text: // 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_zero_example_1
  
  /*`This example demonstrates calculating zeros of the Bessel, Neumann and 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/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.
  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,
                                  OutputIterator out_it); // iterator into container 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,
                                  OutputIterator out_it,
                                  const Policy& pol); // iterator into container for zeros.
  
  */
  //]  [/bessel_zero_example_1]
  
  //[bessel_zero_example_iterator_1]
  /*`We use the `cyl_bessel_j_zero` output iterator parameter `out_it`
  to create a sum of 1/zeros[super 2] by defining a custom output iterator:
  */
  
  template <class T>
  struct output_summation_iterator
  {
     output_summation_iterator(T* p) : p_sum(p)
     {}
     output_summation_iterator& operator*()
     { return *this; }
      output_summation_iterator& operator++()
     { return *this; }
     output_summation_iterator& operator++(int)
     { return *this; }
     output_summation_iterator& operator = (T const& val)
     {
       *p_sum += 1./ (val * val); // Summing 1/zero^2.
       return *this;
     }
  private:
     T* p_sum;
  };
  
  
  //] [/bessel_zero_example_iterator_1]
  
  int main()
  {
    try
    {
  //[bessel_zero_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 can be 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 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;
  
      double inf = std::numeric_limits<double>::infinity();
      double nan = std::numeric_limits<double>::quiet_NaN();
  
      std::cout << "boost::math::cyl_bessel_j_zero(-1.0, 0) " << std::endl;
      double dodgy_root = boost::math::cyl_bessel_j_zero(-1.0, 0, 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 five `double` roots of J[sub v] for integral order 2.
  
  showing the same results 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 generate 50 decimal digit roots of J[sub v] for non-integral order `v = 71/19`.
  
  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"));
  
  /*`The Neumann function zeros are evaluated very similarly:
  */
      using boost::math::cyl_neumann_zero;
  
      double zn = cyl_neumann_zero(2., 1);
  
      std::cout << "cyl_neumann_zero(2., 1) = " << std::endl;
      //double zn0 = zn;
      //    std::cout << "zn0 = " << std::endl;
      //    std::cout << zn0 << std::endl;
      //
      std::cout << zn << std::endl;
      //  std::cout << cyl_neumann_zero(2., 1) << std::endl;
  
      std::vector<float> nzeros(3); // Space for 3 zeros.
      cyl_neumann_zero<float>(2.F, 1, nzeros.size(), nzeros.begin());
  
      std::cout << "cyl_neumann_zero<float>(2.F, 1, " << std::endl;
      // Print the zeros to the output stream.
      std::copy(nzeros.begin(), nzeros.end(),
                std::ostream_iterator<float>(std::cout, "\n"));
  
      std::cout << cyl_neumann_zero(static_cast<float_type>(220)/100, 1) << std::endl;
      // 3.6154383428745996706772556069431792744372398748422
  
  /*`Finally we show how the output iterator can be used to compute a sum of zeros.
  
  (See [@http://dx.doi.org/10.1017/S2040618500034067 Ian N. Sneddon, Infinite Sums of Bessel Zeros],
  page 150 equation 40).
  */
  //] [/bessel_zero_example_2]
  
      {
  //[bessel_zero_example_iterator_2]
  /*`The sum is calculated for many values, converging on the analytical exact value of `1/8`.
  */
      using boost::math::cyl_bessel_j_zero;
      double nu = 1.;
      double sum = 0;
      output_summation_iterator<double> it(&sum);  // sum of 1/zeros^2
      cyl_bessel_j_zero(nu, 1, 10000, it);
  
      double s = 1/(4 * (nu + 1)); // 0.125 = 1/8 is exact analytical solution.
      std::cout << std::setprecision(6) << "nu = " << nu << ", sum = " << sum
        << ", exact = " << s << std::endl;
      // nu = 1.00000, sum = 0.124990, exact = 0.125000
  //] [/bessel_zero_example_iterator_2]
      }
    }
    catch (std::exception& ex)
    {
      std::cout << "Thrown exception " << ex.what() << std::endl;
    }
  
  //[bessel_zero_example_iterator_3]
  
  /*`Examples below show effect of 'bad' arguments that throw a `domain_error` exception.
  */
    try
    { // Try a negative rank m.
      std::cout << "boost::math::cyl_bessel_j_zero(-1.F, -1) " << std::endl;
      float dodgy_root = boost::math::cyl_bessel_j_zero(-1.F, -1);
      std::cout << "boost::math::cyl_bessel_j_zero(-1.F, -1) " << dodgy_root << std::endl;
      // Throw exception Error in function boost::math::cyl_bessel_j_zero<double>(double, int):
      // Order argument is -1, but must be >= 0 !
    }
    catch (std::exception& ex)
    {
      std::cout << "Throw exception " << ex.what() << std::endl;
    }
  
  /*`[note The type shown is the type [*after promotion],
  using __precision_policy and __promotion_policy, from `float` to `double` in this case.]
  
  In this example the promotion goes:
  
  # Arguments are `float` and `int`.
  # Treat `int` "as if" it were a `double`, so arguments are `float` and `double`.
  # Common type is `double` - so that's the precision we want (and the type that will be returned).
  # Evaluate internally as `long double` for full `double` precision.
  
  See full code for other examples that promote from `double` to `long double`.
  
  */
  
  //] [/bessel_zero_example_iterator_3]
      try
    { // order v = inf
       std::cout << "boost::math::cyl_bessel_j_zero(infF, 1) " << std::endl;
       float infF = std::numeric_limits<float>::infinity();
       float inf_root = boost::math::cyl_bessel_j_zero(infF, 1);
        std::cout << "boost::math::cyl_bessel_j_zero(infF, 1) " << inf_root << std::endl;
       //  boost::math::cyl_bessel_j_zero(-1.F, -1) 
       //Thrown exception Error in function boost::math::cyl_bessel_j_zero<double>(double, int):
       // Requested the -1'th zero, but the rank must be positive !
    }
    catch (std::exception& ex)
    {
      std::cout << "Thrown exception " << ex.what() << std::endl;
    }
    try
    { // order v = inf
       double inf = std::numeric_limits<double>::infinity();
       double inf_root = boost::math::cyl_bessel_j_zero(inf, 1);
       std::cout << "boost::math::cyl_bessel_j_zero(inf, 1) " << inf_root << std::endl;
       // Throw exception Error in function boost::math::cyl_bessel_j_zero<long double>(long double, unsigned):
       // Order argument is 1.#INF, but must be finite >= 0 !
    }
    catch (std::exception& ex)
    {
      std::cout << "Thrown exception " << ex.what() << std::endl;
    }
  
    try
    { // order v = NaN
       double nan = std::numeric_limits<double>::quiet_NaN();
       double nan_root = boost::math::cyl_bessel_j_zero(nan, 1);
       std::cout << "boost::math::cyl_bessel_j_zero(nan, 1) " << nan_root << std::endl;
       // Throw exception Error in function boost::math::cyl_bessel_j_zero<long double>(long double, unsigned):
       // Order argument is 1.#QNAN, but must be finite >= 0 !
    }
    catch (std::exception& ex)
    {
      std::cout << "Thrown exception " << ex.what() << std::endl;
    }
  
    try
    {   // Try a negative m.
      double dodgy_root = boost::math::cyl_bessel_j_zero(0.0, -1);
      //  warning C4146: unary minus operator applied to unsigned type, result still unsigned.
      std::cout << "boost::math::cyl_bessel_j_zero(0.0, -1) " << dodgy_root << std::endl;
      //  boost::math::cyl_bessel_j_zero(0.0, -1) 6.74652e+009
      // This *should* fail because m is unreasonably large.
  
    }
    catch (std::exception& ex)
    {
      std::cout << "Thrown exception " << ex.what() << std::endl;
    }
  
    try
    { // m = inf
       double inf = std::numeric_limits<double>::infinity();
       double inf_root = boost::math::cyl_bessel_j_zero(0.0, inf);
       // warning C4244: 'argument' : conversion from 'double' to 'int', possible loss of data.
       std::cout << "boost::math::cyl_bessel_j_zero(0.0, inf) " << inf_root << std::endl;
       // Throw exception Error in function boost::math::cyl_bessel_j_zero<long double>(long double, int):
       // Requested the 0'th zero, but must be > 0 !
  
    }
    catch (std::exception& ex)
    {
      std::cout << "Thrown exception " << ex.what() << std::endl;
    }
  
    try
    { // m = NaN
       std::cout << "boost::math::cyl_bessel_j_zero(0.0, nan) " << std::endl ;
       double nan = std::numeric_limits<double>::quiet_NaN();
       double nan_root = boost::math::cyl_bessel_j_zero(0.0, nan);
       // warning C4244: 'argument' : conversion from 'double' to 'int', possible loss of data.
       std::cout << nan_root << std::endl;
       // Throw exception Error in function boost::math::cyl_bessel_j_zero<long double>(long double, int):
       // Requested the 0'th zero, but must be > 0 !
    }
    catch (std::exception& ex)
    {
      std::cout << "Thrown exception " << ex.what() << std::endl;
    }
  
   } // int main()
  
  /*
  Mathematica: Table[N[BesselJZero[71/19, n], 50], {n, 1, 20, 1}]
  
  7.2731751938316489503185694262290765588963196701623
  10.724858308883141732536172745851416647110749599085
  14.018504599452388106120459558042660282427471931581
  17.25249845917041718216248716654977734919590383861
  20.456678874044517595180234083894285885460502077814
  23.64363089714234522494551422714731959985405172504
  26.819671140255087745421311470965019261522390519297
  29.988343117423674742679141796661432043878868194142
  33.151796897690520871250862469973445265444791966114
  36.3114160002162074157243540350393860813165201842
  39.468132467505236587945197808083337887765967032029
  42.622597801391236474855034831297954018844433480227
  45.775281464536847753390206207806726581495950012439
  48.926530489173566198367766817478553992471739894799
  52.076607045343002794279746041878924876873478063472
  55.225712944912571393594224327817265689059002890192
  58.374006101538886436775188150439025201735151418932
  61.521611873000965273726742659353136266390944103571
  64.66863105379093036834648221487366079456596628716
  67.815145619696290925556791375555951165111460585458
  
  Mathematica: Table[N[BesselKZero[2, n], 50], {n, 1, 5, 1}]
  n |
  1 | 3.3842417671495934727014260185379031127323883259329
  2 | 6.7938075132682675382911671098369487124493222183854
  3 | 10.023477979360037978505391792081418280789658279097
  
  
  */
  
   /*
  [bessel_zero_output]
  
    boost::math::cyl_bessel_j_zero(0.0, 1) 2.40483
    boost::math::cyl_bessel_j_zero(0.0, 1) 2.40482555769577
    boost::math::cyl_bessel_j_zero(-1.0, 1) 1.#QNAN
    boost::math::cyl_bessel_j_zero(inf, 1) 1.#QNAN
    boost::math::cyl_bessel_j_zero(nan, 1) 1.#QNAN
    5.13562230184068
    8.41724414039986
    11.6198411721491
    14.7959517823513
    17.9598194949878
  
    x = 3.7368421052631578947368421052631578947368421052632, r = 7.2731751938316489503185694262290765588963196701623
    x = 3.7368421052631578947368421052631578947368421052632, r = 67.815145619696290925556791375555951165111460585458
    7.2731751938316489503185694262290765588963196701623
    10.724858308883141732536172745851416647110749599085
    14.018504599452388106120459558042660282427471931581
    cyl_neumann_zero(2., 1) = 3.3842417671495935000000000000000000000000000000000
    3.3842418193817139000000000000000000000000000000000
    6.7938075065612793000000000000000000000000000000000
    10.023477554321289000000000000000000000000000000000
    3.6154383428745996706772556069431792744372398748422
    nu = 1.00000, sum = 0.124990, exact = 0.125000
    Throw exception Error in function boost::math::cyl_bessel_j_zero<double>(double, int): Order argument is -1, but must be >= 0 !
    Throw exception Error in function boost::math::cyl_bessel_j_zero<long double>(long double, int): Order argument is 1.#INF, but must be finite >= 0 !
    Throw exception Error in function boost::math::cyl_bessel_j_zero<long double>(long double, int): Order argument is 1.#QNAN, but must be finite >= 0 !
    Throw exception Error in function boost::math::cyl_bessel_j_zero<long double>(long double, int): Requested the -1'th zero, but must be > 0 !
    Throw exception Error in function boost::math::cyl_bessel_j_zero<long double>(long double, int): Requested the -2147483648'th zero, but must be > 0 !
    Throw exception Error in function boost::math::cyl_bessel_j_zero<long double>(long double, int): Requested the -2147483648'th zero, but must be > 0 !
  
  
  ] [/bessel_zero_output]
  */
  
  