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_Y0_HPP
7 #define BOOST_MATH_BESSEL_Y0_HPP
13 #include <boost/math/special_functions/detail/bessel_j0.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 zero
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_y0(T x, const Policy&);
29 template <class T, class Policy>
30 struct bessel_y0_initializer
40 bessel_y0(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_y0_initializer<T, Policy>::init bessel_y0_initializer<T, Policy>::initializer;
54 template <typename T, typename Policy>
55 T bessel_y0(T x, const Policy& pol)
57 bessel_y0_initializer<T, Policy>::force_instantiate();
59 static const T P1[] = {
60 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0723538782003176831e+11)),
61 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.3716255451260504098e+09)),
62 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.0422274357376619816e+08)),
63 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.1287548474401797963e+06)),
64 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0102532948020907590e+04)),
65 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8402381979244993524e+01)),
67 static const T Q1[] = {
68 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.8873865738997033405e+11)),
69 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.1617187777290363573e+09)),
70 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.5662956624278251596e+07)),
71 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.3889393209447253406e+05)),
72 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6475986689240190091e+02)),
73 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
75 static const T P2[] = {
76 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.2213976967566192242e+13)),
77 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.5107435206722644429e+11)),
78 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.3600098638603061642e+10)),
79 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.9590439394619619534e+08)),
80 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.6905288611678631510e+06)),
81 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.4566865832663635920e+04)),
82 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7427031242901594547e+01)),
84 static const T Q2[] = {
85 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.3386146580707264428e+14)),
86 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.4266824419412347550e+12)),
87 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4015103849971240096e+10)),
88 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3960202770986831075e+08)),
89 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0669982352539552018e+05)),
90 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.3030857612070288823e+02)),
91 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
93 static const T P3[] = {
94 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.0728726905150210443e+15)),
95 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.7016641869173237784e+14)),
96 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2829912364088687306e+11)),
97 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.9363051266772083678e+11)),
98 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1958827170518100757e+09)),
99 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0085539923498211426e+07)),
100 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1363534169313901632e+04)),
101 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.7439661319197499338e+01)),
103 static const T Q3[] = {
104 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4563724628846457519e+17)),
105 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.9272425569640309819e+15)),
106 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2598377924042897629e+13)),
107 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.6926121104209825246e+10)),
108 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.4727219475672302327e+08)),
109 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.3924739209768057030e+05)),
110 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.7903362168128450017e+02)),
111 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
113 static const T PC[] = {
114 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684302e+04)),
115 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1345386639580765797e+04)),
116 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1170523380864944322e+04)),
117 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4806486443249270347e+03)),
118 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5376201909008354296e+02)),
119 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.8961548424210455236e-01)),
121 static const T QC[] = {
122 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684318e+04)),
123 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1370412495510416640e+04)),
124 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1215350561880115730e+04)),
125 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.5028735138235608207e+03)),
126 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5711159858080893649e+02)),
127 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
129 static const T PS[] = {
130 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.9226600200800094098e+01)),
131 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8591953644342993800e+02)),
132 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.1183429920482737611e+02)),
133 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.2300261666214198472e+01)),
134 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2441026745835638459e+00)),
135 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.8033303048680751817e-03)),
137 static const T QS[] = {
138 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.7105024128512061905e+03)),
139 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1951131543434613647e+04)),
140 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.2642780169211018836e+03)),
141 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4887231232283756582e+03)),
142 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.0593769594993125859e+01)),
143 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
145 static const T x1 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.9357696627916752158e-01)),
146 x2 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.9576784193148578684e+00)),
147 x3 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.0860510603017726976e+00)),
148 x11 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.280e+02)),
149 x12 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.9519662791675215849e-03)),
150 x21 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0130e+03)),
151 x22 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.4716931485786837568e-04)),
152 x31 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8140e+03)),
153 x32 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1356030177269762362e-04))
155 T value, factor, r, rc, rs;
158 using namespace boost::math::tools;
159 using namespace boost::math::constants;
161 static const char* function = "boost::math::bessel_y0<%1%>(%1%,%1%)";
165 return policies::raise_domain_error<T>(function,
166 "Got x = %1% but x must be non-negative, complex result not supported.", x, pol);
170 return -policies::raise_overflow_error<T>(function, 0, pol);
172 if (x <= 3) // x in (0, 3]
175 T z = 2 * log(x/x1) * bessel_j0(x) / pi<T>();
176 r = evaluate_rational(P1, Q1, y);
177 factor = (x + x1) * ((x - x11/256) - x12);
178 value = z + factor * r;
180 else if (x <= 5.5f) // x in (3, 5.5]
183 T z = 2 * log(x/x2) * bessel_j0(x) / pi<T>();
184 r = evaluate_rational(P2, Q2, y);
185 factor = (x + x2) * ((x - x21/256) - x22);
186 value = z + factor * r;
188 else if (x <= 8) // x in (5.5, 8]
191 T z = 2 * log(x/x3) * bessel_j0(x) / pi<T>();
192 r = evaluate_rational(P3, Q3, y);
193 factor = (x + x3) * ((x - x31/256) - x32);
194 value = z + factor * r;
196 else // x in (8, \infty)
200 rc = evaluate_rational(PC, QC, y2);
201 rs = evaluate_rational(PS, QS, y2);
202 factor = constants::one_div_root_pi<T>() / sqrt(x);
204 // The following code is really just:
206 // T z = x - 0.25f * pi<T>();
207 // value = factor * (rc * sin(z) + y * rs * cos(z));
209 // But using the sin/cos addition formulae and constant values for
210 // sin/cos of PI/4 which then cancel part of the "factor" term as they're all
215 value = factor * (rc * (sx - cx) + y * rs * (cx + sx));
223 #endif // BOOST_MATH_BESSEL_Y0_HPP