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libs/math/example/bessel_zeros_example_1.cpp

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+
+// 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 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
+
+*/
+