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libs/math/example/airy_zeros_example.cpp


// Copyright Paul A. Bristow 2013.

// (See accompanying file LICENSE_1_0.txt or

#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]

//[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

*/

`