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/math/tools/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 <typename T, typename Policy>
42T bessel_y0(T x, const Policy&)
43{
44 static const T P1[] = {
45 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0723538782003176831e+11)),
46 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.3716255451260504098e+09)),
47 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.0422274357376619816e+08)),
48 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.1287548474401797963e+06)),
49 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0102532948020907590e+04)),
50 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8402381979244993524e+01)),
51 };
52 static const T Q1[] = {
53 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.8873865738997033405e+11)),
54 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.1617187777290363573e+09)),
55 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.5662956624278251596e+07)),
56 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.3889393209447253406e+05)),
57 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6475986689240190091e+02)),
58 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
59 };
60 static const T P2[] = {
61 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.2213976967566192242e+13)),
62 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.5107435206722644429e+11)),
63 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.3600098638603061642e+10)),
64 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.9590439394619619534e+08)),
65 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.6905288611678631510e+06)),
66 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.4566865832663635920e+04)),
67 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7427031242901594547e+01)),
68 };
69 static const T Q2[] = {
70 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.3386146580707264428e+14)),
71 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.4266824419412347550e+12)),
72 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4015103849971240096e+10)),
73 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3960202770986831075e+08)),
74 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0669982352539552018e+05)),
75 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.3030857612070288823e+02)),
76 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
77 };
78 static const T P3[] = {
79 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.0728726905150210443e+15)),
80 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.7016641869173237784e+14)),
81 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2829912364088687306e+11)),
82 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.9363051266772083678e+11)),
83 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1958827170518100757e+09)),
84 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0085539923498211426e+07)),
85 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1363534169313901632e+04)),
86 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.7439661319197499338e+01)),
87 };
88 static const T Q3[] = {
89 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4563724628846457519e+17)),
90 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.9272425569640309819e+15)),
91 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2598377924042897629e+13)),
92 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.6926121104209825246e+10)),
93 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.4727219475672302327e+08)),
94 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.3924739209768057030e+05)),
95 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.7903362168128450017e+02)),
96 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
97 };
98 static const T PC[] = {
99 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684302e+04)),
100 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1345386639580765797e+04)),
101 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1170523380864944322e+04)),
102 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4806486443249270347e+03)),
103 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5376201909008354296e+02)),
104 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.8961548424210455236e-01)),
105 };
106 static const T QC[] = {
107 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684318e+04)),
108 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1370412495510416640e+04)),
109 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1215350561880115730e+04)),
110 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.5028735138235608207e+03)),
111 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5711159858080893649e+02)),
112 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
113 };
114 static const T PS[] = {
115 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.9226600200800094098e+01)),
116 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8591953644342993800e+02)),
117 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.1183429920482737611e+02)),
118 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.2300261666214198472e+01)),
119 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2441026745835638459e+00)),
120 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.8033303048680751817e-03)),
121 };
122 static const T QS[] = {
123 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.7105024128512061905e+03)),
124 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1951131543434613647e+04)),
125 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.2642780169211018836e+03)),
126 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4887231232283756582e+03)),
127 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.0593769594993125859e+01)),
128 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
129 };
130 static const T x1 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.9357696627916752158e-01)),
131 x2 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.9576784193148578684e+00)),
132 x3 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.0860510603017726976e+00)),
133 x11 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.280e+02)),
134 x12 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.9519662791675215849e-03)),
135 x21 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0130e+03)),
136 x22 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.4716931485786837568e-04)),
137 x31 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8140e+03)),
138 x32 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1356030177269762362e-04))
139 ;
140 T value, factor, r, rc, rs;
141
142 BOOST_MATH_STD_USING
143 using namespace boost::math::tools;
144 using namespace boost::math::constants;
145
146 BOOST_MATH_ASSERT(x > 0);
147
148 if (x <= 3) // x in (0, 3]
149 {
150 T y = x * x;
151 T z = 2 * log(x/x1) * bessel_j0(x) / pi<T>();
152 r = evaluate_rational(P1, Q1, y);
153 factor = (x + x1) * ((x - x11/256) - x12);
154 value = z + factor * r;
155 }
156 else if (x <= 5.5f) // x in (3, 5.5]
157 {
158 T y = x * x;
159 T z = 2 * log(x/x2) * bessel_j0(x) / pi<T>();
160 r = evaluate_rational(P2, Q2, y);
161 factor = (x + x2) * ((x - x21/256) - x22);
162 value = z + factor * r;
163 }
164 else if (x <= 8) // x in (5.5, 8]
165 {
166 T y = x * x;
167 T z = 2 * log(x/x3) * bessel_j0(x) / pi<T>();
168 r = evaluate_rational(P3, Q3, y);
169 factor = (x + x3) * ((x - x31/256) - x32);
170 value = z + factor * r;
171 }
172 else // x in (8, \infty)
173 {
174 T y = 8 / x;
175 T y2 = y * y;
176 rc = evaluate_rational(PC, QC, y2);
177 rs = evaluate_rational(PS, QS, y2);
178 factor = constants::one_div_root_pi<T>() / sqrt(x);
179 //
180 // The following code is really just:
181 //
182 // T z = x - 0.25f * pi<T>();
183 // value = factor * (rc * sin(z) + y * rs * cos(z));
184 //
185 // But using the sin/cos addition formulae and constant values for
186 // sin/cos of PI/4 which then cancel part of the "factor" term as they're all
187 // 1 / sqrt(2):
188 //
189 T sx = sin(x);
190 T cx = cos(x);
191 value = factor * (rc * (sx - cx) + y * rs * (cx + sx));
192 }
193
194 return value;
195}
196
197}}} // namespaces
198
199#ifdef _MSC_VER
200#pragma warning(pop)
201#endif
202
203#endif // BOOST_MATH_BESSEL_Y0_HPP
204
205

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