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 | |