EQ2EMu/EQ2/source/depends/boost_1_72_0/libs/math/test/test_trapezoidal.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 trapezoidal_quadrature

#include <complex>
#include <boost/config.hpp>
//#include <boost/multiprecision/mpc.hpp>
#include <boost/test/included/unit_test.hpp>
#include <boost/test/tools/floating_point_comparison.hpp>
#include <boost/math/concepts/real_concept.hpp>
#include <boost/math/special_functions/bessel.hpp>
#include <boost/math/quadrature/trapezoidal.hpp>
#include <boost/multiprecision/cpp_bin_float.hpp>
#include <boost/multiprecision/cpp_dec_float.hpp>
#ifdef BOOST_HAS_FLOAT128
#include <boost/multiprecision/complex128.hpp>
#endif
using boost::multiprecision::cpp_bin_float_50;
using boost::multiprecision::cpp_bin_float_100;
using boost::math::quadrature::trapezoidal;

// These tests come from:
// https://doi.org/10.1023/A:1025524324969
// "Computing special functions by using quadrature rules",  Gil, Segura, and Temme.
template<class Complex>
void test_complex_bessel()
{
    std::cout << "Testing that complex-valued integrands are integrated correctly by the adaptive trapezoidal routine on type " << boost::typeindex::type_id<Complex>().pretty_name()  << "\n";
    typedef typename Complex::value_type Real;
    Complex z{2, 3};
    int n = 2;
    using boost::math::constants::pi;
    auto bessel_integrand = [&n, &z](Real theta)->Complex
    {
        using std::cos;
        using std::sin;
        Real t1 = sin(theta);
        Real t2 = - n*theta;
        Complex arg = z*t1 + t2;
        return cos(arg)/pi<Real>();
    };

    using boost::math::quadrature::trapezoidal;

    Real a = 0;
    Real b = pi<Real>();
    Complex Jnz = trapezoidal<decltype(bessel_integrand), Real>(bessel_integrand, a, b);
    // N[BesselJ[2, 2 + 3 I], 143]
    // 1.257674591970511077630764085052638490387449039392695959943027966195657681586539389134094087028482099931927725892... +
    // 2.318771368505683055818032722011594415038779144567369903204833213112457210243098545874099591376455981793627257060... i
    Real Jnzx = boost::lexical_cast<Real>("1.257674591970511077630764085052638490387449039392695959943027966195657681586539389134094087028482099931927725892");
    Real Jnzy = boost::lexical_cast<Real>("2.318771368505683055818032722011594415038779144567369903204833213112457210243098545874099591376455981793627257060");
    Real tol = 10*std::numeric_limits<Real>::epsilon();
    BOOST_CHECK_CLOSE_FRACTION(Jnz.real(), Jnzx, tol);
    BOOST_CHECK_CLOSE_FRACTION(Jnz.imag(), Jnzy, tol);
}

template<class Complex>
void test_I0_complex()
{
    std::cout << "Testing that complex-argument I0 is calculated correctly by the adaptive trapezoidal routine on type " << boost::typeindex::type_id<Complex>().pretty_name()  << "\n";
    typedef typename Complex::value_type Real;
    Complex z{2, 3};
    using boost::math::constants::pi;
    auto I0 = [&z](Real theta)->Complex
    {
        using std::cos;
        using std::exp;
        return exp(z*cos(theta))/pi<Real>();
    };

    using boost::math::quadrature::trapezoidal;

    Real a = 0;
    Real b = pi<Real>();
    Complex I0z = trapezoidal<decltype(I0), Real>(I0, a, b);
    // N[BesselI[0, 2 + 3 I], 143]
    // -1.24923487960742219637619681391438589436703710701063561548156438052154090067526565701278826317992172207565649925713468090525951417141982808439560899101
    // 0.947983792057734776114060623981442199525094227418764823692296622398838765371662384207319492908490909109393495109183270208372778907692930675595924819922 i
    Real I0zx = boost::lexical_cast<Real>("-1.24923487960742219637619681391438589436703710701063561548156438052154090067526565701278826317992172207565649925713468090525951417141982808439560899101");
    Real I0zy = boost::lexical_cast<Real>("0.947983792057734776114060623981442199525094227418764823692296622398838765371662384207319492908490909109393495109183270208372778907692930675595924819922");
    Real tol = 10*std::numeric_limits<Real>::epsilon();
    BOOST_CHECK_CLOSE_FRACTION(I0z.real(), I0zx, tol);
    BOOST_CHECK_CLOSE_FRACTION(I0z.imag(), I0zy, tol);
}


template<class Complex>
void test_erfc()
{
    std::cout << "Testing that complex-argument erfc is calculated correctly by the adaptive trapezoidal routine on type " << boost::typeindex::type_id<Complex>().pretty_name()  << "\n";
    typedef typename Complex::value_type Real;
    Complex z{2, -1};
    using boost::math::constants::pi;
    using boost::math::constants::two_pi;
    auto erfc = [&z](Real theta)->Complex
    {
        using std::exp;
        using std::tan;
        Real t = tan(theta/2);
        Complex arg = -z*z*(1+t*t);
        return exp(arg)/two_pi<Real>();
    };

    using boost::math::quadrature::trapezoidal;

    Real a = -pi<Real>();
    Real b = pi<Real>();
    Complex erfcz = trapezoidal<decltype(erfc), Real>(erfc, a, b, boost::math::tools::root_epsilon<Real>(), 17);
    // N[Erfc[2-i], 150]
    //-0.00360634272565175091291182820541914235532928536595056623793472801084629874817202107825472707423984408881473019087931573313969503235634965264302640170177
    // - 0.0112590060288150250764009156316482248536651598819882163212627394923365188251633729432967232423246312345152595958230197778555210858871376231770868078020 i
    Real erfczx = boost::lexical_cast<Real>("-0.00360634272565175091291182820541914235532928536595056623793472801084629874817202107825472707423984408881473019087931573313969503235634965264302640170177");
    Real erfczy = boost::lexical_cast<Real>("-0.0112590060288150250764009156316482248536651598819882163212627394923365188251633729432967232423246312345152595958230197778555210858871376231770868078020");
    Real tol = 5000*std::numeric_limits<Real>::epsilon();
    BOOST_CHECK_CLOSE_FRACTION(erfcz.real(), erfczx, tol);
    BOOST_CHECK_CLOSE_FRACTION(erfcz.imag(), erfczy, tol);
}


template<class Real>
void test_constant()
{
    std::cout << "Testing constants are integrated correctly by the adaptive trapezoidal routine on type " << boost::typeindex::type_id<Real>().pretty_name()  << "\n";

    auto f = [](Real)->Real { return boost::math::constants::half<Real>(); };
    Real Q = trapezoidal<decltype(f), Real>(f, (Real) 0.0, (Real) 10.0);
    BOOST_CHECK_CLOSE(Q, 5.0, 100*std::numeric_limits<Real>::epsilon());
}


template<class Real>
void test_rational_periodic()
{
    using boost::math::constants::two_pi;
    using boost::math::constants::third;
    std::cout << "Testing that rational periodic functions are integrated correctly by trapezoidal rule on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";

    auto f = [](Real x)->Real { return 1/(5 - 4*cos(x)); };

    Real tol = 100*boost::math::tools::epsilon<Real>();
    Real Q = trapezoidal(f, (Real) 0.0, two_pi<Real>(), tol);

    BOOST_CHECK_CLOSE_FRACTION(Q, two_pi<Real>()*third<Real>(), 10*tol);
}

template<class Real>
void test_bump_function()
{
    std::cout << "Testing that bump functions are integrated correctly by trapezoidal rule on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
    auto f = [](Real x)->Real {
        if( x>= 1 || x <= -1)
        {
            return (Real) 0;
        }
        return (Real) exp(-(Real) 1/(1-x*x));
    };
    Real tol = boost::math::tools::epsilon<Real>();
    Real Q = trapezoidal(f, (Real) -1, (Real) 1, tol);
    // 2*NIntegrate[Exp[-(1/(1 - x^2))], {x, 0, 1}, WorkingPrecision -> 210]
    Real Q_exp = boost::lexical_cast<Real>("0.44399381616807943782304892117055266376120178904569749730748455394704");
    BOOST_CHECK_CLOSE_FRACTION(Q, Q_exp, 30*tol);
}

template<class Real>
void test_zero_function()
{
    std::cout << "Testing that zero functions are integrated correctly by trapezoidal rule on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
    auto f = [](Real)->Real { return (Real) 0;};
    Real tol = 100* boost::math::tools::epsilon<Real>();
    Real Q = trapezoidal(f, (Real) -1, (Real) 1, tol);
    BOOST_CHECK_SMALL(Q, 100*tol);
}

template<class Real>
void test_sinsq()
{
    std::cout << "Testing that sin(x)^2 is integrated correctly by the trapezoidal rule on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
    auto f = [](Real x)->Real { return sin(10*x)*sin(10*x); };
    Real tol = 100* boost::math::tools::epsilon<Real>();
    Real Q = trapezoidal(f, (Real) 0, (Real) boost::math::constants::pi<Real>(), tol);
    BOOST_CHECK_CLOSE_FRACTION(Q, boost::math::constants::half_pi<Real>(), tol);

}

template<class Real>
void test_slowly_converging()
{
    using std::sqrt;
    std::cout << "Testing that non-periodic functions are correctly integrated by the trapezoidal rule, even if slowly, on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
    // This function is not periodic, so it should not be fast to converge:
    auto f = [](Real x)->Real { using std::sqrt;  return sqrt(1 - x*x); };

    Real tol = sqrt(sqrt(boost::math::tools::epsilon<Real>()));
    Real error_estimate;
    Real Q = trapezoidal(f, (Real) 0, (Real) 1, tol, 15, &error_estimate);
    BOOST_CHECK_CLOSE_FRACTION(Q, boost::math::constants::half_pi<Real>()/2, 10*tol);
}

template<class Real>
void test_rational_sin()
{
    using std::pow;
    using std::sin;
    using boost::math::constants::two_pi;
    using boost::math::constants::half;
    std::cout << "Testing that a rational sin function is integrated correctly by the trapezoidal rule on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
    Real a = 5;
    auto f = [=](Real x)->Real { using std::sin;  Real t = a + sin(x); return 1.0f / (t*t); };

    Real expected = two_pi<Real>()*a/pow(a*a - 1, 3*half<Real>());
    Real tol = 100* boost::math::tools::epsilon<Real>();
    Real Q = trapezoidal(f, (Real) 0, (Real) boost::math::constants::two_pi<Real>(), tol);
    BOOST_CHECK_CLOSE_FRACTION(Q, expected, tol);
}

BOOST_AUTO_TEST_CASE(trapezoidal_quadrature)
{
    test_constant<float>();
    test_constant<double>();
    test_constant<long double>();
    test_constant<boost::math::concepts::real_concept>();
    test_constant<cpp_bin_float_50>();
    test_constant<cpp_bin_float_100>();

    test_rational_periodic<float>();
    test_rational_periodic<double>();
    test_rational_periodic<long double>();
    test_rational_periodic<boost::math::concepts::real_concept>();
    test_rational_periodic<cpp_bin_float_50>();
    test_rational_periodic<cpp_bin_float_100>();

    test_bump_function<float>();
    test_bump_function<double>();
    test_bump_function<long double>();
    test_rational_periodic<boost::math::concepts::real_concept>();
    test_rational_periodic<cpp_bin_float_50>();

    test_zero_function<float>();
    test_zero_function<double>();
    test_zero_function<long double>();
    test_zero_function<boost::math::concepts::real_concept>();
    test_zero_function<cpp_bin_float_50>();
    test_zero_function<cpp_bin_float_100>();

    test_sinsq<float>();
    test_sinsq<double>();
    test_sinsq<long double>();
    test_sinsq<boost::math::concepts::real_concept>();
    test_sinsq<cpp_bin_float_50>();
    test_sinsq<cpp_bin_float_100>();

    test_slowly_converging<float>();
    test_slowly_converging<double>();
    test_slowly_converging<long double>();
    test_slowly_converging<boost::math::concepts::real_concept>();

    test_rational_sin<float>();
    test_rational_sin<double>();
    test_rational_sin<long double>();
    //test_rational_sin<boost::math::concepts::real_concept>();
    test_rational_sin<cpp_bin_float_50>();

    test_complex_bessel<std::complex<float>>();
    test_complex_bessel<std::complex<double>>();
    test_complex_bessel<std::complex<long double>>();
    //test_complex_bessel<boost::multiprecision::mpc_complex_100>();
    test_I0_complex<std::complex<float>>();
    test_I0_complex<std::complex<double>>();
    test_I0_complex<std::complex<long double>>();
    //test_I0_complex<boost::multiprecision::mpc_complex_100>();
    test_erfc<std::complex<float>>();
    test_erfc<std::complex<double>>();
    test_erfc<std::complex<long double>>();
    //test_erfc<boost::multiprecision::number<boost::multiprecision::mpc_complex_backend<20>>>();
    //test_erfc<boost::multiprecision::number<boost::multiprecision::mpc_complex_backend<30>>>();
    //test_erfc<boost::multiprecision::mpc_complex_50>();
    //test_erfc<boost::multiprecision::mpc_complex_100>();

#ifdef BOOST_HAS_FLOAT128
    test_complex_bessel<boost::multiprecision::complex128>();
    test_I0_complex<boost::multiprecision::complex128>();
    test_erfc<boost::multiprecision::complex128>();
#endif

}