosp; | |
109 | BOOST_MATH_INSTRUMENT_VARIABLE(sinp); | |
110 | BOOST_MATH_INSTRUMENT_VARIABLE(cosp); | |
111 | if(sinp > tools::min_value<T>()) | |
112 | { | |
113 | // | |
114 | // Use http://dlmf.nist.gov/19.25#E5, note that | |
115 | // c-1 simplifies to cot^2(rphi) which avoid cancellation: | |
116 | // | |
117 | T c = 1 / sinp; | |
118 | result = rphi == 0 ? static_cast<T>(0) : static_cast<T>(s * ellint_rf_imp(T(cosp / sinp), T(c - k * k), c, pol)); | |
119 | } | |
120 | else | |
121 | result = s * sin(rphi); | |
122 | BOOST_MATH_INSTRUMENT_VARIABLE(result); | |
123 | if(m != 0) | |
124 | { | |
125 | result += m * ellint_k_imp(k, pol); | |
126 | BOOST_MATH_INSTRUMENT_VARIABLE(result); | |
127 | } | |
128 | } | |
129 | return invert ? T(-result) : result; | |
130 | } | |
131 | ||
132 | // Complete elliptic integral (Legendre form) of the first kind | |
133 | template <typename T, typename Policy> | |
134 | T ellint_k_imp(T k, const Policy& pol) | |
135 | { | |
136 | BOOST_MATH_STD_USING | |
137 | using namespace boost::math::tools; | |
138 | ||
139 | static const char* function = "boost::math::ellint_k<%1%>(%1%)"; | |
140 | ||
141 | if (abs(k) > 1) | |
142 | { | |
143 | return policies::raise_domain_error<T>(function, | |
144 | "Got k = %1%, function requires |k| <= 1", k, pol); | |
145 | } | |
146 | if (abs(k) == 1) | |
147 | { | |
148 | return policies::raise_overflow_error<T>(function, 0, pol); | |
149 | } | |
150 | ||
151 | T x = 0; | |
152 | T y = 1 - k * k; | |
153 | T z = 1; | |
154 | T value = ellint_rf_imp(x, y, z, pol); | |
155 | ||
156 | return value; | |
157 | } | |
158 | ||
159 | template <typename T, typename Policy> | |
160 | inline typename tools::promote_args<T>::type ellint_1(T k, const Policy& pol, const mpl::true_&) | |
161 | { | |
162 | typedef typename tools::promote_args<T>::type result_type; | |
163 | typedef typename policies::evaluation<result_type, Policy>::type value_type; | |
164 | return policies::checked_narrowing_cast<result_type, Policy>(detail::ellint_k_imp(static_cast<value_type>(k), pol), "boost::math::ellint_1<%1%>(%1%)"); | |
165 | } | |
166 | ||
167 | template <class T1, class T2> | |
168 | inline typename tools::promote_args<T1, T2>::type ellint_1(T1 k, T2 phi, const mpl::false_&) | |
169 | { | |
170 | return boost::math::ellint_1(k, phi, policies::policy<>()); | |
171 | } | |
172 | ||
173 | } | |
174 | ||
175 | // Complete elliptic integral (Legendre form) of the first kind | |
176 | template <typename T> | |
177 | inline typename tools::promote_args<T>::type ellint_1(T k) | |
178 | { | |
179 | return ellint_1(k, policies::policy<>()); | |
180 | } | |
181 | ||
182 | // Elliptic integral (Legendre form) of the first kind | |
183 | template <class T1, class T2, class Policy> | |
184 | inline typename tools::promote_args<T1, T2>::type ellint_1(T1 k, T2 phi, const Policy& pol) | |
185 | { | |
186 | typedef typename tools::promote_args<T1, T2>::type result_type; | |
187 | typedef typename policies::evaluation<result_type, Policy>::type value_type; | |
188 | return policies::checked_narrowing_cast<result_type, Policy>(detail::ellint_f_imp(static_cast<value_type>(phi), static_cast<value_type>(k), pol), "boost::math::ellint_1<%1%>(%1%,%1%)"); | |
189 | } | |
190 | ||
191 | template <class T1, class T2> | |
192 | inline typename tools::promote_args<T1, T2>::type ellint_1(T1 k, T2 phi) | |
193 | { | |
194 | typedef typename policies::is_policy<T2>::type tag_type; | |
195 | return detail::ellint_1(k, phi, tag_type()); | |
196 | } | |
197 | ||
198 | }} // namespaces | |
199 | ||
200 | #endif // BOOST_MATH_ELLINT_1_HPP | |
201 |