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_JN_HPP
7#define BOOST_MATH_BESSEL_JN_HPP
8
9#ifdef _MSC_VER
10#pragma once
11#endif
12
13#include <boost/math/special_functions/detail/bessel_j0.hpp>
14#include <boost/math/special_functions/detail/bessel_j1.hpp>
15#include <boost/math/special_functions/detail/bessel_jy.hpp>
16#include <boost/math/special_functions/detail/bessel_jy_asym.hpp>
17#include <boost/math/special_functions/detail/bessel_jy_series.hpp>
18
19// Bessel function of the first kind of integer order
20// J_n(z) is the minimal solution
21// n < abs(z), forward recurrence stable and usable
22// n >= abs(z), forward recurrence unstable, use Miller's algorithm
23
24namespace boost { namespace math { namespace detail{
25
26template <typename T, typename Policy>
27T bessel_jn(int n, T x, const Policy& pol)
28{
29 T value(0), factor, current, prev, next;
30
31 BOOST_MATH_STD_USING
32
33 //
34 // Reflection has to come first:
35 //
36 if (n < 0)
37 {
38 factor = static_cast<T>((n & 0x1) ? -1 : 1); // J_{-n}(z) = (-1)^n J_n(z)
39 n = -n;
40 }
41 else
42 {
43 factor = 1;
44 }
45 if(x < 0)
46 {
47 factor *= (n & 0x1) ? -1 : 1; // J_{n}(-z) = (-1)^n J_n(z)
48 x = -x;
49 }
50 //
51 // Special cases:
52 //
53 if(asymptotic_bessel_large_x_limit(T(n), x))
54 return factor * asymptotic_bessel_j_large_x_2<T>(T(n), x);
55 if (n == 0)
56 {
57 return factor * bessel_j0(x);
58 }
59 if (n == 1)
60 {
61 return factor * bessel_j1(x);
62 }
63
64 if (x == 0) // n >= 2
65 {
66 return static_cast<T>(0);
67 }
68
69 BOOST_ASSERT(n > 1);
70 T scale = 1;
71 if (n < abs(x)) // forward recurrence
72 {
73 prev = bessel_j0(x);
74 current = bessel_j1(x);
75 policies::check_series_iterations<T>("boost::math::bessel_j_n<%1%>(%1%,%1%)", n, pol);
76 for (int k = 1; k < n; k++)
77 {
78 T fact = 2 * k / x;
79 //
80 // rescale if we would overflow or underflow:
81 //
82 if((fabs(fact) > 1) && ((tools::max_value<T>() - fabs(prev)) / fabs(fact) < fabs(current)))
83 {
84 scale /= current;
85 prev /= current;
86 current = 1;
87 }
88 value = fact * current - prev;
89 prev = current;
90 current = value;
91 }
92 }
93 else if((x < 1) || (n > x * x / 4) || (x < 5))
94 {
95 return factor * bessel_j_small_z_series(T(n), x, pol);
96 }
97 else // backward recurrence
98 {
99 T fn; int s; // fn = J_(n+1) / J_n
100 // |x| <= n, fast convergence for continued fraction CF1
101 boost::math::detail::CF1_jy(static_cast<T>(n), x, &fn, &s, pol);
102 prev = fn;
103 current = 1;
104 // Check recursion won't go on too far:
105 policies::check_series_iterations<T>("boost::math::bessel_j_n<%1%>(%1%,%1%)", n, pol);
106 for (int k = n; k > 0; k--)
107 {
108 T fact = 2 * k / x;
109 if((fabs(fact) > 1) && ((tools::max_value<T>() - fabs(prev)) / fabs(fact) < fabs(current)))
110 {
111 prev /= current;
112 scale /= current;
113 current = 1;
114 }
115 next = fact * current - prev;
116 prev = current;
117 current = next;
118 }
119 value = bessel_j0(x) / current; // normalization
120 scale = 1 / scale;
121 }
122 value *= factor;
123
124 if(tools::max_value<T>() * scale < fabs(value))
125 return policies::raise_overflow_error<T>("boost::math::bessel_jn<%1%>(%1%,%1%)", 0, pol);
126
127 return value / scale;
128}
129
130}}} // namespaces
131
132#endif // BOOST_MATH_BESSEL_JN_HPP
133
134

source code of include/boost/math/special_functions/detail/bessel_jn.hpp