1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
|
// Copyright (c) 2006 Xiaogang Zhang
// Copyright (c) 2006 John Maddock
// Use, modification and distribution are subject to the
// Boost Software License, Version 1.0. (See accompanying file
// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
//
// History:
// XZ wrote the original of this file as part of the Google
// Summer of Code 2006. JM modified it to fit into the
// Boost.Math conceptual framework better, and to ensure
// that the code continues to work no matter how many digits
// type T has.
#ifndef BOOST_MATH_ELLINT_1_HPP
#define BOOST_MATH_ELLINT_1_HPP
#ifdef _MSC_VER
#pragma once
#endif
#include <boost/math/special_functions/math_fwd.hpp>
#include <boost/math/special_functions/ellint_rf.hpp>
#include <boost/math/constants/constants.hpp>
#include <boost/math/policies/error_handling.hpp>
#include <boost/math/tools/workaround.hpp>
#include <boost/math/special_functions/round.hpp>
// Elliptic integrals (complete and incomplete) of the first kind
// Carlson, Numerische Mathematik, vol 33, 1 (1979)
namespace boost { namespace math {
template <class T1, class T2, class Policy>
typename tools::promote_args<T1, T2>::type ellint_1(T1 k, T2 phi, const Policy& pol);
namespace detail{
template <typename T, typename Policy>
T ellint_k_imp(T k, const Policy& pol);
// Elliptic integral (Legendre form) of the first kind
template <typename T, typename Policy>
T ellint_f_imp(T phi, T k, const Policy& pol)
{
BOOST_MATH_STD_USING
using namespace boost::math::tools;
using namespace boost::math::constants;
static const char* function = "boost::math::ellint_f<%1%>(%1%,%1%)";
BOOST_MATH_INSTRUMENT_VARIABLE(phi);
BOOST_MATH_INSTRUMENT_VARIABLE(k);
BOOST_MATH_INSTRUMENT_VARIABLE(function);
if (abs(k) > 1)
{
return policies::raise_domain_error<T>(function,
"Got k = %1%, function requires |k| <= 1", k, pol);
}
bool invert = false;
if(phi < 0)
{
BOOST_MATH_INSTRUMENT_VARIABLE(phi);
phi = fabs(phi);
invert = true;
}
T result;
if(phi >= tools::max_value<T>())
{
// Need to handle infinity as a special case:
result = policies::raise_overflow_error<T>(function, 0, pol);
BOOST_MATH_INSTRUMENT_VARIABLE(result);
}
else if(phi > 1 / tools::epsilon<T>())
{
// Phi is so large that phi%pi is necessarily zero (or garbage),
// just return the second part of the duplication formula:
result = 2 * phi * ellint_k_imp(k, pol) / constants::pi<T>();
BOOST_MATH_INSTRUMENT_VARIABLE(result);
}
else
{
// Carlson's algorithm works only for |phi| <= pi/2,
// use the integrand's periodicity to normalize phi
//
// Xiaogang's original code used a cast to long long here
// but that fails if T has more digits than a long long,
// so rewritten to use fmod instead:
//
BOOST_MATH_INSTRUMENT_CODE("pi/2 = " << constants::pi<T>() / 2);
T rphi = boost::math::tools::fmod_workaround(phi, T(constants::half_pi<T>()));
BOOST_MATH_INSTRUMENT_VARIABLE(rphi);
T m = boost::math::round((phi - rphi) / constants::half_pi<T>());
BOOST_MATH_INSTRUMENT_VARIABLE(m);
int s = 1;
if(boost::math::tools::fmod_workaround(m, T(2)) > 0.5)
{
m += 1;
s = -1;
rphi = constants::half_pi<T>() - rphi;
BOOST_MATH_INSTRUMENT_VARIABLE(rphi);
}
T sinp = sin(rphi);
T cosp = cos(rphi);
BOOST_MATH_INSTRUMENT_VARIABLE(sinp);
BOOST_MATH_INSTRUMENT_VARIABLE(cosp);
result = s * sinp * ellint_rf_imp(T(cosp * cosp), T(1 - k * k * sinp * sinp), T(1), pol);
BOOST_MATH_INSTRUMENT_VARIABLE(result);
if(m != 0)
{
result += m * ellint_k_imp(k, pol);
BOOST_MATH_INSTRUMENT_VARIABLE(result);
}
}
return invert ? T(-result) : result;
}
// Complete elliptic integral (Legendre form) of the first kind
template <typename T, typename Policy>
T ellint_k_imp(T k, const Policy& pol)
{
BOOST_MATH_STD_USING
using namespace boost::math::tools;
static const char* function = "boost::math::ellint_k<%1%>(%1%)";
if (abs(k) > 1)
{
return policies::raise_domain_error<T>(function,
"Got k = %1%, function requires |k| <= 1", k, pol);
}
if (abs(k) == 1)
{
return policies::raise_overflow_error<T>(function, 0, pol);
}
T x = 0;
T y = 1 - k * k;
T z = 1;
T value = ellint_rf_imp(x, y, z, pol);
return value;
}
template <typename T, typename Policy>
inline typename tools::promote_args<T>::type ellint_1(T k, const Policy& pol, const mpl::true_&)
{
typedef typename tools::promote_args<T>::type result_type;
typedef typename policies::evaluation<result_type, Policy>::type value_type;
return policies::checked_narrowing_cast<result_type, Policy>(detail::ellint_k_imp(static_cast<value_type>(k), pol), "boost::math::ellint_1<%1%>(%1%)");
}
template <class T1, class T2>
inline typename tools::promote_args<T1, T2>::type ellint_1(T1 k, T2 phi, const mpl::false_&)
{
return boost::math::ellint_1(k, phi, policies::policy<>());
}
}
// Complete elliptic integral (Legendre form) of the first kind
template <typename T>
inline typename tools::promote_args<T>::type ellint_1(T k)
{
return ellint_1(k, policies::policy<>());
}
// Elliptic integral (Legendre form) of the first kind
template <class T1, class T2, class Policy>
inline typename tools::promote_args<T1, T2>::type ellint_1(T1 k, T2 phi, const Policy& pol)
{
typedef typename tools::promote_args<T1, T2>::type result_type;
typedef typename policies::evaluation<result_type, Policy>::type value_type;
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%)");
}
template <class T1, class T2>
inline typename tools::promote_args<T1, T2>::type ellint_1(T1 k, T2 phi)
{
typedef typename policies::is_policy<T2>::type tag_type;
return detail::ellint_1(k, phi, tag_type());
}
}} // namespaces
#endif // BOOST_MATH_ELLINT_1_HPP
|
