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)
6 #ifndef BOOST_MATH_BESSEL_Y1_HPP
7 #define BOOST_MATH_BESSEL_Y1_HPP
13 #include <boost/math/special_functions/detail/bessel_j1.hpp>
14 #include <boost/math/constants/constants.hpp>
15 #include <boost/math/tools/rational.hpp>
16 #include <boost/math/tools/big_constant.hpp>
17 #include <boost/math/policies/error_handling.hpp>
18 #include <boost/assert.hpp>
20 // Bessel function of the second kind of order one
21 // x <= 8, minimax rational approximations on root-bracketing intervals
22 // x > 8, Hankel asymptotic expansion in Hart, Computer Approximations, 1968
24 namespace boost { namespace math { namespace detail{
26 template <typename T, typename Policy>
27 T bessel_y1(T x, const Policy&);
29 template <class T, class Policy>
30 struct bessel_y1_initializer
40 bessel_y1(T(1), Policy());
42 void force_instantiate()const{}
44 static const init initializer;
45 static void force_instantiate()
47 initializer.force_instantiate();
51 template <class T, class Policy>
52 const typename bessel_y1_initializer<T, Policy>::init bessel_y1_initializer<T, Policy>::initializer;
54 template <typename T, typename Policy>
55 T bessel_y1(T x, const Policy& pol)
57 bessel_y1_initializer<T, Policy>::force_instantiate();
59 static const T P1[] = {
60 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0535726612579544093e+13)),
61 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.4708611716525426053e+12)),
62 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.7595974497819597599e+11)),
63 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.2144548214502560419e+09)),
64 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.9157479997408395984e+07)),
65 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2157953222280260820e+05)),
66 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.1714424660046133456e+02)),
68 static const T Q1[] = {
69 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.0737873921079286084e+14)),
70 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1272286200406461981e+12)),
71 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.7800352738690585613e+10)),
72 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.2250435122182963220e+08)),
73 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.8136470753052572164e+05)),
74 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.2079908168393867438e+02)),
75 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
77 static const T P2[] = {
78 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1514276357909013326e+19)),
79 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.6808094574724204577e+18)),
80 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.3638408497043134724e+16)),
81 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0686275289804744814e+15)),
82 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.9530713129741981618e+13)),
83 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.7453673962438488783e+11)),
84 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.1957961912070617006e+09)),
85 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.9153806858264202986e+06)),
86 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2337180442012953128e+03)),
88 static const T Q2[] = {
89 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.3321844313316185697e+20)),
90 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.6968198822857178911e+18)),
91 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.0837179548112881950e+16)),
92 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1187010065856971027e+14)),
93 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.0221766852960403645e+11)),
94 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.3550318087088919566e+08)),
95 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0453748201934079734e+06)),
96 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.2855164849321609336e+03)),
97 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
99 static const T PC[] = {
100 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -4.4357578167941278571e+06)),
101 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -9.9422465050776411957e+06)),
102 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.6033732483649391093e+06)),
103 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.5235293511811373833e+06)),
104 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0982405543459346727e+05)),
105 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.6116166443246101165e+03)),
106 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.0)),
108 static const T QC[] = {
109 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -4.4357578167941278568e+06)),
110 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -9.9341243899345856590e+06)),
111 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.5853394797230870728e+06)),
112 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.5118095066341608816e+06)),
113 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0726385991103820119e+05)),
114 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.4550094401904961825e+03)),
115 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
117 static const T PS[] = {
118 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.3220913409857223519e+04)),
119 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.5145160675335701966e+04)),
120 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6178836581270835179e+04)),
121 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8494262873223866797e+04)),
122 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7063754290207680021e+03)),
123 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.5265133846636032186e+01)),
124 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.0)),
126 static const T QS[] = {
127 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.0871281941028743574e+05)),
128 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8194580422439972989e+06)),
129 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4194606696037208929e+06)),
130 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0029443582266975117e+05)),
131 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.7890229745772202641e+04)),
132 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.6383677696049909675e+02)),
133 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
135 static const T x1 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1971413260310170351e+00)),
136 x2 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.4296810407941351328e+00)),
137 x11 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.620e+02)),
138 x12 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8288260310170351490e-03)),
139 x21 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3900e+03)),
140 x22 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.4592058648672279948e-06))
142 T value, factor, r, rc, rs;
145 using namespace boost::math::tools;
146 using namespace boost::math::constants;
150 return policies::raise_domain_error<T>("bost::math::bessel_y1<%1%>(%1%,%1%)",
151 "Got x == %1%, but x must be > 0, complex result not supported.", x, pol);
153 if (x <= 4) // x in (0, 4]
156 T z = 2 * log(x/x1) * bessel_j1(x) / pi<T>();
157 r = evaluate_rational(P1, Q1, y);
158 factor = (x + x1) * ((x - x11/256) - x12) / x;
159 value = z + factor * r;
161 else if (x <= 8) // x in (4, 8]
164 T z = 2 * log(x/x2) * bessel_j1(x) / pi<T>();
165 r = evaluate_rational(P2, Q2, y);
166 factor = (x + x2) * ((x - x21/256) - x22) / x;
167 value = z + factor * r;
169 else // x in (8, \infty)
173 rc = evaluate_rational(PC, QC, y2);
174 rs = evaluate_rational(PS, QS, y2);
175 factor = 1 / (sqrt(x) * root_pi<T>());
177 // This code is really just:
179 // T z = x - 0.75f * pi<T>();
180 // value = factor * (rc * sin(z) + y * rs * cos(z));
182 // But using the sin/cos addition rules, plus constants for sin/cos of 3PI/4
183 // which then cancel out with corresponding terms in "factor".
187 value = factor * (y * rs * (sx - cx) - rc * (sx + cx));
195 #endif // BOOST_MATH_BESSEL_Y1_HPP