1// Copyright (c) 2013 Christopher Kormanyos
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// This work is based on an earlier work:
7// "Algorithm 910: A Portable C++ Multiple-Precision System for Special-Function Calculations",
8// in ACM TOMS, {VOL 37, ISSUE 4, (February 2011)} (C) ACM, 2011. http://doi.acm.org/10.1145/1916461.1916469
9//
10// This header contains implementation details for estimating the zeros
11// of the Airy functions airy_ai and airy_bi on the negative real axis.
12//
13#ifndef BOOST_MATH_AIRY_AI_BI_ZERO_2013_01_20_HPP_
14 #define BOOST_MATH_AIRY_AI_BI_ZERO_2013_01_20_HPP_
15
16 #include <boost/math/constants/constants.hpp>
17 #include <boost/math/special_functions/cbrt.hpp>
18
19 namespace boost { namespace math {
20 namespace detail
21 {
22 // Forward declarations of the needed Airy function implementations.
23 template <class T, class Policy>
24 T airy_ai_imp(T x, const Policy& pol);
25 template <class T, class Policy>
26 T airy_bi_imp(T x, const Policy& pol);
27 template <class T, class Policy>
28 T airy_ai_prime_imp(T x, const Policy& pol);
29 template <class T, class Policy>
30 T airy_bi_prime_imp(T x, const Policy& pol);
31
32 namespace airy_zero
33 {
34 template<class T>
35 T equation_as_10_4_105(const T& z)
36 {
37 const T one_over_z (T(1) / z);
38 const T one_over_z_squared(one_over_z * one_over_z);
39
40 const T z_pow_third (boost::math::cbrt(z));
41 const T z_pow_two_thirds(z_pow_third * z_pow_third);
42
43 // Implement the top line of Eq. 10.4.105.
44 const T fz(z_pow_two_thirds * ((((( + (T(162375596875.0) / 334430208UL)
45 * one_over_z_squared - ( T(108056875.0) / 6967296UL))
46 * one_over_z_squared + ( T(77125UL) / 82944UL))
47 * one_over_z_squared - ( T(5U) / 36U))
48 * one_over_z_squared + ( T(5U) / 48U))
49 * one_over_z_squared + (1)));
50
51 return fz;
52 }
53
54 namespace airy_ai_zero_detail
55 {
56 template<class T>
57 T initial_guess(const int m)
58 {
59 T guess;
60
61 switch(m)
62 {
63 case 0: { guess = T(0); break; }
64 case 1: { guess = T(-2.33810741045976703849); break; }
65 case 2: { guess = T(-4.08794944413097061664); break; }
66 case 3: { guess = T(-5.52055982809555105913); break; }
67 case 4: { guess = T(-6.78670809007175899878); break; }
68 case 5: { guess = T(-7.94413358712085312314); break; }
69 case 6: { guess = T(-9.02265085334098038016); break; }
70 case 7: { guess = T(-10.0401743415580859306); break; }
71 case 8: { guess = T(-11.0085243037332628932); break; }
72 case 9: { guess = T(-11.9360155632362625170); break; }
73 case 10:{ guess = T(-12.8287767528657572004); break; }
74 default:
75 {
76 const T t(((boost::math::constants::pi<T>() * 3) * ((T(m) * 4) - 1)) / 8);
77 guess = -boost::math::detail::airy_zero::equation_as_10_4_105(t);
78 break;
79 }
80 }
81
82 return guess;
83 }
84
85 template<class T, class Policy>
86 class function_object_ai_and_ai_prime
87 {
88 public:
89 function_object_ai_and_ai_prime(const Policy& pol) : my_pol(pol) { }
90
91 boost::math::tuple<T, T> operator()(const T& x) const
92 {
93 // Return a tuple containing both Ai(x) and Ai'(x).
94 return boost::math::make_tuple(
95 boost::math::detail::airy_ai_imp (x, my_pol),
96 boost::math::detail::airy_ai_prime_imp(x, my_pol));
97 }
98
99 private:
100 const Policy& my_pol;
101 const function_object_ai_and_ai_prime& operator=(const function_object_ai_and_ai_prime&);
102 };
103 } // namespace airy_ai_zero_detail
104
105 namespace airy_bi_zero_detail
106 {
107 template<class T>
108 T initial_guess(const int m)
109 {
110 T guess;
111
112 switch(m)
113 {
114 case 0: { guess = T(0); break; }
115 case 1: { guess = T(-1.17371322270912792492); break; }
116 case 2: { guess = T(-3.27109330283635271568); break; }
117 case 3: { guess = T(-4.83073784166201593267); break; }
118 case 4: { guess = T(-6.16985212831025125983); break; }
119 case 5: { guess = T(-7.37676207936776371360); break; }
120 case 6: { guess = T(-8.49194884650938801345); break; }
121 case 7: { guess = T(-9.53819437934623888663); break; }
122 case 8: { guess = T(-10.5299135067053579244); break; }
123 case 9: { guess = T(-11.4769535512787794379); break; }
124 case 10: { guess = T(-12.3864171385827387456); break; }
125 default:
126 {
127 const T t(((boost::math::constants::pi<T>() * 3) * ((T(m) * 4) - 3)) / 8);
128 guess = -boost::math::detail::airy_zero::equation_as_10_4_105(t);
129 break;
130 }
131 }
132
133 return guess;
134 }
135
136 template<class T, class Policy>
137 class function_object_bi_and_bi_prime
138 {
139 public:
140 function_object_bi_and_bi_prime(const Policy& pol) : my_pol(pol) { }
141
142 boost::math::tuple<T, T> operator()(const T& x) const
143 {
144 // Return a tuple containing both Bi(x) and Bi'(x).
145 return boost::math::make_tuple(
146 boost::math::detail::airy_bi_imp (x, my_pol),
147 boost::math::detail::airy_bi_prime_imp(x, my_pol));
148 }
149
150 private:
151 const Policy& my_pol;
152 const function_object_bi_and_bi_prime& operator=(const function_object_bi_and_bi_prime&);
153 };
154 } // namespace airy_bi_zero_detail
155 } // namespace airy_zero
156 } // namespace detail
157 } // namespace math
158 } // namespaces boost
159
160#endif // BOOST_MATH_AIRY_AI_BI_ZERO_2013_01_20_HPP_
161

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