EQ2EMu/EQ2/source/depends/boost_1_72_0/libs/math/test/chebyshev_transform_test.cpp

/*
 * 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)
 */
#define BOOST_TEST_MODULE chebyshev_transform_test

#include <boost/cstdfloat.hpp>
#include <boost/type_index.hpp>
#include <boost/test/included/unit_test.hpp>
#include <boost/test/tools/floating_point_comparison.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/multiprecision/cpp_bin_float.hpp>
#include <boost/multiprecision/cpp_dec_float.hpp>

#if !defined(TEST1) && !defined(TEST2) && !defined(TEST3) && !defined(TEST4)
#  define TEST1
#  define TEST2
#  define TEST3
#  define TEST4
#endif

using boost::multiprecision::cpp_bin_float_quad;
using boost::multiprecision::cpp_bin_float_50;
using boost::multiprecision::cpp_bin_float_100;
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 = 10*std::numeric_limits<Real>::epsilon();
    auto f = [](Real x) { 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);
        if (abs(s) < tol)
        {
            BOOST_CHECK_SMALL(cheb(x), 100*tol);
            BOOST_CHECK_CLOSE_FRACTION(c, cheb.prime(x), 100*tol);
        }
        else
        {
            BOOST_CHECK_CLOSE_FRACTION(s, cheb(x), 100*tol);
            if (abs(c) < tol)
            {
                BOOST_CHECK_SMALL(cheb.prime(x), 100*tol);
            }
            else
            {
                BOOST_CHECK_CLOSE_FRACTION(c, cheb.prime(x), 100*tol);
            }
        }
        x += static_cast<Real>(1)/static_cast<Real>(1 << 7);
    }

    Real Q = cheb.integrate();

    BOOST_CHECK_CLOSE_FRACTION(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 = 500*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; }
        if (s < tol)
        {
            BOOST_CHECK_SMALL(cheb(x), tol);
        }
        else
        {
            BOOST_CHECK_CLOSE_FRACTION(s, cheb(x), tol);
        }

        if (abs(ds) < tol)
        {
            BOOST_CHECK_SMALL(cheb.prime(x), 5 * tol);
        }
        else
        {
            BOOST_CHECK_CLOSE_FRACTION(ds, cheb.prime(x), 300*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::lexical_cast<Real>("0.94608307036718301494135331382317965781233795473811179047145477356668");
    BOOST_CHECK_CLOSE_FRACTION(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 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-.5)*(x-0.5));};
    Real a = -1;
    Real b = 1;
    chebyshev_transform<Real> cheb1(f1, a, b);
    chebyshev_transform<Real> cheb2(f2, a, b, tol);
    //chebyshev_transform<Real> cheb3(f3, a, b, tol);

    Real x = a;
    while (x < b)
    {
        //Real s = f1(x);
        if (sizeof(Real) == sizeof(float))
        {
           BOOST_CHECK_CLOSE_FRACTION(f1(x), cheb1(x), 4e-3);
        }
        else
        {
           BOOST_CHECK_CLOSE_FRACTION(f1(x), cheb1(x), 1.3e-5);
        }
        BOOST_CHECK_CLOSE_FRACTION(f2(x), cheb2(x), 6e-3);
        //BOOST_CHECK_CLOSE_FRACTION(f3(x), cheb3(x), 100*tol);
        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);
    BOOST_CHECK_CLOSE_FRACTION(cheb0.coefficients()[0], 2, tol);

    Real x = -1;
    while (x < 1)
    {
        BOOST_CHECK_CLOSE_FRACTION(cheb0(x), 1, tol);
        BOOST_CHECK_SMALL(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);
    BOOST_CHECK_CLOSE_FRACTION(cheb1.coefficients()[1], 1, tol);

    x = -1;
    while (x < 1)
    {
        if (x == 0)
        {
            BOOST_CHECK_SMALL(cheb1(x), tol);
        }
        else
        {
            BOOST_CHECK_CLOSE_FRACTION(cheb1(x), x, tol);
        }
        BOOST_CHECK_CLOSE_FRACTION(cheb1.prime(x), 1, 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);
    BOOST_CHECK_CLOSE_FRACTION(cheb2.coefficients()[2], 1, tol);

    x = -1;
    while (x < 1)
    {
        BOOST_CHECK_CLOSE_FRACTION(cheb2(x), t2(x), tol);
        if (x != 0)
        {
            BOOST_CHECK_CLOSE_FRACTION(cheb2.prime(x), 4*x, tol);
        }
        else
        {
            BOOST_CHECK_SMALL(cheb2.prime(x), tol);
        }
        x += static_cast<Real>(1)/static_cast<Real>(1 << 7);
    }
}

BOOST_AUTO_TEST_CASE(chebyshev_transform_test)
{
#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
}