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#if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128)
23//
24// This is the only way we can avoid
25// warning: non-standard suffix on floating constant [-Wpedantic]
26// when building with -Wall -pedantic. Neither __extension__
27// nor #pragma diagnostic ignored work :(
28//
29#pragma GCC system_header
30#endif
31
32// Bessel function of the second kind of order zero
33// x <= 8, minimax rational approximations on root-bracketing intervals
34// x > 8, Hankel asymptotic expansion in Hart, Computer Approximations, 1968
35
36namespace boost { namespace math { namespace detail{
37
38template <typename T, typename Policy>
39T bessel_y0(T x, const Policy&);
40
41template <class T, class Policy>
42struct bessel_y0_initializer
43{
44 struct init
45 {
46 init()
47 {
48 do_init();
49 }
50 static void do_init()
51 {
52 bessel_y0(T(1), Policy());
53 }
54 void force_instantiate()const{}
55 };
56 static const init initializer;
57 static void force_instantiate()
58 {
59 initializer.force_instantiate();
60 }
61};
62
63template <class T, class Policy>
64const typename bessel_y0_initializer<T, Policy>::init bessel_y0_initializer<T, Policy>::initializer;
65
66template <typename T, typename Policy>
67T bessel_y0(T x, const Policy& pol)
68{
69 bessel_y0_initializer<T, Policy>::force_instantiate();
70
71 static const T P1[] = {
72 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0723538782003176831e+11)),
73 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.3716255451260504098e+09)),
74 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.0422274357376619816e+08)),
75 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.1287548474401797963e+06)),
76 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0102532948020907590e+04)),
77 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8402381979244993524e+01)),
78 };
79 static const T Q1[] = {
80 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.8873865738997033405e+11)),
81 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.1617187777290363573e+09)),
82 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.5662956624278251596e+07)),
83 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.3889393209447253406e+05)),
84 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6475986689240190091e+02)),
85 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
86 };
87 static const T P2[] = {
88 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.2213976967566192242e+13)),
89 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.5107435206722644429e+11)),
90 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.3600098638603061642e+10)),
91 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.9590439394619619534e+08)),
92 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.6905288611678631510e+06)),
93 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.4566865832663635920e+04)),
94 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7427031242901594547e+01)),
95 };
96 static const T Q2[] = {
97 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.3386146580707264428e+14)),
98 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.4266824419412347550e+12)),
99 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4015103849971240096e+10)),
100 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3960202770986831075e+08)),
101 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0669982352539552018e+05)),
102 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.3030857612070288823e+02)),
103 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
104 };
105 static const T P3[] = {
106 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.0728726905150210443e+15)),
107 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.7016641869173237784e+14)),
108 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2829912364088687306e+11)),
109 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.9363051266772083678e+11)),
110 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1958827170518100757e+09)),
111 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0085539923498211426e+07)),
112 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1363534169313901632e+04)),
113 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.7439661319197499338e+01)),
114 };
115 static const T Q3[] = {
116 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4563724628846457519e+17)),
117 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.9272425569640309819e+15)),
118 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2598377924042897629e+13)),
119 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.6926121104209825246e+10)),
120 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.4727219475672302327e+08)),
121 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.3924739209768057030e+05)),
122 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.7903362168128450017e+02)),
123 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
124 };
125 static const T PC[] = {
126 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684302e+04)),
127 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1345386639580765797e+04)),
128 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1170523380864944322e+04)),
129 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4806486443249270347e+03)),
130 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5376201909008354296e+02)),
131 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.8961548424210455236e-01)),
132 };
133 static const T QC[] = {
134 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684318e+04)),
135 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1370412495510416640e+04)),
136 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1215350561880115730e+04)),
137 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.5028735138235608207e+03)),
138 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5711159858080893649e+02)),
139 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
140 };
141 static const T PS[] = {
142 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.9226600200800094098e+01)),
143 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8591953644342993800e+02)),
144 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.1183429920482737611e+02)),
145 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.2300261666214198472e+01)),
146 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2441026745835638459e+00)),
147 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.8033303048680751817e-03)),
148 };
149 static const T QS[] = {
150 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.7105024128512061905e+03)),
151 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1951131543434613647e+04)),
152 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.2642780169211018836e+03)),
153 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4887231232283756582e+03)),
154 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.0593769594993125859e+01)),
155 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
156 };
157 static const T x1 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.9357696627916752158e-01)),
158 x2 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.9576784193148578684e+00)),
159 x3 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.0860510603017726976e+00)),
160 x11 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.280e+02)),
161 x12 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.9519662791675215849e-03)),
162 x21 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0130e+03)),
163 x22 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.4716931485786837568e-04)),
164 x31 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8140e+03)),
165 x32 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1356030177269762362e-04))
166 ;
167 T value, factor, r, rc, rs;
168
169 BOOST_MATH_STD_USING
170 using namespace boost::math::tools;
171 using namespace boost::math::constants;
172
173 static const char* function = "boost::math::bessel_y0<%1%>(%1%,%1%)";
174
175 if (x < 0)
176 {
177 return policies::raise_domain_error<T>(function,
178 "Got x = %1% but x must be non-negative, complex result not supported.", x, pol);
179 }
180 if (x == 0)
181 {
182 return -policies::raise_overflow_error<T>(function, 0, pol);
183 }
184 if (x <= 3) // x in (0, 3]
185 {
186 T y = x * x;
187 T z = 2 * log(x/x1) * bessel_j0(x) / pi<T>();
188 r = evaluate_rational(P1, Q1, y);
189 factor = (x + x1) * ((x - x11/256) - x12);
190 value = z + factor * r;
191 }
192 else if (x <= 5.5f) // x in (3, 5.5]
193 {
194 T y = x * x;
195 T z = 2 * log(x/x2) * bessel_j0(x) / pi<T>();
196 r = evaluate_rational(P2, Q2, y);
197 factor = (x + x2) * ((x - x21/256) - x22);
198 value = z + factor * r;
199 }
200 else if (x <= 8) // x in (5.5, 8]
201 {
202 T y = x * x;
203 T z = 2 * log(x/x3) * bessel_j0(x) / pi<T>();
204 r = evaluate_rational(P3, Q3, y);
205 factor = (x + x3) * ((x - x31/256) - x32);
206 value = z + factor * r;
207 }
208 else // x in (8, \infty)
209 {
210 T y = 8 / x;
211 T y2 = y * y;
212 rc = evaluate_rational(PC, QC, y2);
213 rs = evaluate_rational(PS, QS, y2);
214 factor = constants::one_div_root_pi<T>() / sqrt(x);
215 //
216 // The following code is really just:
217 //
218 // T z = x - 0.25f * pi<T>();
219 // value = factor * (rc * sin(z) + y * rs * cos(z));
220 //
221 // But using the sin/cos addition formulae and constant values for
222 // sin/cos of PI/4 which then cancel part of the "factor" term as they're all
223 // 1 / sqrt(2):
224 //
225 T sx = sin(x);
226 T cx = cos(x);
227 value = factor * (rc * (sx - cx) + y * rs * (cx + sx));
228 }
229
230 return value;
231}
232
233}}} // namespaces
234
235#ifdef _MSC_VER
236#pragma warning(pop)
237#endif
238
239#endif // BOOST_MATH_BESSEL_Y0_HPP
240
241

source code of include/boost/math/special_functions/detail/bessel_y0.hpp