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 //[bessel_zeros_example_1
28
29 /*`This example demonstrates calculating zeros of the Bessel and Neumann 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/bessel.hpp>
44
45 //`This file includes the forward declaration signatures for the zero-finding functions:
46
47 // #include <boost/math/special_functions/math_fwd.hpp>
48
49 /*`but more details are in the full documentation, for example at
50 [@http://www.boost.org/doc/libs/1_53_0/libs/math/doc/sf_and_dist/html/math_toolkit/special/bessel/bessel_over.html Boost.Math Bessel functions].
51 */
52
53 /*`This example shows obtaining both a single zero of the Bessel function,
54 and then placing multiple zeros into a container like `std::vector` by providing an iterator.
55 */
56 //] [/bessel_zeros_example_1]
57
58 /*The signature of the single value function is:
59
60 template <class T>
61 inline typename detail::bessel_traits<T, T, policies::policy<> >::result_type
62 cyl_bessel_j_zero(
63 T v, // Floating-point value for Jv.
64 int m); // start index.
65
66 The result type is controlled by the floating-point type of parameter `v`
67 (but subject to the usual __precision_policy and __promotion_policy).
68
69 The signature of multiple zeros function is:
70
71 template <class T, class OutputIterator>
72 inline OutputIterator cyl_bessel_j_zero(
73 T v, // Floating-point value for Jv.
74 int start_index, // 1-based start index.
75 unsigned number_of_zeros, // How many zeros to generate
76 OutputIterator out_it); // Destination for zeros.
77
78 There is also a version which allows control of the __policy_section for error handling and precision.
79
80 template <class T, class OutputIterator, class Policy>
81 inline OutputIterator cyl_bessel_j_zero(
82 T v, // Floating-point value for Jv.
83 int start_index, // 1-based start index.
84 unsigned number_of_zeros, // How many zeros to generate
85 OutputIterator out_it, // Destination for zeros.
86 const Policy& pol); // Policy to use.
87 */
88
main()89 int main()
90 {
91 try
92 {
93 //[bessel_zeros_example_2
94
95 /*`[tip It is always wise to place code using Boost.Math inside try'n'catch blocks;
96 this will ensure that helpful error messages are shown when exceptional conditions arise.]
97
98 First, evaluate a single Bessel zero.
99
100 The precision is controlled by the float-point type of template parameter `T` of `v`
101 so this example has `double` precision, at least 15 but up to 17 decimal digits (for the common 64-bit double).
102 */
103 // double root = boost::math::cyl_bessel_j_zero(0.0, 1);
104 // // Displaying with default precision of 6 decimal digits:
105 // std::cout << "boost::math::cyl_bessel_j_zero(0.0, 1) " << root << std::endl; // 2.40483
106 // // And with all the guaranteed (15) digits:
107 // std::cout.precision(std::numeric_limits<double>::digits10);
108 // std::cout << "boost::math::cyl_bessel_j_zero(0.0, 1) " << root << std::endl; // 2.40482555769577
109 /*`But note that because the parameter `v` controls the precision of the result,
110 `v` [*must be a floating-point type].
111 So if you provide an integer type, say 0, rather than 0.0, then it will fail to compile thus:
112 ``
113 root = boost::math::cyl_bessel_j_zero(0, 1);
114 ``
115 with this error message
116 ``
117 error C2338: Order must be a floating-point type.
118 ``
119
120 Optionally, we can use a policy to ignore errors, C-style, returning some value,
121 perhaps infinity or NaN, or the best that can be done. (See __user_error_handling).
122
123 To create a (possibly unwise!) policy `ignore_all_policy` that ignores all errors:
124 */
125
126 typedef boost::math::policies::policy<
127 boost::math::policies::domain_error<boost::math::policies::ignore_error>,
128 boost::math::policies::overflow_error<boost::math::policies::ignore_error>,
129 boost::math::policies::underflow_error<boost::math::policies::ignore_error>,
130 boost::math::policies::denorm_error<boost::math::policies::ignore_error>,
131 boost::math::policies::pole_error<boost::math::policies::ignore_error>,
132 boost::math::policies::evaluation_error<boost::math::policies::ignore_error>
133 > ignore_all_policy;
134 //`Examples of use of this `ignore_all_policy` are
135
136 double inf = std::numeric_limits<double>::infinity();
137 double nan = std::numeric_limits<double>::quiet_NaN();
138
139 double dodgy_root = boost::math::cyl_bessel_j_zero(-1.0, 1, ignore_all_policy());
140 std::cout << "boost::math::cyl_bessel_j_zero(-1.0, 1) " << dodgy_root << std::endl; // 1.#QNAN
141 double inf_root = boost::math::cyl_bessel_j_zero(inf, 1, ignore_all_policy());
142 std::cout << "boost::math::cyl_bessel_j_zero(inf, 1) " << inf_root << std::endl; // 1.#QNAN
143 double nan_root = boost::math::cyl_bessel_j_zero(nan, 1, ignore_all_policy());
144 std::cout << "boost::math::cyl_bessel_j_zero(nan, 1) " << nan_root << std::endl; // 1.#QNAN
145
146 /*`Another version of `cyl_bessel_j_zero` allows calculation of multiple zeros with one call,
147 placing the results in a container, often `std::vector`.
148 For example, generate and display the first five `double` roots of J[sub v] for integral order 2,
149 as column ['J[sub 2](x)] in table 1 of
150 [@ http://mathworld.wolfram.com/BesselFunctionZeros.html Wolfram Bessel Function Zeros].
151 */
152 unsigned int n_roots = 5U;
153 std::vector<double> roots;
154 boost::math::cyl_bessel_j_zero(2.0, 1, n_roots, std::back_inserter(roots));
155 std::copy(roots.begin(),
156 roots.end(),
157 std::ostream_iterator<double>(std::cout, "\n"));
158
159 /*`Or we can use Boost.Multiprecision to generate 50 decimal digit roots of ['J[sub v]]
160 for non-integral order `v= 71/19 == 3.736842`, expressed as an exact-integer fraction
161 to generate the most accurate value possible for all floating-point types.
162
163 We set the precision of the output stream, and show trailing zeros to display a fixed 50 decimal digits.
164 */
165 std::cout.precision(std::numeric_limits<float_type>::digits10); // 50 decimal digits.
166 std::cout << std::showpoint << std::endl; // Show trailing zeros.
167
168 float_type x = float_type(71) / 19;
169 float_type r = boost::math::cyl_bessel_j_zero(x, 1); // 1st root.
170 std::cout << "x = " << x << ", r = " << r << std::endl;
171
172 r = boost::math::cyl_bessel_j_zero(x, 20U); // 20th root.
173 std::cout << "x = " << x << ", r = " << r << std::endl;
174
175 std::vector<float_type> zeros;
176 boost::math::cyl_bessel_j_zero(x, 1, 3, std::back_inserter(zeros));
177
178 std::cout << "cyl_bessel_j_zeros" << std::endl;
179 // Print the roots to the output stream.
180 std::copy(zeros.begin(), zeros.end(),
181 std::ostream_iterator<float_type>(std::cout, "\n"));
182 //] [/bessel_zeros_example_2]
183 }
184 catch (std::exception const& ex)
185 {
186 std::cout << "Thrown exception " << ex.what() << std::endl;
187 }
188
189 } // int main()
190
191 /*
192
193 Output:
194
195 Description: Autorun "J:\Cpp\big_number\Debug\bessel_zeros_example_1.exe"
196 boost::math::cyl_bessel_j_zero(-1.0, 1) 3.83171
197 boost::math::cyl_bessel_j_zero(inf, 1) 1.#QNAN
198 boost::math::cyl_bessel_j_zero(nan, 1) 1.#QNAN
199 5.13562
200 8.41724
201 11.6198
202 14.796
203 17.9598
204
205 x = 3.7368421052631578947368421052631578947368421052632, r = 7.2731751938316489503185694262290765588963196701623
206 x = 3.7368421052631578947368421052631578947368421052632, r = 67.815145619696290925556791375555951165111460585458
207 cyl_bessel_j_zeros
208 7.2731751938316489503185694262290765588963196701623
209 10.724858308883141732536172745851416647110749599085
210 14.018504599452388106120459558042660282427471931581
211
212 */
213
214