1
2 // Copyright Christopher Kormanyos 2013.
3 // Copyright Paul A. Bristow 2013.
4 // Copyright John Maddock 2013.
5
6 // Distributed under the Boost Software License, Version 1.0.
7 // (See accompanying file LICENSE_1_0.txt or
8 // copy at http://www.boost.org/LICENSE_1_0.txt).
9
10 #ifdef _MSC_VER
11 # pragma warning (disable : 4512) // assignment operator could not be generated.
12 # pragma warning (disable : 4996) // assignment operator could not be generated.
13 #endif
14
15 #include <iostream>
16 #include <limits>
17 #include <vector>
18 #include <algorithm>
19 #include <iomanip>
20 #include <iterator>
21
22 // Weisstein, Eric W. "Bessel Function Zeros." From MathWorld--A Wolfram Web Resource.
23 // http://mathworld.wolfram.com/BesselFunctionZeros.html
24 // Test values can be calculated using [@wolframalpha.com WolframAplha]
25 // See also http://dlmf.nist.gov/10.21
26
27 //[airy_zeros_example_1
28
29 /*`This example demonstrates calculating zeros of the Airy functions.
30 It also shows how Boost.Math and Boost.Multiprecision can be combined to provide
31 a many decimal digit precision. For 50 decimal digit precision we need to include
32 */
33
34 #include <boost/multiprecision/cpp_dec_float.hpp>
35
36 /*`and a `typedef` for `float_type` may be convenient
37 (allowing a quick switch to re-compute at built-in `double` or other precision)
38 */
39 typedef boost::multiprecision::cpp_dec_float_50 float_type;
40
41 //`To use the functions for finding zeros of the functions we need
42
43 #include <boost/math/special_functions/airy.hpp>
44
45 /*`This example shows obtaining both a single zero of the Airy functions,
46 and then placing multiple zeros into a container like `std::vector` by providing an iterator.
47 The signature of the single-value Airy Ai function is:
48
49 template <class T>
50 T airy_ai_zero(unsigned m); // 1-based index of the zero.
51
52 The signature of multiple zeros Airy Ai function is:
53
54 template <class T, class OutputIterator>
55 OutputIterator airy_ai_zero(
56 unsigned start_index, // 1-based index of the zero.
57 unsigned number_of_zeros, // How many zeros to generate.
58 OutputIterator out_it); // Destination for zeros.
59
60 There are also versions which allows control of the __policy_section for error handling and precision.
61
62 template <class T, class OutputIterator, class Policy>
63 OutputIterator airy_ai_zero(
64 unsigned start_index, // 1-based index of the zero.
65 unsigned number_of_zeros, // How many zeros to generate.
66 OutputIterator out_it, // Destination for zeros.
67 const Policy& pol); // Policy to use.
68 */
69 //] [/airy_zeros_example_1]
70
main()71 int main()
72 {
73 try
74 {
75 //[airy_zeros_example_2
76
77 /*`[tip It is always wise to place code using Boost.Math inside `try'n'catch` blocks;
78 this will ensure that helpful error messages are shown when exceptional conditions arise.]
79
80 First, evaluate a single Airy zero.
81
82 The precision is controlled by the template parameter `T`,
83 so this example has `double` precision, at least 15 but up to 17 decimal digits
84 (for the common 64-bit double).
85 */
86 double aiz1 = boost::math::airy_ai_zero<double>(1);
87 std::cout << "boost::math::airy_ai_zero<double>(1) = " << aiz1 << std::endl;
88 double aiz2 = boost::math::airy_ai_zero<double>(2);
89 std::cout << "boost::math::airy_ai_zero<double>(2) = " << aiz2 << std::endl;
90 double biz3 = boost::math::airy_bi_zero<double>(3);
91 std::cout << "boost::math::airy_bi_zero<double>(3) = " << biz3 << std::endl;
92
93 /*`Other versions of `airy_ai_zero` and `airy_bi_zero`
94 allow calculation of multiple zeros with one call,
95 placing the results in a container, often `std::vector`.
96 For example, generate and display the first five `double` roots
97 [@http://mathworld.wolfram.com/AiryFunctionZeros.html Wolfram Airy Functions Zeros].
98 */
99 unsigned int n_roots = 5U;
100 std::vector<double> roots;
101 boost::math::airy_ai_zero<double>(1U, n_roots, std::back_inserter(roots));
102 std::cout << "airy_ai_zeros:" << std::endl;
103 std::copy(roots.begin(),
104 roots.end(),
105 std::ostream_iterator<double>(std::cout, "\n"));
106
107 /*`The first few real roots of Ai(x) are approximately -2.33811, -4.08795, -5.52056, -6.7867144, -7.94413, -9.02265 ...
108
109 Or we can use Boost.Multiprecision to generate 50 decimal digit roots.
110
111 We set the precision of the output stream, and show trailing zeros to display a fixed 50 decimal digits.
112 */
113 std::cout.precision(std::numeric_limits<float_type>::digits10); // float_type has 50 decimal digits.
114 std::cout << std::showpoint << std::endl; // Show trailing zeros too.
115
116 unsigned int m = 1U;
117 float_type r = boost::math::airy_ai_zero<double>(1U); // 1st root.
118 std::cout << "boost::math::airy_bi_zero<float_type>(" << m << ") = " << r << std::endl;
119 r = boost::math::airy_ai_zero<float_type>(1U); // 1st root.
120 std::cout << "boost::math::airy_bi_zero<float_type>(" << m << ") = " << r << std::endl;
121 m = 7U;
122 r = boost::math::airy_bi_zero<float_type>(7U); // 7th root.
123 std::cout << "boost::math::airy_bi_zero<float_type>(" << m << ") = " << r << std::endl;
124
125 std::vector<float_type> zeros;
126 boost::math::airy_ai_zero<float_type>(1U, 3, std::back_inserter(zeros));
127 std::cout << "airy_ai_zeros:" << std::endl;
128 // Print the roots to the output stream.
129 std::copy(zeros.begin(), zeros.end(),
130 std::ostream_iterator<float_type>(std::cout, "\n"));
131 //] [/airy_zeros_example_2]
132 }
133 catch (std::exception ex)
134 {
135 std::cout << "Thrown exception " << ex.what() << std::endl;
136 }
137
138 } // int main()
139
140 /*
141
142 Output:
143
144 Description: Autorun "J:\Cpp\big_number\Debug\airy_zeros_example.exe"
145 boost::math::airy_ai_zero<double>(1) = -2.33811
146 boost::math::airy_ai_zero<double>(2) = -4.08795
147 boost::math::airy_bi_zero<double>(3) = -4.83074
148
149 airy_ai_zeros:
150 -2.33811
151 -4.08795
152 -5.52056
153 -6.78671
154 -7.94413
155
156 boost::math::airy_bi_zero<float_type>(1) = -2.3381074104597665552773833042010664939880371093750
157 boost::math::airy_bi_zero<float_type>(1) = -2.3381074104597670384891972524467354406385502908783
158 boost::math::airy_bi_zero<float_type>(7) = -9.5381943793462388866329885451560196208390720763825
159
160 airy_ai_zeros:
161 -2.3381074104597670384891972524467354406385502908783
162 -4.0879494441309706166369887014573910602247646991085
163 -5.5205598280955510591298555129312935737972142806175
164
165
166
167 */
168
169