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 // Bessel function of the second kind of order one
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_y1(T x, const Policy&);
30 
31 template <class T, class Policy>
32 struct bessel_y1_initializer
33 {
34    struct init
35    {
initboost::math::detail::bessel_y1_initializer::init36       init()
37       {
38          do_init();
39       }
do_initboost::math::detail::bessel_y1_initializer::init40       static void do_init()
41       {
42          bessel_y1(T(1), Policy());
43       }
force_instantiateboost::math::detail::bessel_y1_initializer::init44       void force_instantiate()const{}
45    };
46    static const init initializer;
force_instantiateboost::math::detail::bessel_y1_initializer47    static void force_instantiate()
48    {
49       initializer.force_instantiate();
50    }
51 };
52 
53 template <class T, class Policy>
54 const typename bessel_y1_initializer<T, Policy>::init bessel_y1_initializer<T, Policy>::initializer;
55 
56 template <typename T, typename Policy>
57 T bessel_y1(T x, const Policy& pol)
58 {
59     bessel_y1_initializer<T, Policy>::force_instantiate();
60 
61     static const T P1[] = {
62          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0535726612579544093e+13)),
63          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.4708611716525426053e+12)),
64         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.7595974497819597599e+11)),
65          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.2144548214502560419e+09)),
66         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.9157479997408395984e+07)),
67          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2157953222280260820e+05)),
68         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.1714424660046133456e+02)),
69     };
70     static const T Q1[] = {
71          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.0737873921079286084e+14)),
72          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1272286200406461981e+12)),
73          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.7800352738690585613e+10)),
74          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.2250435122182963220e+08)),
75          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.8136470753052572164e+05)),
76          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.2079908168393867438e+02)),
77          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
78     };
79     static const T P2[] = {
80          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1514276357909013326e+19)),
81         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.6808094574724204577e+18)),
82         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.3638408497043134724e+16)),
83          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0686275289804744814e+15)),
84         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.9530713129741981618e+13)),
85          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.7453673962438488783e+11)),
86         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.1957961912070617006e+09)),
87          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.9153806858264202986e+06)),
88         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2337180442012953128e+03)),
89     };
90     static const T Q2[] = {
91          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.3321844313316185697e+20)),
92          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.6968198822857178911e+18)),
93          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.0837179548112881950e+16)),
94          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1187010065856971027e+14)),
95          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.0221766852960403645e+11)),
96          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.3550318087088919566e+08)),
97          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0453748201934079734e+06)),
98          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.2855164849321609336e+03)),
99          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
100     };
101     static const T PC[] = {
102         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -4.4357578167941278571e+06)),
103         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -9.9422465050776411957e+06)),
104         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.6033732483649391093e+06)),
105         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.5235293511811373833e+06)),
106         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0982405543459346727e+05)),
107         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.6116166443246101165e+03)),
108          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.0)),
109     };
110     static const T QC[] = {
111         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -4.4357578167941278568e+06)),
112         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -9.9341243899345856590e+06)),
113         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.5853394797230870728e+06)),
114         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.5118095066341608816e+06)),
115         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0726385991103820119e+05)),
116         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.4550094401904961825e+03)),
117          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
118     };
119     static const T PS[] = {
120          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.3220913409857223519e+04)),
121          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.5145160675335701966e+04)),
122          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6178836581270835179e+04)),
123          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8494262873223866797e+04)),
124          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7063754290207680021e+03)),
125          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.5265133846636032186e+01)),
126          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.0)),
127     };
128     static const T QS[] = {
129          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.0871281941028743574e+05)),
130          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8194580422439972989e+06)),
131          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4194606696037208929e+06)),
132          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0029443582266975117e+05)),
133          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.7890229745772202641e+04)),
134          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.6383677696049909675e+02)),
135          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
136     };
137     static const T x1  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1971413260310170351e+00)),
138                    x2  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.4296810407941351328e+00)),
139                    x11 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.620e+02)),
140                    x12 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8288260310170351490e-03)),
141                    x21 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3900e+03)),
142                    x22 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.4592058648672279948e-06))
143     ;
144     T value, factor, r, rc, rs;
145 
146     BOOST_MATH_STD_USING
147     using namespace boost::math::tools;
148     using namespace boost::math::constants;
149 
150     if (x <= 0)
151     {
152        return policies::raise_domain_error<T>("bost::math::bessel_y1<%1%>(%1%,%1%)",
153             "Got x == %1%, but x must be > 0, complex result not supported.", x, pol);
154     }
155     if (x <= 4)                       // x in (0, 4]
156     {
157         T y = x * x;
158         T z = 2 * log(x/x1) * bessel_j1(x) / pi<T>();
159         r = evaluate_rational(P1, Q1, y);
160         factor = (x + x1) * ((x - x11/256) - x12) / x;
161         value = z + factor * r;
162     }
163     else if (x <= 8)                  // x in (4, 8]
164     {
165         T y = x * x;
166         T z = 2 * log(x/x2) * bessel_j1(x) / pi<T>();
167         r = evaluate_rational(P2, Q2, y);
168         factor = (x + x2) * ((x - x21/256) - x22) / x;
169         value = z + factor * r;
170     }
171     else                                // x in (8, \infty)
172     {
173         T y = 8 / x;
174         T y2 = y * y;
175         rc = evaluate_rational(PC, QC, y2);
176         rs = evaluate_rational(PS, QS, y2);
177         factor = 1 / (sqrt(x) * root_pi<T>());
178         //
179         // This code is really just:
180         //
181         // T z = x - 0.75f * pi<T>();
182         // value = factor * (rc * sin(z) + y * rs * cos(z));
183         //
184         // But using the sin/cos addition rules, plus constants for sin/cos of 3PI/4
185         // which then cancel out with corresponding terms in "factor".
186         //
187         T sx = sin(x);
188         T cx = cos(x);
189         value = factor * (y * rs * (sx - cx) - rc * (sx + cx));
190     }
191 
192     return value;
193 }
194 
195 }}} // namespaces
196 
197 #ifdef _MSC_VER
198 #pragma warning(pop)
199 #endif
200 
201 #endif // BOOST_MATH_BESSEL_Y1_HPP
202 
203