libs/math/example/airy_zeros_example.cpp
// 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 //[airy_zeros_example_1 /*`This example demonstrates calculating zeros of the Airy 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/airy.hpp> /*`This example shows obtaining both a single zero of the Airy functions, and then placing multiple zeros into a container like `std::vector` by providing an iterator. The signature of the single-value Airy Ai function is: template <class T> T airy_ai_zero(unsigned m); // 1-based index of the zero. The signature of multiple zeros Airy Ai function is: template <class T, class OutputIterator> OutputIterator airy_ai_zero( unsigned start_index, // 1-based index of the zero. unsigned number_of_zeros, // How many zeros to generate. OutputIterator out_it); // Destination for zeros. There are also versions which allows control of the __policy_section for error handling and precision. template <class T, class OutputIterator, class Policy> OutputIterator airy_ai_zero( unsigned start_index, // 1-based index of the zero. unsigned number_of_zeros, // How many zeros to generate. OutputIterator out_it, // Destination for zeros. const Policy& pol); // Policy to use. */ //] [/airy_zeros_example_1] int main() { try { //[airy_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 Airy zero. The precision is controlled by the template parameter `T`, so this example has `double` precision, at least 15 but up to 17 decimal digits (for the common 64-bit double). */ double aiz1 = boost::math::airy_ai_zero<double>(1); std::cout << "boost::math::airy_ai_zero<double>(1) = " << aiz1 << std::endl; double aiz2 = boost::math::airy_ai_zero<double>(2); std::cout << "boost::math::airy_ai_zero<double>(2) = " << aiz2 << std::endl; double biz3 = boost::math::airy_bi_zero<double>(3); std::cout << "boost::math::airy_bi_zero<double>(3) = " << biz3 << std::endl; /*`Other versions of `airy_ai_zero` and `airy_bi_zero` allow 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 [@http://mathworld.wolfram.com/AiryFunctionZeros.html Wolfram Airy Functions Zeros]. */ unsigned int n_roots = 5U; std::vector<double> roots; boost::math::airy_ai_zero<double>(1U, n_roots, std::back_inserter(roots)); std::cout << "airy_ai_zeros:" << std::endl; std::copy(roots.begin(), roots.end(), std::ostream_iterator<double>(std::cout, "\n")); /*`The first few real roots of Ai(x) are approximately -2.33811, -4.08795, -5.52056, -6.7867144, -7.94413, -9.02265 ... Or we can use Boost.Multiprecision to generate 50 decimal digit roots. 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); // float_type has 50 decimal digits. std::cout << std::showpoint << std::endl; // Show trailing zeros too. unsigned int m = 1U; float_type r = boost::math::airy_ai_zero<float_type>(1U); // 1st root. std::cout << "boost::math::airy_bi_zero<float_type>(" << m << ") = " << r << std::endl; m = 2; r = boost::math::airy_ai_zero<float_type>(2U); // 2nd root. std::cout << "boost::math::airy_bi_zero<float_type>(" << m << ") = " << r << std::endl; m = 7U; r = boost::math::airy_bi_zero<float_type>(7U); // 7th root. std::cout << "boost::math::airy_bi_zero<float_type>(" << m << ") = " << r << std::endl; std::vector<float_type> zeros; boost::math::airy_ai_zero<float_type>(1U, 3, std::back_inserter(zeros)); std::cout << "airy_ai_zeros:" << std::endl; // Print the roots to the output stream. std::copy(zeros.begin(), zeros.end(), std::ostream_iterator<float_type>(std::cout, "\n")); //] [/airy_zeros_example_2] } catch (std::exception ex) { std::cout << "Thrown exception " << ex.what() << std::endl; } } // int main() /* Output: Description: Autorun "J:\Cpp\big_number\Debug\airy_zeros_example.exe" boost::math::airy_ai_zero<double>(1) = -2.33811 boost::math::airy_ai_zero<double>(2) = -4.08795 boost::math::airy_bi_zero<double>(3) = -4.83074 airy_ai_zeros: -2.33811 -4.08795 -5.52056 -6.78671 -7.94413 boost::math::airy_bi_zero<float_type>(1) = -2.3381074104597670384891972524467354406385401456711 boost::math::airy_bi_zero<float_type>(2) = -4.0879494441309706166369887014573910602247646991085 boost::math::airy_bi_zero<float_type>(7) = -9.5381943793462388866329885451560196208390720763825 airy_ai_zeros: -2.3381074104597670384891972524467354406385401456711 -4.0879494441309706166369887014573910602247646991085 -5.5205598280955510591298555129312935737972142806175 */