123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213 |
- // Copyright Christopher Kormanyos 2013.
- // Copyright Paul A. Bristow 2013.
- // Copyright John Maddock 2013.
- // Distributed under the Boost Software License, Version 1.0.
- // (See accompanying file LICENSE_1_0.txt or
- // copy at http://www.boost.org/LICENSE_1_0.txt).
- #ifdef _MSC_VER
- # pragma warning (disable : 4512) // assignment operator could not be generated.
- # pragma warning (disable : 4996) // assignment operator could not be generated.
- #endif
- #include <iostream>
- #include <limits>
- #include <vector>
- #include <algorithm>
- #include <iomanip>
- #include <iterator>
- // Weisstein, Eric W. "Bessel Function Zeros." From MathWorld--A Wolfram Web Resource.
- // http://mathworld.wolfram.com/BesselFunctionZeros.html
- // Test values can be calculated using [@wolframalpha.com WolframAplha]
- // See also http://dlmf.nist.gov/10.21
- //[bessel_zeros_example_1
- /*`This example demonstrates calculating zeros of the Bessel and Neumann functions.
- It also shows how Boost.Math and Boost.Multiprecision can be combined to provide
- a many decimal digit precision. For 50 decimal digit precision we need to include
- */
- #include <boost/multiprecision/cpp_dec_float.hpp>
- /*`and a `typedef` for `float_type` may be convenient
- (allowing a quick switch to re-compute at built-in `double` or other precision)
- */
- typedef boost::multiprecision::cpp_dec_float_50 float_type;
- //`To use the functions for finding zeros of the functions we need
- #include <boost/math/special_functions/bessel.hpp>
- //`This file includes the forward declaration signatures for the zero-finding functions:
- // #include <boost/math/special_functions/math_fwd.hpp>
- /*`but more details are in the full documentation, for example at
- [@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].
- */
- /*`This example shows obtaining both a single zero of the Bessel function,
- and then placing multiple zeros into a container like `std::vector` by providing an iterator.
- */
- //] [/bessel_zeros_example_1]
- /*The signature of the single value function is:
- template <class T>
- inline typename detail::bessel_traits<T, T, policies::policy<> >::result_type
- cyl_bessel_j_zero(
- T v, // Floating-point value for Jv.
- int m); // start index.
- The result type is controlled by the floating-point type of parameter `v`
- (but subject to the usual __precision_policy and __promotion_policy).
- The signature of multiple zeros function is:
- template <class T, class OutputIterator>
- inline OutputIterator cyl_bessel_j_zero(
- T v, // Floating-point value for Jv.
- int start_index, // 1-based start index.
- unsigned number_of_zeros, // How many zeros to generate
- OutputIterator out_it); // Destination for zeros.
- There is also a version which allows control of the __policy_section for error handling and precision.
- template <class T, class OutputIterator, class Policy>
- inline OutputIterator cyl_bessel_j_zero(
- T v, // Floating-point value for Jv.
- int start_index, // 1-based start index.
- unsigned number_of_zeros, // How many zeros to generate
- OutputIterator out_it, // Destination for zeros.
- const Policy& pol); // Policy to use.
- */
- int main()
- {
- try
- {
- //[bessel_zeros_example_2
- /*`[tip It is always wise to place code using Boost.Math inside try'n'catch blocks;
- this will ensure that helpful error messages are shown when exceptional conditions arise.]
- First, evaluate a single Bessel zero.
- The precision is controlled by the float-point type of template parameter `T` of `v`
- so this example has `double` precision, at least 15 but up to 17 decimal digits (for the common 64-bit double).
- */
- // double root = boost::math::cyl_bessel_j_zero(0.0, 1);
- // // Displaying with default precision of 6 decimal digits:
- // std::cout << "boost::math::cyl_bessel_j_zero(0.0, 1) " << root << std::endl; // 2.40483
- // // And with all the guaranteed (15) digits:
- // std::cout.precision(std::numeric_limits<double>::digits10);
- // std::cout << "boost::math::cyl_bessel_j_zero(0.0, 1) " << root << std::endl; // 2.40482555769577
- /*`But note that because the parameter `v` controls the precision of the result,
- `v` [*must be a floating-point type].
- So if you provide an integer type, say 0, rather than 0.0, then it will fail to compile thus:
- ``
- root = boost::math::cyl_bessel_j_zero(0, 1);
- ``
- with this error message
- ``
- error C2338: Order must be a floating-point type.
- ``
- Optionally, we can use a policy to ignore errors, C-style, returning some value,
- perhaps infinity or NaN, or the best that can be done. (See __user_error_handling).
- To create a (possibly unwise!) policy `ignore_all_policy` that ignores all errors:
- */
- typedef boost::math::policies::policy<
- boost::math::policies::domain_error<boost::math::policies::ignore_error>,
- boost::math::policies::overflow_error<boost::math::policies::ignore_error>,
- boost::math::policies::underflow_error<boost::math::policies::ignore_error>,
- boost::math::policies::denorm_error<boost::math::policies::ignore_error>,
- boost::math::policies::pole_error<boost::math::policies::ignore_error>,
- boost::math::policies::evaluation_error<boost::math::policies::ignore_error>
- > ignore_all_policy;
- //`Examples of use of this `ignore_all_policy` are
- double inf = std::numeric_limits<double>::infinity();
- double nan = std::numeric_limits<double>::quiet_NaN();
- double dodgy_root = boost::math::cyl_bessel_j_zero(-1.0, 1, ignore_all_policy());
- std::cout << "boost::math::cyl_bessel_j_zero(-1.0, 1) " << dodgy_root << std::endl; // 1.#QNAN
- double inf_root = boost::math::cyl_bessel_j_zero(inf, 1, ignore_all_policy());
- std::cout << "boost::math::cyl_bessel_j_zero(inf, 1) " << inf_root << std::endl; // 1.#QNAN
- double nan_root = boost::math::cyl_bessel_j_zero(nan, 1, ignore_all_policy());
- std::cout << "boost::math::cyl_bessel_j_zero(nan, 1) " << nan_root << std::endl; // 1.#QNAN
- /*`Another version of `cyl_bessel_j_zero` allows calculation of multiple zeros with one call,
- placing the results in a container, often `std::vector`.
- For example, generate and display the first five `double` roots of J[sub v] for integral order 2,
- as column ['J[sub 2](x)] in table 1 of
- [@ http://mathworld.wolfram.com/BesselFunctionZeros.html Wolfram Bessel Function Zeros].
- */
- unsigned int n_roots = 5U;
- std::vector<double> roots;
- boost::math::cyl_bessel_j_zero(2.0, 1, n_roots, std::back_inserter(roots));
- std::copy(roots.begin(),
- roots.end(),
- std::ostream_iterator<double>(std::cout, "\n"));
- /*`Or we can use Boost.Multiprecision to generate 50 decimal digit roots of ['J[sub v]]
- for non-integral order `v= 71/19 == 3.736842`, expressed as an exact-integer fraction
- to generate the most accurate value possible for all floating-point types.
- We set the precision of the output stream, and show trailing zeros to display a fixed 50 decimal digits.
- */
- std::cout.precision(std::numeric_limits<float_type>::digits10); // 50 decimal digits.
- std::cout << std::showpoint << std::endl; // Show trailing zeros.
- float_type x = float_type(71) / 19;
- float_type r = boost::math::cyl_bessel_j_zero(x, 1); // 1st root.
- std::cout << "x = " << x << ", r = " << r << std::endl;
- r = boost::math::cyl_bessel_j_zero(x, 20U); // 20th root.
- std::cout << "x = " << x << ", r = " << r << std::endl;
- std::vector<float_type> zeros;
- boost::math::cyl_bessel_j_zero(x, 1, 3, std::back_inserter(zeros));
- std::cout << "cyl_bessel_j_zeros" << std::endl;
- // Print the roots to the output stream.
- std::copy(zeros.begin(), zeros.end(),
- std::ostream_iterator<float_type>(std::cout, "\n"));
- //] [/bessel_zeros_example_2]
- }
- catch (std::exception const& ex)
- {
- std::cout << "Thrown exception " << ex.what() << std::endl;
- }
- } // int main()
- /*
- Output:
- Description: Autorun "J:\Cpp\big_number\Debug\bessel_zeros_example_1.exe"
- boost::math::cyl_bessel_j_zero(-1.0, 1) 3.83171
- boost::math::cyl_bessel_j_zero(inf, 1) 1.#QNAN
- boost::math::cyl_bessel_j_zero(nan, 1) 1.#QNAN
- 5.13562
- 8.41724
- 11.6198
- 14.796
- 17.9598
-
- x = 3.7368421052631578947368421052631578947368421052632, r = 7.2731751938316489503185694262290765588963196701623
- x = 3.7368421052631578947368421052631578947368421052632, r = 67.815145619696290925556791375555951165111460585458
- cyl_bessel_j_zeros
- 7.2731751938316489503185694262290765588963196701623
- 10.724858308883141732536172745851416647110749599085
- 14.018504599452388106120459558042660282427471931581
- */
|