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214 lines
8.8 KiB
C++
214 lines
8.8 KiB
C++
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// Copyright Christopher Kormanyos 2013.
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// Copyright Paul A. Bristow 2013.
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// Copyright John Maddock 2013.
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// Distributed under the Boost Software License, Version 1.0.
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// (See accompanying file LICENSE_1_0.txt or
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// copy at http://www.boost.org/LICENSE_1_0.txt).
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#ifdef _MSC_VER
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# pragma warning (disable : 4512) // assignment operator could not be generated.
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# pragma warning (disable : 4996) // assignment operator could not be generated.
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#endif
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#include <iostream>
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#include <limits>
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#include <vector>
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#include <algorithm>
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#include <iomanip>
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#include <iterator>
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// Weisstein, Eric W. "Bessel Function Zeros." From MathWorld--A Wolfram Web Resource.
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// http://mathworld.wolfram.com/BesselFunctionZeros.html
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// Test values can be calculated using [@wolframalpha.com WolframAplha]
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// See also http://dlmf.nist.gov/10.21
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//[bessel_zeros_example_1
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/*`This example demonstrates calculating zeros of the Bessel and Neumann functions.
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It also shows how Boost.Math and Boost.Multiprecision can be combined to provide
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a many decimal digit precision. For 50 decimal digit precision we need to include
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*/
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#include <boost/multiprecision/cpp_dec_float.hpp>
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/*`and a `typedef` for `float_type` may be convenient
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(allowing a quick switch to re-compute at built-in `double` or other precision)
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*/
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typedef boost::multiprecision::cpp_dec_float_50 float_type;
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//`To use the functions for finding zeros of the functions we need
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#include <boost/math/special_functions/bessel.hpp>
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//`This file includes the forward declaration signatures for the zero-finding functions:
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// #include <boost/math/special_functions/math_fwd.hpp>
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/*`but more details are in the full documentation, for example at
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[@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].
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*/
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/*`This example shows obtaining both a single zero of the Bessel function,
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and then placing multiple zeros into a container like `std::vector` by providing an iterator.
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*/
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//] [/bessel_zeros_example_1]
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/*The signature of the single value function is:
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template <class T>
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inline typename detail::bessel_traits<T, T, policies::policy<> >::result_type
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cyl_bessel_j_zero(
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T v, // Floating-point value for Jv.
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int m); // start index.
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The result type is controlled by the floating-point type of parameter `v`
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(but subject to the usual __precision_policy and __promotion_policy).
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The signature of multiple zeros function is:
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template <class T, class OutputIterator>
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inline OutputIterator cyl_bessel_j_zero(
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T v, // Floating-point value for Jv.
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int start_index, // 1-based start index.
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unsigned number_of_zeros, // How many zeros to generate
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OutputIterator out_it); // Destination for zeros.
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There is also a version which allows control of the __policy_section for error handling and precision.
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template <class T, class OutputIterator, class Policy>
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inline OutputIterator cyl_bessel_j_zero(
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T v, // Floating-point value for Jv.
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int start_index, // 1-based start index.
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unsigned number_of_zeros, // How many zeros to generate
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OutputIterator out_it, // Destination for zeros.
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const Policy& pol); // Policy to use.
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*/
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int main()
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{
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try
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{
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//[bessel_zeros_example_2
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/*`[tip It is always wise to place code using Boost.Math inside try'n'catch blocks;
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this will ensure that helpful error messages are shown when exceptional conditions arise.]
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First, evaluate a single Bessel zero.
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The precision is controlled by the float-point type of template parameter `T` of `v`
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so this example has `double` precision, at least 15 but up to 17 decimal digits (for the common 64-bit double).
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*/
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// double root = boost::math::cyl_bessel_j_zero(0.0, 1);
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// // Displaying with default precision of 6 decimal digits:
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// std::cout << "boost::math::cyl_bessel_j_zero(0.0, 1) " << root << std::endl; // 2.40483
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// // And with all the guaranteed (15) digits:
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// std::cout.precision(std::numeric_limits<double>::digits10);
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// std::cout << "boost::math::cyl_bessel_j_zero(0.0, 1) " << root << std::endl; // 2.40482555769577
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/*`But note that because the parameter `v` controls the precision of the result,
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`v` [*must be a floating-point type].
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So if you provide an integer type, say 0, rather than 0.0, then it will fail to compile thus:
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``
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root = boost::math::cyl_bessel_j_zero(0, 1);
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``
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with this error message
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``
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error C2338: Order must be a floating-point type.
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``
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Optionally, we can use a policy to ignore errors, C-style, returning some value,
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perhaps infinity or NaN, or the best that can be done. (See __user_error_handling).
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To create a (possibly unwise!) policy `ignore_all_policy` that ignores all errors:
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*/
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typedef boost::math::policies::policy<
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boost::math::policies::domain_error<boost::math::policies::ignore_error>,
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boost::math::policies::overflow_error<boost::math::policies::ignore_error>,
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boost::math::policies::underflow_error<boost::math::policies::ignore_error>,
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boost::math::policies::denorm_error<boost::math::policies::ignore_error>,
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boost::math::policies::pole_error<boost::math::policies::ignore_error>,
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boost::math::policies::evaluation_error<boost::math::policies::ignore_error>
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> ignore_all_policy;
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//`Examples of use of this `ignore_all_policy` are
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double inf = std::numeric_limits<double>::infinity();
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double nan = std::numeric_limits<double>::quiet_NaN();
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double dodgy_root = boost::math::cyl_bessel_j_zero(-1.0, 1, ignore_all_policy());
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std::cout << "boost::math::cyl_bessel_j_zero(-1.0, 1) " << dodgy_root << std::endl; // 1.#QNAN
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double inf_root = boost::math::cyl_bessel_j_zero(inf, 1, ignore_all_policy());
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std::cout << "boost::math::cyl_bessel_j_zero(inf, 1) " << inf_root << std::endl; // 1.#QNAN
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double nan_root = boost::math::cyl_bessel_j_zero(nan, 1, ignore_all_policy());
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std::cout << "boost::math::cyl_bessel_j_zero(nan, 1) " << nan_root << std::endl; // 1.#QNAN
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/*`Another version of `cyl_bessel_j_zero` allows calculation of multiple zeros with one call,
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placing the results in a container, often `std::vector`.
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For example, generate and display the first five `double` roots of J[sub v] for integral order 2,
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as column ['J[sub 2](x)] in table 1 of
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[@ http://mathworld.wolfram.com/BesselFunctionZeros.html Wolfram Bessel Function Zeros].
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*/
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unsigned int n_roots = 5U;
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std::vector<double> roots;
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boost::math::cyl_bessel_j_zero(2.0, 1, n_roots, std::back_inserter(roots));
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std::copy(roots.begin(),
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roots.end(),
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std::ostream_iterator<double>(std::cout, "\n"));
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/*`Or we can use Boost.Multiprecision to generate 50 decimal digit roots of ['J[sub v]]
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for non-integral order `v= 71/19 == 3.736842`, expressed as an exact-integer fraction
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to generate the most accurate value possible for all floating-point types.
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We set the precision of the output stream, and show trailing zeros to display a fixed 50 decimal digits.
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*/
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std::cout.precision(std::numeric_limits<float_type>::digits10); // 50 decimal digits.
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std::cout << std::showpoint << std::endl; // Show trailing zeros.
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float_type x = float_type(71) / 19;
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float_type r = boost::math::cyl_bessel_j_zero(x, 1); // 1st root.
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std::cout << "x = " << x << ", r = " << r << std::endl;
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r = boost::math::cyl_bessel_j_zero(x, 20U); // 20th root.
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std::cout << "x = " << x << ", r = " << r << std::endl;
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std::vector<float_type> zeros;
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boost::math::cyl_bessel_j_zero(x, 1, 3, std::back_inserter(zeros));
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std::cout << "cyl_bessel_j_zeros" << std::endl;
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// Print the roots to the output stream.
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std::copy(zeros.begin(), zeros.end(),
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std::ostream_iterator<float_type>(std::cout, "\n"));
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//] [/bessel_zeros_example_2]
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}
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catch (std::exception ex)
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{
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std::cout << "Thrown exception " << ex.what() << std::endl;
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}
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} // int main()
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/*
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Output:
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Description: Autorun "J:\Cpp\big_number\Debug\bessel_zeros_example_1.exe"
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boost::math::cyl_bessel_j_zero(-1.0, 1) 3.83171
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boost::math::cyl_bessel_j_zero(inf, 1) 1.#QNAN
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boost::math::cyl_bessel_j_zero(nan, 1) 1.#QNAN
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5.13562
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8.41724
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11.6198
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14.796
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17.9598
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x = 3.7368421052631578947368421052631578947368421052632, r = 7.2731751938316489503185694262290765588963196701623
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x = 3.7368421052631578947368421052631578947368421052632, r = 67.815145619696290925556791375555951165111460585458
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cyl_bessel_j_zeros
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7.2731751938316489503185694262290765588963196701623
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10.724858308883141732536172745851416647110749599085
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14.018504599452388106120459558042660282427471931581
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*/
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