Adding upstream version 14.2.21.upstream/14.2.21upstream

Signed-off-by: Daniel Baumann <daniel.baumann@progress-linux.org>

1 files changed, 447 insertions, 0 deletions

diff --git a/src/boost/libs/math/example/bessel_zeros_example.cpp b/src/boost/libs/math/example/bessel_zeros_example.cpp
new file mode 100644
index 00000000..0d3ec3cc
--- /dev/null
+++ b/src/boost/libs/math/example/bessel_zeros_example.cpp
@@ -0,0 +1,447 @@
+// 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 [@https://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]
+*/
+