264 lines
7.4 KiB
C++
264 lines
7.4 KiB
C++
/*
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* Copyright Nick Thompson, 2017
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* Use, modification and distribution are subject to the
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* Boost Software License, Version 1.0. (See accompanying file
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* LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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*/
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#define BOOST_TEST_MODULE chebyshev_transform_test
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#include <boost/cstdfloat.hpp>
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#include <boost/type_index.hpp>
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#include <boost/test/included/unit_test.hpp>
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#include <boost/test/tools/floating_point_comparison.hpp>
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#include <boost/math/special_functions/chebyshev.hpp>
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#include <boost/math/special_functions/chebyshev_transform.hpp>
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#include <boost/math/special_functions/sinc.hpp>
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#include <boost/multiprecision/cpp_bin_float.hpp>
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#include <boost/multiprecision/cpp_dec_float.hpp>
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#if !defined(TEST1) && !defined(TEST2) && !defined(TEST3) && !defined(TEST4)
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# define TEST1
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# define TEST2
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# define TEST3
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# define TEST4
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#endif
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using boost::multiprecision::cpp_bin_float_quad;
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using boost::multiprecision::cpp_bin_float_50;
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using boost::multiprecision::cpp_bin_float_100;
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using boost::math::chebyshev_t;
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using boost::math::chebyshev_t_prime;
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using boost::math::chebyshev_u;
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using boost::math::chebyshev_transform;
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template<class Real>
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void test_sin_chebyshev_transform()
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{
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using boost::math::chebyshev_transform;
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using boost::math::constants::half_pi;
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using std::sin;
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using std::cos;
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using std::abs;
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Real tol = 10*std::numeric_limits<Real>::epsilon();
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auto f = [](Real x) { return sin(x); };
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Real a = 0;
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Real b = 1;
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chebyshev_transform<Real> cheb(f, a, b, tol);
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Real x = a;
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while (x < b)
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{
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Real s = sin(x);
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Real c = cos(x);
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if (abs(s) < tol)
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{
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BOOST_CHECK_SMALL(cheb(x), 100*tol);
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BOOST_CHECK_CLOSE_FRACTION(c, cheb.prime(x), 100*tol);
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}
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else
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{
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BOOST_CHECK_CLOSE_FRACTION(s, cheb(x), 100*tol);
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if (abs(c) < tol)
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{
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BOOST_CHECK_SMALL(cheb.prime(x), 100*tol);
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}
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else
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{
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BOOST_CHECK_CLOSE_FRACTION(c, cheb.prime(x), 100*tol);
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}
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}
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x += static_cast<Real>(1)/static_cast<Real>(1 << 7);
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}
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Real Q = cheb.integrate();
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BOOST_CHECK_CLOSE_FRACTION(1 - cos(static_cast<Real>(1)), Q, 100*tol);
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}
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template<class Real>
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void test_sinc_chebyshev_transform()
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{
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using std::cos;
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using std::sin;
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using std::abs;
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using boost::math::sinc_pi;
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using boost::math::chebyshev_transform;
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using boost::math::constants::half_pi;
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Real tol = 500*std::numeric_limits<Real>::epsilon();
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auto f = [](Real x) { return boost::math::sinc_pi(x); };
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Real a = 0;
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Real b = 1;
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chebyshev_transform<Real> cheb(f, a, b, tol/50);
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Real x = a;
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while (x < b)
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{
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Real s = sinc_pi(x);
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Real ds = (cos(x)-sinc_pi(x))/x;
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if (x == 0) { ds = 0; }
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if (s < tol)
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{
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BOOST_CHECK_SMALL(cheb(x), tol);
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}
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else
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{
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BOOST_CHECK_CLOSE_FRACTION(s, cheb(x), tol);
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}
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if (abs(ds) < tol)
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{
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BOOST_CHECK_SMALL(cheb.prime(x), 5 * tol);
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}
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else
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{
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BOOST_CHECK_CLOSE_FRACTION(ds, cheb.prime(x), 300*tol);
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}
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x += static_cast<Real>(1)/static_cast<Real>(1 << 7);
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}
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Real Q = cheb.integrate();
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//NIntegrate[Sinc[x], {x, 0, 1}, WorkingPrecision -> 200, AccuracyGoal -> 150, PrecisionGoal -> 150, MaxRecursion -> 150]
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Real Q_exp = boost::lexical_cast<Real>("0.94608307036718301494135331382317965781233795473811179047145477356668");
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BOOST_CHECK_CLOSE_FRACTION(Q_exp, Q, tol);
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}
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//Examples taken from "Approximation Theory and Approximation Practice", by Trefethen
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template<class Real>
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void test_atap_examples()
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{
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using std::sin;
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using boost::math::constants::half;
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using boost::math::sinc_pi;
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using boost::math::chebyshev_transform;
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using boost::math::constants::half_pi;
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Real tol = 10*std::numeric_limits<Real>::epsilon();
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auto f1 = [](Real x) { return ((0 < x) - (x < 0)) - x/2; };
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auto f2 = [](Real x) { Real t = sin(6*x); Real s = sin(x + exp(2*x));
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Real u = (0 < s) - (s < 0);
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return t + u; };
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auto f3 = [](Real x) { return sin(6*x) + sin(60*exp(x)); };
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auto f4 = [](Real x) { return 1/(1+1000*(x+half<Real>())*(x+half<Real>())) + 1/sqrt(1+1000*(x-.5)*(x-0.5));};
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Real a = -1;
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Real b = 1;
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chebyshev_transform<Real> cheb1(f1, a, b);
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chebyshev_transform<Real> cheb2(f2, a, b, tol);
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//chebyshev_transform<Real> cheb3(f3, a, b, tol);
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Real x = a;
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while (x < b)
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{
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//Real s = f1(x);
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if (sizeof(Real) == sizeof(float))
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{
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BOOST_CHECK_CLOSE_FRACTION(f1(x), cheb1(x), 4e-3);
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}
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else
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{
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BOOST_CHECK_CLOSE_FRACTION(f1(x), cheb1(x), 1.3e-5);
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}
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BOOST_CHECK_CLOSE_FRACTION(f2(x), cheb2(x), 6e-3);
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//BOOST_CHECK_CLOSE_FRACTION(f3(x), cheb3(x), 100*tol);
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x += static_cast<Real>(1)/static_cast<Real>(1 << 7);
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}
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}
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//Validate that the Chebyshev polynomials are well approximated by the Chebyshev transform.
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template<class Real>
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void test_chebyshev_chebyshev_transform()
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{
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Real tol = 500*std::numeric_limits<Real>::epsilon();
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// T_0 = 1:
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auto t0 = [](Real) { return 1; };
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chebyshev_transform<Real> cheb0(t0, -1, 1);
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BOOST_CHECK_CLOSE_FRACTION(cheb0.coefficients()[0], 2, tol);
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Real x = -1;
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while (x < 1)
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{
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BOOST_CHECK_CLOSE_FRACTION(cheb0(x), 1, tol);
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BOOST_CHECK_SMALL(cheb0.prime(x), tol);
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x += static_cast<Real>(1)/static_cast<Real>(1 << 7);
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}
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// T_1 = x:
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auto t1 = [](Real x) { return x; };
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chebyshev_transform<Real> cheb1(t1, -1, 1);
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BOOST_CHECK_CLOSE_FRACTION(cheb1.coefficients()[1], 1, tol);
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x = -1;
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while (x < 1)
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{
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if (x == 0)
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{
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BOOST_CHECK_SMALL(cheb1(x), tol);
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}
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else
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{
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BOOST_CHECK_CLOSE_FRACTION(cheb1(x), x, tol);
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}
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BOOST_CHECK_CLOSE_FRACTION(cheb1.prime(x), 1, tol);
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x += static_cast<Real>(1)/static_cast<Real>(1 << 7);
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}
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auto t2 = [](Real x) { return 2*x*x-1; };
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chebyshev_transform<Real> cheb2(t2, -1, 1);
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BOOST_CHECK_CLOSE_FRACTION(cheb2.coefficients()[2], 1, tol);
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x = -1;
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while (x < 1)
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{
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BOOST_CHECK_CLOSE_FRACTION(cheb2(x), t2(x), tol);
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if (x != 0)
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{
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BOOST_CHECK_CLOSE_FRACTION(cheb2.prime(x), 4*x, tol);
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}
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else
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{
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BOOST_CHECK_SMALL(cheb2.prime(x), tol);
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}
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x += static_cast<Real>(1)/static_cast<Real>(1 << 7);
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}
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}
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BOOST_AUTO_TEST_CASE(chebyshev_transform_test)
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{
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#ifdef TEST1
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test_chebyshev_chebyshev_transform<float>();
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test_sin_chebyshev_transform<float>();
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test_atap_examples<float>();
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test_sinc_chebyshev_transform<float>();
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#endif
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#ifdef TEST2
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test_chebyshev_chebyshev_transform<double>();
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test_sin_chebyshev_transform<double>();
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test_atap_examples<double>();
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test_sinc_chebyshev_transform<double>();
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#endif
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#ifdef TEST3
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test_chebyshev_chebyshev_transform<long double>();
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test_sin_chebyshev_transform<long double>();
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test_atap_examples<long double>();
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test_sinc_chebyshev_transform<long double>();
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#endif
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#ifdef TEST4
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#ifdef BOOST_HAS_FLOAT128
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test_chebyshev_chebyshev_transform<__float128>();
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test_sin_chebyshev_transform<__float128>();
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test_atap_examples<__float128>();
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test_sinc_chebyshev_transform<__float128>();
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#endif
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#endif
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}
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