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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_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]
- */
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