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_Y1_HPP
7 #define BOOST_MATH_BESSEL_Y1_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_j1.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 dianostic ignored work :(
28 //
29 #pragma GCC system_header
30 #endif
31 
32 // Bessel function of the second kind of order one
33 // x <= 8, minimax rational approximations on root-bracketing intervals
34 // x > 8, Hankel asymptotic expansion in Hart, Computer Approximations, 1968
35 
36 namespace boost { namespace math { namespace detail{
37 
38 template <typename T, typename Policy>
39 T bessel_y1(T x, const Policy&);
40 
41 template <class T, class Policy>
42 struct bessel_y1_initializer
43 {
44    struct init
45    {
initboost::math::detail::bessel_y1_initializer::init46       init()
47       {
48          do_init();
49       }
do_initboost::math::detail::bessel_y1_initializer::init50       static void do_init()
51       {
52          bessel_y1(T(1), Policy());
53       }
force_instantiateboost::math::detail::bessel_y1_initializer::init54       void force_instantiate()const{}
55    };
56    static const init initializer;
force_instantiateboost::math::detail::bessel_y1_initializer57    static void force_instantiate()
58    {
59       initializer.force_instantiate();
60    }
61 };
62 
63 template <class T, class Policy>
64 const typename bessel_y1_initializer<T, Policy>::init bessel_y1_initializer<T, Policy>::initializer;
65 
66 template <typename T, typename Policy>
67 T bessel_y1(T x, const Policy& pol)
68 {
69     bessel_y1_initializer<T, Policy>::force_instantiate();
70 
71     static const T P1[] = {
72          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0535726612579544093e+13)),
73          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.4708611716525426053e+12)),
74         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.7595974497819597599e+11)),
75          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.2144548214502560419e+09)),
76         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.9157479997408395984e+07)),
77          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2157953222280260820e+05)),
78         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.1714424660046133456e+02)),
79     };
80     static const T Q1[] = {
81          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.0737873921079286084e+14)),
82          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1272286200406461981e+12)),
83          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.7800352738690585613e+10)),
84          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.2250435122182963220e+08)),
85          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.8136470753052572164e+05)),
86          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.2079908168393867438e+02)),
87          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
88     };
89     static const T P2[] = {
90          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1514276357909013326e+19)),
91         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.6808094574724204577e+18)),
92         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.3638408497043134724e+16)),
93          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0686275289804744814e+15)),
94         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.9530713129741981618e+13)),
95          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.7453673962438488783e+11)),
96         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.1957961912070617006e+09)),
97          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.9153806858264202986e+06)),
98         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2337180442012953128e+03)),
99     };
100     static const T Q2[] = {
101          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.3321844313316185697e+20)),
102          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.6968198822857178911e+18)),
103          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.0837179548112881950e+16)),
104          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1187010065856971027e+14)),
105          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.0221766852960403645e+11)),
106          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.3550318087088919566e+08)),
107          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0453748201934079734e+06)),
108          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.2855164849321609336e+03)),
109          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
110     };
111     static const T PC[] = {
112         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -4.4357578167941278571e+06)),
113         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -9.9422465050776411957e+06)),
114         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.6033732483649391093e+06)),
115         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.5235293511811373833e+06)),
116         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0982405543459346727e+05)),
117         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.6116166443246101165e+03)),
118          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.0)),
119     };
120     static const T QC[] = {
121         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -4.4357578167941278568e+06)),
122         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -9.9341243899345856590e+06)),
123         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.5853394797230870728e+06)),
124         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.5118095066341608816e+06)),
125         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0726385991103820119e+05)),
126         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.4550094401904961825e+03)),
127          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
128     };
129     static const T PS[] = {
130          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.3220913409857223519e+04)),
131          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.5145160675335701966e+04)),
132          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6178836581270835179e+04)),
133          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8494262873223866797e+04)),
134          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7063754290207680021e+03)),
135          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.5265133846636032186e+01)),
136          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.0)),
137     };
138     static const T QS[] = {
139          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.0871281941028743574e+05)),
140          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8194580422439972989e+06)),
141          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4194606696037208929e+06)),
142          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0029443582266975117e+05)),
143          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.7890229745772202641e+04)),
144          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.6383677696049909675e+02)),
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, 2.1971413260310170351e+00)),
148                    x2  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.4296810407941351328e+00)),
149                    x11 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.620e+02)),
150                    x12 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8288260310170351490e-03)),
151                    x21 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3900e+03)),
152                    x22 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.4592058648672279948e-06))
153     ;
154     T value, factor, r, rc, rs;
155 
156     BOOST_MATH_STD_USING
157     using namespace boost::math::tools;
158     using namespace boost::math::constants;
159 
160     if (x <= 0)
161     {
162        return policies::raise_domain_error<T>("bost::math::bessel_y1<%1%>(%1%,%1%)",
163             "Got x == %1%, but x must be > 0, complex result not supported.", x, pol);
164     }
165     if (x <= 4)                       // x in (0, 4]
166     {
167         T y = x * x;
168         T z = 2 * log(x/x1) * bessel_j1(x) / pi<T>();
169         r = evaluate_rational(P1, Q1, y);
170         factor = (x + x1) * ((x - x11/256) - x12) / x;
171         value = z + factor * r;
172     }
173     else if (x <= 8)                  // x in (4, 8]
174     {
175         T y = x * x;
176         T z = 2 * log(x/x2) * bessel_j1(x) / pi<T>();
177         r = evaluate_rational(P2, Q2, y);
178         factor = (x + x2) * ((x - x21/256) - x22) / x;
179         value = z + factor * r;
180     }
181     else                                // x in (8, \infty)
182     {
183         T y = 8 / x;
184         T y2 = y * y;
185         rc = evaluate_rational(PC, QC, y2);
186         rs = evaluate_rational(PS, QS, y2);
187         factor = 1 / (sqrt(x) * root_pi<T>());
188         //
189         // This code is really just:
190         //
191         // T z = x - 0.75f * pi<T>();
192         // value = factor * (rc * sin(z) + y * rs * cos(z));
193         //
194         // But using the sin/cos addition rules, plus constants for sin/cos of 3PI/4
195         // which then cancel out with corresponding terms in "factor".
196         //
197         T sx = sin(x);
198         T cx = cos(x);
199         value = factor * (y * rs * (sx - cx) - rc * (sx + cx));
200     }
201 
202     return value;
203 }
204 
205 }}} // namespaces
206 
207 #ifdef _MSC_VER
208 #pragma warning(pop)
209 #endif
210 
211 #endif // BOOST_MATH_BESSEL_Y1_HPP
212 
213