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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_Y0_HPP | |
7 | #define BOOST_MATH_BESSEL_Y0_HPP | |
8 | ||
9 | #ifdef _MSC_VER | |
10 | #pragma once | |
11 | #pragma warning(push) | |
12 | #pragma warning(disable:4702) // Unreachable code (release mode only warning) | |
13 | #endif | |
14 | ||
15 | #include <boost/math/special_functions/detail/bessel_j0.hpp> | |
16 | #include <boost/math/constants/constants.hpp> | |
17 | #include <boost/math/tools/rational.hpp> | |
18 | #include <boost/math/tools/big_constant.hpp> | |
19 | #include <boost/math/policies/error_handling.hpp> | |
20 | #include <boost/assert.hpp> | |
21 | ||
22 | // Bessel function of the second kind of order zero | |
23 | // x <= 8, minimax rational approximations on root-bracketing intervals | |
24 | // x > 8, Hankel asymptotic expansion in Hart, Computer Approximations, 1968 | |
25 | ||
26 | namespace boost { namespace math { namespace detail{ | |
27 | ||
28 | template <typename T, typename Policy> | |
29 | T bessel_y0(T x, const Policy&); | |
30 | ||
31 | template <class T, class Policy> | |
32 | struct bessel_y0_initializer | |
33 | { | |
34 | struct init | |
35 | { | |
36 | init() | |
37 | { | |
38 | do_init(); | |
39 | } | |
40 | static void do_init() | |
41 | { | |
42 | bessel_y0(T(1), Policy()); | |
43 | } | |
44 | void force_instantiate()const{} | |
45 | }; | |
46 | static const init initializer; | |
47 | static void force_instantiate() | |
48 | { | |
49 | initializer.force_instantiate(); | |
50 | } | |
51 | }; | |
52 | ||
53 | template <class T, class Policy> | |
54 | const typename bessel_y0_initializer<T, Policy>::init bessel_y0_initializer<T, Policy>::initializer; | |
55 | ||
56 | template <typename T, typename Policy> | |
57 | T bessel_y0(T x, const Policy& pol) | |
58 | { | |
59 | bessel_y0_initializer<T, Policy>::force_instantiate(); | |
60 | ||
61 | static const T P1[] = { | |
62 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0723538782003176831e+11)), | |
63 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.3716255451260504098e+09)), | |
64 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.0422274357376619816e+08)), | |
65 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.1287548474401797963e+06)), | |
66 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0102532948020907590e+04)), | |
67 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8402381979244993524e+01)), | |
68 | }; | |
69 | static const T Q1[] = { | |
70 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.8873865738997033405e+11)), | |
71 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.1617187777290363573e+09)), | |
72 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.5662956624278251596e+07)), | |
73 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.3889393209447253406e+05)), | |
74 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6475986689240190091e+02)), | |
75 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)), | |
76 | }; | |
77 | static const T P2[] = { | |
78 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.2213976967566192242e+13)), | |
79 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.5107435206722644429e+11)), | |
80 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.3600098638603061642e+10)), | |
81 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.9590439394619619534e+08)), | |
82 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.6905288611678631510e+06)), | |
83 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.4566865832663635920e+04)), | |
84 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7427031242901594547e+01)), | |
85 | }; | |
86 | static const T Q2[] = { | |
87 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.3386146580707264428e+14)), | |
88 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.4266824419412347550e+12)), | |
89 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4015103849971240096e+10)), | |
90 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3960202770986831075e+08)), | |
91 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0669982352539552018e+05)), | |
92 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.3030857612070288823e+02)), | |
93 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)), | |
94 | }; | |
95 | static const T P3[] = { | |
96 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.0728726905150210443e+15)), | |
97 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.7016641869173237784e+14)), | |
98 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2829912364088687306e+11)), | |
99 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.9363051266772083678e+11)), | |
100 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1958827170518100757e+09)), | |
101 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0085539923498211426e+07)), | |
102 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1363534169313901632e+04)), | |
103 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.7439661319197499338e+01)), | |
104 | }; | |
105 | static const T Q3[] = { | |
106 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4563724628846457519e+17)), | |
107 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.9272425569640309819e+15)), | |
108 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2598377924042897629e+13)), | |
109 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.6926121104209825246e+10)), | |
110 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.4727219475672302327e+08)), | |
111 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.3924739209768057030e+05)), | |
112 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.7903362168128450017e+02)), | |
113 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)), | |
114 | }; | |
115 | static const T PC[] = { | |
116 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684302e+04)), | |
117 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1345386639580765797e+04)), | |
118 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1170523380864944322e+04)), | |
119 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4806486443249270347e+03)), | |
120 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5376201909008354296e+02)), | |
121 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.8961548424210455236e-01)), | |
122 | }; | |
123 | static const T QC[] = { | |
124 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684318e+04)), | |
125 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1370412495510416640e+04)), | |
126 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1215350561880115730e+04)), | |
127 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.5028735138235608207e+03)), | |
128 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5711159858080893649e+02)), | |
129 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)), | |
130 | }; | |
131 | static const T PS[] = { | |
132 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.9226600200800094098e+01)), | |
133 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8591953644342993800e+02)), | |
134 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.1183429920482737611e+02)), | |
135 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.2300261666214198472e+01)), | |
136 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2441026745835638459e+00)), | |
137 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.8033303048680751817e-03)), | |
138 | }; | |
139 | static const T QS[] = { | |
140 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.7105024128512061905e+03)), | |
141 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1951131543434613647e+04)), | |
142 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.2642780169211018836e+03)), | |
143 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4887231232283756582e+03)), | |
144 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.0593769594993125859e+01)), | |
145 | static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)), | |
146 | }; | |
147 | static const T x1 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.9357696627916752158e-01)), | |
148 | x2 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.9576784193148578684e+00)), | |
149 | x3 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.0860510603017726976e+00)), | |
150 | x11 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.280e+02)), | |
151 | x12 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.9519662791675215849e-03)), | |
152 | x21 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0130e+03)), | |
153 | x22 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.4716931485786837568e-04)), | |
154 | x31 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8140e+03)), | |
155 | x32 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1356030177269762362e-04)) | |
156 | ; | |
157 | T value, factor, r, rc, rs; | |
158 | ||
159 | BOOST_MATH_STD_USING | |
160 | using namespace boost::math::tools; | |
161 | using namespace boost::math::constants; | |
162 | ||
163 | static const char* function = "boost::math::bessel_y0<%1%>(%1%,%1%)"; | |
164 | ||
165 | if (x < 0) | |
166 | { | |
167 | return policies::raise_domain_error<T>(function, | |
168 | "Got x = %1% but x must be non-negative, complex result not supported.", x, pol); | |
169 | } | |
170 | if (x == 0) | |
171 | { | |
172 | return -policies::raise_overflow_error<T>(function, 0, pol); | |
173 | } | |
174 | if (x <= 3) // x in (0, 3] | |
175 | { | |
176 | T y = x * x; | |
177 | T z = 2 * log(x/x1) * bessel_j0(x) / pi<T>(); | |
178 | r = evaluate_rational(P1, Q1, y); | |
179 | factor = (x + x1) * ((x - x11/256) - x12); | |
180 | value = z + factor * r; | |
181 | } | |
182 | else if (x <= 5.5f) // x in (3, 5.5] | |
183 | { | |
184 | T y = x * x; | |
185 | T z = 2 * log(x/x2) * bessel_j0(x) / pi<T>(); | |
186 | r = evaluate_rational(P2, Q2, y); | |
187 | factor = (x + x2) * ((x - x21/256) - x22); | |
188 | value = z + factor * r; | |
189 | } | |
190 | else if (x <= 8) // x in (5.5, 8] | |
191 | { | |
192 | T y = x * x; | |
193 | T z = 2 * log(x/x3) * bessel_j0(x) / pi<T>(); | |
194 | r = evaluate_rational(P3, Q3, y); | |
195 | factor = (x + x3) * ((x - x31/256) - x32); | |
196 | value = z + factor * r; | |
197 | } | |
198 | else // x in (8, \infty) | |
199 | { | |
200 | T y = 8 / x; | |
201 | T y2 = y * y; | |
202 | rc = evaluate_rational(PC, QC, y2); | |
203 | rs = evaluate_rational(PS, QS, y2); | |
204 | factor = constants::one_div_root_pi<T>() / sqrt(x); | |
205 | // | |
206 | // The following code is really just: | |
207 | // | |
208 | // T z = x - 0.25f * pi<T>(); | |
209 | // value = factor * (rc * sin(z) + y * rs * cos(z)); | |
210 | // | |
211 | // But using the sin/cos addition formulae and constant values for | |
212 | // sin/cos of PI/4 which then cancel part of the "factor" term as they're all | |
213 | // 1 / sqrt(2): | |
214 | // | |
215 | T sx = sin(x); | |
216 | T cx = cos(x); | |
217 | value = factor * (rc * (sx - cx) + y * rs * (cx + sx)); | |
218 | } | |
219 | ||
220 | return value; | |
221 | } | |
222 | ||
223 | }}} // namespaces | |
224 | ||
225 | #ifdef _MSC_VER | |
226 | #pragma warning(pop) | |
227 | #endif | |
228 | ||
229 | #endif // BOOST_MATH_BESSEL_Y0_HPP | |
230 |