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/*
* Copyright Nick Thompson, 2017
* Use, modification and distribution are subject to 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)
*/
#include "math_unit_test.hpp"
#include <boost/type_index.hpp>
#include <boost/math/special_functions/chebyshev.hpp>
#include <boost/math/special_functions/chebyshev_transform.hpp>
#include <boost/math/special_functions/sinc.hpp>
#include <boost/math/tools/test_value.hpp>
#if !defined(TEST1) && !defined(TEST2) && !defined(TEST3) && !defined(TEST4)
# define TEST1
# define TEST2
# define TEST3
# define TEST4
#endif
using boost::math::chebyshev_t;
using boost::math::chebyshev_t_prime;
using boost::math::chebyshev_u;
using boost::math::chebyshev_transform;
template<class Real>
void test_sin_chebyshev_transform()
{
using boost::math::chebyshev_transform;
using boost::math::constants::half_pi;
using std::sin;
using std::cos;
using std::abs;
Real tol = std::numeric_limits<Real>::epsilon();
auto f = [](Real x)->Real { return sin(x); };
Real a = 0;
Real b = 1;
chebyshev_transform<Real> cheb(f, a, b, tol);
Real x = a;
while (x < b)
{
Real s = sin(x);
Real c = cos(x);
CHECK_ABSOLUTE_ERROR(s, cheb(x), tol);
CHECK_ABSOLUTE_ERROR(c, cheb.prime(x), 150*tol);
x += static_cast<Real>(1)/static_cast<Real>(1 << 7);
}
Real Q = cheb.integrate();
CHECK_ABSOLUTE_ERROR(1 - cos(static_cast<Real>(1)), Q, 100*tol);
}
template<class Real>
void test_sinc_chebyshev_transform()
{
using std::cos;
using std::sin;
using std::abs;
using boost::math::sinc_pi;
using boost::math::chebyshev_transform;
using boost::math::constants::half_pi;
Real tol = 100*std::numeric_limits<Real>::epsilon();
auto f = [](Real x) { return boost::math::sinc_pi(x); };
Real a = 0;
Real b = 1;
chebyshev_transform<Real> cheb(f, a, b, tol/50);
Real x = a;
while (x < b)
{
Real s = sinc_pi(x);
Real ds = (cos(x)-sinc_pi(x))/x;
if (x == 0) { ds = 0; }
CHECK_ABSOLUTE_ERROR(s, cheb(x), tol);
CHECK_ABSOLUTE_ERROR(ds, cheb.prime(x), 10*tol);
x += static_cast<Real>(1)/static_cast<Real>(1 << 7);
}
Real Q = cheb.integrate();
//NIntegrate[Sinc[x], {x, 0, 1}, WorkingPrecision -> 200, AccuracyGoal -> 150, PrecisionGoal -> 150, MaxRecursion -> 150]
Real Q_exp = BOOST_MATH_TEST_VALUE(Real, 0.94608307036718301494135331382317965781233795473811179047145477356668);
CHECK_ABSOLUTE_ERROR(Q_exp, Q, tol);
}
//Examples taken from "Approximation Theory and Approximation Practice", by Trefethen
template<class Real>
void test_atap_examples()
{
using std::sin;
using std::exp;
using std::sqrt;
using boost::math::constants::half;
using boost::math::sinc_pi;
using boost::math::chebyshev_transform;
using boost::math::constants::half_pi;
Real tol = 10*std::numeric_limits<Real>::epsilon();
auto f1 = [](Real x) { return ((0 < x) - (x < 0)) - x/2; };
auto f2 = [](Real x) { Real t = sin(6*x); Real s = sin(x + exp(2*x));
Real u = (0 < s) - (s < 0);
return t + u; };
//auto f3 = [](Real x) { return sin(6*x) + sin(60*exp(x)); };
//auto f4 = [](Real x) { return 1/(1+1000*(x+half<Real>())*(x+half<Real>())) + 1/sqrt(1+1000*(x-Real(1)/Real(2))*(x-Real(1)/Real(2)));};
Real a = -1;
Real b = 1;
chebyshev_transform<Real> cheb1(f1, a, b, tol);
chebyshev_transform<Real> cheb2(f2, a, b, tol);
//chebyshev_transform<Real> cheb3(f3, a, b, tol);
Real x = a;
while (x < b)
{
// f1 and f2 are not differentiable; standard convergence rate theorems don't apply.
// Basically, the max refinements are always hit; so the error is not related to the precision of the type.
Real acceptable_error = sqrt(tol);
Real acceptable_error_2 = 9e-4;
if (std::is_same<Real, long double>::value)
{
acceptable_error = 1.6e-5;
}
if (std::is_same<Real, double>::value)
{
acceptable_error *= 500;
}
CHECK_ABSOLUTE_ERROR(f1(x), cheb1(x), acceptable_error);
CHECK_ABSOLUTE_ERROR(f2(x), cheb2(x), acceptable_error_2);
x += static_cast<Real>(1)/static_cast<Real>(1 << 7);
}
}
//Validate that the Chebyshev polynomials are well approximated by the Chebyshev transform.
template<class Real>
void test_chebyshev_chebyshev_transform()
{
Real tol = 500*std::numeric_limits<Real>::epsilon();
// T_0 = 1:
auto t0 = [](Real) { return 1; };
chebyshev_transform<Real> cheb0(t0, -1, 1);
CHECK_ABSOLUTE_ERROR(2, cheb0.coefficients()[0], tol);
Real x = -1;
while (x < 1)
{
CHECK_ABSOLUTE_ERROR(1, cheb0(x), tol);
CHECK_ABSOLUTE_ERROR(Real(0), cheb0.prime(x), tol);
x += static_cast<Real>(1)/static_cast<Real>(1 << 7);
}
// T_1 = x:
auto t1 = [](Real x) { return x; };
chebyshev_transform<Real> cheb1(t1, -1, 1);
CHECK_ABSOLUTE_ERROR(Real(1), cheb1.coefficients()[1], tol);
x = -1;
while (x < 1)
{
CHECK_ABSOLUTE_ERROR(x, cheb1(x), tol);
CHECK_ABSOLUTE_ERROR(Real(1), cheb1.prime(x), tol);
x += static_cast<Real>(1)/static_cast<Real>(1 << 7);
}
auto t2 = [](Real x) { return 2*x*x-1; };
chebyshev_transform<Real> cheb2(t2, -1, 1);
CHECK_ABSOLUTE_ERROR(Real(1), cheb2.coefficients()[2], tol);
x = -1;
while (x < 1)
{
CHECK_ABSOLUTE_ERROR(t2(x), cheb2(x), tol);
CHECK_ABSOLUTE_ERROR(4*x, cheb2.prime(x), tol);
x += static_cast<Real>(1)/static_cast<Real>(1 << 7);
}
}
int main()
{
#ifdef TEST1
test_chebyshev_chebyshev_transform<float>();
test_sin_chebyshev_transform<float>();
test_atap_examples<float>();
test_sinc_chebyshev_transform<float>();
#endif
#ifdef TEST2
test_chebyshev_chebyshev_transform<double>();
test_sin_chebyshev_transform<double>();
test_atap_examples<double>();
test_sinc_chebyshev_transform<double>();
#endif
#ifdef TEST3
test_chebyshev_chebyshev_transform<long double>();
test_sin_chebyshev_transform<long double>();
test_atap_examples<long double>();
test_sinc_chebyshev_transform<long double>();
#endif
#ifdef TEST4
#ifdef BOOST_HAS_FLOAT128
test_chebyshev_chebyshev_transform<__float128>();
test_sin_chebyshev_transform<__float128>();
test_atap_examples<__float128>();
test_sinc_chebyshev_transform<__float128>();
#endif
#endif
return boost::math::test::report_errors();
}
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