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1 | // Copyright (c) 2006 Xiaogang Zhang |
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 | #ifndef BOOST_MATH_BESSEL_J0_HPP | |
7 | #define BOOST_MATH_BESSEL_J0_HPP | |
8 | ||
9 | #ifdef _MSC_VER | |
10 | #pragma once | |
11 | #endif | |
12 | ||
13 | #include <boost/math/constants/constants.hpp> | |
14 | #include <boost/math/tools/rational.hpp> | |
15 | #include <boost/math/tools/big_constant.hpp> | |
16 | #include <boost/assert.hpp> | |
17 | ||
18 | // Bessel function of the first kind of order zero | |
19 | // x <= 8, minimax rational approximations on root-bracketing intervals | |
20 | // x > 8, Hankel asymptotic expansion in Hart, Computer Approximations, 1968 | |
21 | ||
22 | namespace boost { namespace math { namespace detail{ | |
23 | ||
24 | template <typename T> | |
25 | T bessel_j0(T x); | |
26 | ||
27 | template <class T> | |
28 | struct bessel_j0_initializer | |
29 | { | |
30 | struct init | |
31 | { | |
32 | init() | |
33 | { | |
34 | do_init(); | |
35 | } | |
36 | static void do_init() | |
37 | { | |
38 | bessel_j0(T(1)); | |
39 | } | |
40 | void force_instantiate()const{} | |
41 | }; | |
42 | static const init initializer; | |
43 | static void force_instantiate() | |
44 | { | |
45 | initializer.force_instantiate(); | |
46 | } | |
47 | }; | |
48 | ||
49 | template <class T> | |
50 | const typename bessel_j0_initializer<T>::init bessel_j0_initializer<T>::initializer; | |
51 | ||
52 | template <typename T> | |
53 | T bessel_j0(T x) | |
54 | { | |
55 | bessel_j0_initializer<T>::force_instantiate(); | |
56 | ||
57 | static const T P1[] = { | |
58 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -4.1298668500990866786e+11)), | |
59 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.7282507878605942706e+10)), | |
60 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.2140700423540120665e+08)), | |
61 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6302997904833794242e+06)), | |
62 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.6629814655107086448e+04)), | |
63 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0344222815443188943e+02)), | |
64 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2117036164593528341e-01)) | |
65 | }; | |
66 | static const T Q1[] = { | |
67 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.3883787996332290397e+12)), | |
68 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.6328198300859648632e+10)), | |
69 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3985097372263433271e+08)), | |
70 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.5612696224219938200e+05)), | |
71 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.3614022392337710626e+02)), | |
72 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)), | |
73 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.0)) | |
74 | }; | |
75 | static const T P2[] = { | |
76 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8319397969392084011e+03)), | |
77 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2254078161378989535e+04)), | |
78 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -7.2879702464464618998e+03)), | |
79 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0341910641583726701e+04)), | |
80 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1725046279757103576e+04)), | |
81 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.4176707025325087628e+03)), | |
82 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.4321196680624245801e+02)), | |
83 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.8591703355916499363e+01)) | |
84 | }; | |
85 | static const T Q2[] = { | |
86 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.5783478026152301072e+05)), | |
87 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.4599102262586308984e+05)), | |
88 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.4055062591169562211e+04)), | |
89 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8680990008359188352e+04)), | |
90 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.9458766545509337327e+03)), | |
91 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.3307310774649071172e+02)), | |
92 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.5258076240801555057e+01)), | |
93 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)) | |
94 | }; | |
95 | static const T PC[] = { | |
96 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684302e+04)), | |
97 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1345386639580765797e+04)), | |
98 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1170523380864944322e+04)), | |
99 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4806486443249270347e+03)), | |
100 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5376201909008354296e+02)), | |
101 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.8961548424210455236e-01)) | |
102 | }; | |
103 | static const T QC[] = { | |
104 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684318e+04)), | |
105 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1370412495510416640e+04)), | |
106 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1215350561880115730e+04)), | |
107 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.5028735138235608207e+03)), | |
108 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5711159858080893649e+02)), | |
109 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)) | |
110 | }; | |
111 | static const T PS[] = { | |
112 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.9226600200800094098e+01)), | |
113 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8591953644342993800e+02)), | |
114 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.1183429920482737611e+02)), | |
115 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.2300261666214198472e+01)), | |
116 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2441026745835638459e+00)), | |
117 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.8033303048680751817e-03)) | |
118 | }; | |
119 | static const T QS[] = { | |
120 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.7105024128512061905e+03)), | |
121 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1951131543434613647e+04)), | |
122 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.2642780169211018836e+03)), | |
123 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4887231232283756582e+03)), | |
124 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.0593769594993125859e+01)), | |
125 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)) | |
126 | }; | |
127 | static const T x1 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.4048255576957727686e+00)), | |
128 | x2 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.5200781102863106496e+00)), | |
129 | x11 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.160e+02)), | |
130 | x12 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.42444230422723137837e-03)), | |
131 | x21 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4130e+03)), | |
132 | x22 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.46860286310649596604e-04)); | |
133 | ||
134 | T value, factor, r, rc, rs; | |
135 | ||
136 | BOOST_MATH_STD_USING | |
137 | using namespace boost::math::tools; | |
138 | using namespace boost::math::constants; | |
139 | ||
140 | if (x < 0) | |
141 | { | |
142 | x = -x; // even function | |
143 | } | |
144 | if (x == 0) | |
145 | { | |
146 | return static_cast<T>(1); | |
147 | } | |
148 | if (x <= 4) // x in (0, 4] | |
149 | { | |
150 | T y = x * x; | |
151 | BOOST_ASSERT(sizeof(P1) == sizeof(Q1)); | |
152 | r = evaluate_rational(P1, Q1, y); | |
153 | factor = (x + x1) * ((x - x11/256) - x12); | |
154 | value = factor * r; | |
155 | } | |
156 | else if (x <= 8.0) // x in (4, 8] | |
157 | { | |
158 | T y = 1 - (x * x)/64; | |
159 | BOOST_ASSERT(sizeof(P2) == sizeof(Q2)); | |
160 | r = evaluate_rational(P2, Q2, y); | |
161 | factor = (x + x2) * ((x - x21/256) - x22); | |
162 | value = factor * r; | |
163 | } | |
164 | else // x in (8, \infty) | |
165 | { | |
166 | T y = 8 / x; | |
167 | T y2 = y * y; | |
168 | BOOST_ASSERT(sizeof(PC) == sizeof(QC)); | |
169 | BOOST_ASSERT(sizeof(PS) == sizeof(QS)); | |
170 | rc = evaluate_rational(PC, QC, y2); | |
171 | rs = evaluate_rational(PS, QS, y2); | |
172 | factor = constants::one_div_root_pi<T>() / sqrt(x); | |
173 | // | |
174 | // What follows is really just: | |
175 | // | |
176 | // T z = x - pi/4; | |
177 | // value = factor * (rc * cos(z) - y * rs * sin(z)); | |
178 | // | |
179 | // But using the addition formulae for sin and cos, plus | |
180 | // the special values for sin/cos of pi/4. | |
181 | // | |
182 | T sx = sin(x); | |
183 | T cx = cos(x); | |
184 | value = factor * (rc * (cx + sx) - y * rs * (sx - cx)); | |
185 | } | |
186 | ||
187 | return value; | |
188 | } | |
189 | ||
190 | }}} // namespaces | |
191 | ||
192 | #endif // BOOST_MATH_BESSEL_J0_HPP | |
193 |