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math/test/tanh_sinh_quadrature_test.cpp
jzmaddock 238436ee07 Sterling: extend Sterling's approximation to the beta functions. 
Adjust expected error rates accordingly.
Fixes https://github.com/boostorg/math/issues/247.
2019-10-11 11:12:02 +01:00

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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)
#define BOOST_TEST_MODULE tanh_sinh_quadrature_test
#include <boost/config.hpp>
#include <boost/detail/workaround.hpp>
#if !defined(BOOST_NO_CXX11_DECLTYPE) && !defined(BOOST_NO_CXX11_TRAILING_RESULT_TYPES) && !defined(BOOST_NO_SFINAE_EXPR)
#include <boost/math/concepts/real_concept.hpp>
#include <boost/test/included/unit_test.hpp>
#include <boost/test/tools/floating_point_comparison.hpp>
#include <boost/math/quadrature/tanh_sinh.hpp>
#include <boost/math/special_functions/sinc.hpp>
#include <boost/multiprecision/cpp_bin_float.hpp>
#include <boost/multiprecision/cpp_dec_float.hpp>
#include <boost/math/special_functions/next.hpp>
#include <boost/math/special_functions/gamma.hpp>
#include <boost/math/special_functions/beta.hpp>
#include <boost/math/special_functions/ellint_rc.hpp>
#include <boost/math/special_functions/ellint_rj.hpp>
#ifdef BOOST_HAS_FLOAT128
#include <boost/multiprecision/float128.hpp>
#endif
#ifdef _MSC_VER
#pragma warning(disable:4127) // Conditional expression is constant
#endif
#if !defined(TEST1) && !defined(TEST2) && !defined(TEST3) && !defined(TEST4) && !defined(TEST5) && !defined(TEST6) && !defined(TEST7) && !defined(TEST8)\
&& !defined(TEST1A) && !defined(TEST1B) && !defined(TEST2A) && !defined(TEST3A) && !defined(TEST6A) && !defined(TEST9)
# define TEST1
# define TEST2
# define TEST3
# define TEST4
# define TEST5
# define TEST6
# define TEST7
# define TEST8
# define TEST1A
# define TEST1B
# define TEST2A
# define TEST3A
# define TEST6A
# define TEST9
#endif
using std::expm1;
using std::atan;
using std::tan;
using std::log;
using std::log1p;
using std::asinh;
using std::atanh;
using std::sqrt;
using std::isnormal;
using std::abs;
using std::sinh;
using std::tanh;
using std::cosh;
using std::pow;
using std::exp;
using std::sin;
using std::cos;
using std::string;
using boost::multiprecision::cpp_bin_float_50;
using boost::multiprecision::cpp_bin_float_100;
using boost::multiprecision::cpp_dec_float_50;
using boost::multiprecision::cpp_dec_float_100;
using boost::multiprecision::cpp_bin_float_quad;
using boost::math::sinc_pi;
using boost::math::quadrature::tanh_sinh;
using boost::math::quadrature::detail::tanh_sinh_detail;
using boost::math::constants::pi;
using boost::math::constants::half_pi;
using boost::math::constants::two_div_pi;
using boost::math::constants::two_pi;
using boost::math::constants::half;
using boost::math::constants::third;
using boost::math::constants::half;
using boost::math::constants::third;
using boost::math::constants::catalan;
using boost::math::constants::ln_two;
using boost::math::constants::root_two;
using boost::math::constants::root_two_pi;
using boost::math::constants::root_pi;
template <class T>
void print_levels(const T& v, const char* suffix)
{
std::cout << "{\n";
for (unsigned i = 0; i < v.size(); ++i)
{
std::cout << " { ";
for (unsigned j = 0; j < v[i].size(); ++j)
{
std::cout << v[i][j] << suffix << ", ";
}
std::cout << "},\n";
}
std::cout << " };\n";
}
template <class T>
void print_complement_indexes(const T& v)
{
std::cout << "\n {";
for (unsigned i = 0; i < v.size(); ++i)
{
unsigned index = 0;
while (v[i][index] >= 0)
++index;
std::cout << index << ", ";
}
std::cout << "};\n";
}
template <class T>
void print_levels(const std::pair<T, T>& p, const char* suffix = "")
{
std::cout << " static const std::vector<std::vector<Real> > abscissa = ";
print_levels(p.first, suffix);
std::cout << " static const std::vector<std::vector<Real> > weights = ";
print_levels(p.second, suffix);
std::cout << " static const std::vector<unsigned > indexes = ";
print_complement_indexes(p.first);
}
template <class Real>
std::pair<std::vector<std::vector<Real>>, std::vector<std::vector<Real>> > generate_constants(unsigned max_index, unsigned max_rows)
{
using boost::math::constants::half_pi;
using boost::math::constants::two_div_pi;
using boost::math::constants::pi;
auto g = [](Real t) { return tanh(half_pi<Real>()*sinh(t)); };
auto w = [](Real t) { Real cs = cosh(half_pi<Real>() * sinh(t)); return half_pi<Real>() * cosh(t) / (cs * cs); };
auto gc = [](Real t) { Real u2 = half_pi<Real>() * sinh(t); return 1 / (exp(u2) *cosh(u2)); };
auto g_inv = [](float x)->float { return asinh(two_div_pi<float>()*atanh(x)); };
auto gc_inv = [](float x)
{
float l = log(sqrt((2 - x) / x));
return log((sqrt(4 * l * l + pi<float>() * pi<float>()) + 2 * l) / pi<float>());
};
std::vector<std::vector<Real>> abscissa, weights;
std::vector<Real> temp;
float t_crossover = gc_inv(0.5f);
Real h = 1;
for (unsigned i = 0; i < max_index; ++i)
{
temp.push_back((i < t_crossover) ? g(i * h) : -gc(i * h));
}
abscissa.push_back(temp);
temp.clear();
for (unsigned i = 0; i < max_index; ++i)
{
temp.push_back(w(i * h));
}
weights.push_back(temp);
temp.clear();
for (unsigned row = 1; row < max_rows; ++row)
{
h /= 2;
for (Real t = h; t < max_index - 1; t += 2 * h)
temp.push_back((t < t_crossover) ? g(t) : -gc(t));
abscissa.push_back(temp);
temp.clear();
}
h = 1;
for (unsigned row = 1; row < max_rows; ++row)
{
h /= 2;
for (Real t = h; t < max_index - 1; t += 2 * h)
temp.push_back(w(t));
weights.push_back(temp);
temp.clear();
}
return std::make_pair(abscissa, weights);
}
template <class Real>
const tanh_sinh<Real>& get_integrator()
{
static const tanh_sinh<Real> integrator(15);
return integrator;
}
template <class Real>
Real get_convergence_tolerance()
{
return boost::math::tools::root_epsilon<Real>();
}
template<class Real>
void test_linear()
{
std::cout << "Testing linear functions are integrated properly by tanh_sinh on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
Real tol = 10*boost::math::tools::epsilon<Real>();
auto integrator = get_integrator<Real>();
auto f = [](const Real& x)
{
return 5*x + 7;
};
Real error;
Real L1;
Real Q = integrator.integrate(f, (Real) 0, (Real) 1, get_convergence_tolerance<Real>(), &error, &L1);
BOOST_CHECK_CLOSE_FRACTION(Q, 9.5, tol);
BOOST_CHECK_CLOSE_FRACTION(L1, 9.5, tol);
}
template<class Real>
void test_quadratic()
{
std::cout << "Testing quadratic functions are integrated properly by tanh_sinh on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
Real tol = 10*boost::math::tools::epsilon<Real>();
auto integrator = get_integrator<Real>();
auto f = [](const Real& x) { return 5*x*x + 7*x + 12; };
Real error;
Real L1;
Real Q = integrator.integrate(f, (Real) 0, (Real) 1, get_convergence_tolerance<Real>(), &error, &L1);
BOOST_CHECK_CLOSE_FRACTION(Q, (Real) 17 + half<Real>()*third<Real>(), tol);
BOOST_CHECK_CLOSE_FRACTION(L1, (Real) 17 + half<Real>()*third<Real>(), tol);
}
template<class Real>
void test_singular()
{
std::cout << "Testing integration of endpoint singularities on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
Real tol = 10*boost::math::tools::epsilon<Real>();
Real error;
Real L1;
auto integrator = get_integrator<Real>();
auto f = [](const Real& x) { return log(x)*log(1-x); };
Real Q = integrator.integrate(f, (Real) 0, (Real) 1, get_convergence_tolerance<Real>(), &error, &L1);
Real Q_expected = 2 - pi<Real>()*pi<Real>()*half<Real>()*third<Real>();
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol);
}
// Examples taken from
//http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/quadrature.pdf
template<class Real>
void test_ca()
{
std::cout << "Testing integration of C(a) on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
Real tol = 10 * boost::math::tools::epsilon<Real>();
Real error;
Real L1;
auto integrator = get_integrator<Real>();
auto f1 = [](const Real& x) { return atan(x)/(x*(x*x + 1)) ; };
Real Q = integrator.integrate(f1, (Real) 0, (Real) 1, get_convergence_tolerance<Real>(), &error, &L1);
Real Q_expected = pi<Real>()*ln_two<Real>()/8 + catalan<Real>()*half<Real>();
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol);
auto f2 = [](Real x)->Real { Real t0 = x*x + 1; Real t1 = sqrt(t0); return atan(t1)/(t0*t1); };
Q = integrator.integrate(f2, (Real) 0 , (Real) 1, get_convergence_tolerance<Real>(), &error, &L1);
Q_expected = pi<Real>()/4 - pi<Real>()/root_two<Real>() + 3*atan(root_two<Real>())/root_two<Real>();
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol);
auto f5 = [](Real t)->Real { return t*t*log(t)/((t*t - 1)*(t*t*t*t + 1)); };
Q = integrator.integrate(f5, (Real) 0 , (Real) 1);
Q_expected = pi<Real>()*pi<Real>()*(2 - root_two<Real>())/32;
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
// Oh it suffers on this one.
auto f6 = [](Real t)->Real { Real x = log(t); return x*x; };
Q = integrator.integrate(f6, (Real) 0 , (Real) 1, get_convergence_tolerance<Real>(), &error, &L1);
Q_expected = 2;
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, 50*tol);
BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, 50*tol);
// Although it doesn't get to the requested tolerance on this integral, the error bounds can be queried and are reasonable:
tol = sqrt(boost::math::tools::epsilon<Real>());
auto f7 = [](const Real& t) { return sqrt(tan(t)); };
Q = integrator.integrate(f7, (Real) 0 , (Real) half_pi<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
Q_expected = pi<Real>()/root_two<Real>();
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol);
auto f8 = [](const Real& t) { return log(cos(t)); };
Q = integrator.integrate(f8, (Real) 0 , half_pi<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
Q_expected = -pi<Real>()*ln_two<Real>()*half<Real>();
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
BOOST_CHECK_CLOSE_FRACTION(L1, -Q_expected, tol);
}
template<class Real>
void test_three_quadrature_schemes_examples()
{
std::cout << "Testing integral in 'A Comparison of Three High Precision Quadrature Schemes' on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
Real tol = 10 * boost::math::tools::epsilon<Real>();
Real Q;
Real Q_expected;
auto integrator = get_integrator<Real>();
// Example 1:
auto f1 = [](const Real& t) { return t*boost::math::log1p(t); };
Q = integrator.integrate(f1, (Real) 0 , (Real) 1);
Q_expected = half<Real>()*half<Real>();
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
// Example 2:
auto f2 = [](const Real& t) { return t*t*atan(t); };
Q = integrator.integrate(f2, (Real) 0 , (Real) 1);
Q_expected = (pi<Real>() -2 + 2*ln_two<Real>())/12;
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, 2 * tol);
// Example 3:
auto f3 = [](const Real& t) { return exp(t)*cos(t); };
Q = integrator.integrate(f3, (Real) 0, half_pi<Real>());
Q_expected = boost::math::expm1(half_pi<Real>())*half<Real>();
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
// Example 4:
auto f4 = [](Real x)->Real { Real t0 = sqrt(x*x + 2); return atan(t0)/(t0*(x*x+1)); };
Q = integrator.integrate(f4, (Real) 0 , (Real) 1);
Q_expected = 5*pi<Real>()*pi<Real>()/96;
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
// Example 5:
auto f5 = [](const Real& t) { return sqrt(t)*log(t); };
Q = integrator.integrate(f5, (Real) 0 , (Real) 1);
Q_expected = -4/ (Real) 9;
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
// Example 6:
auto f6 = [](const Real& t) { return sqrt(1 - t*t); };
Q = integrator.integrate(f6, (Real) 0 , (Real) 1);
Q_expected = pi<Real>()/4;
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
}
template<class Real>
void test_integration_over_real_line()
{
std::cout << "Testing integrals over entire real line in 'A Comparison of Three High Precision Quadrature Schemes' on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
Real tol = 10 * boost::math::tools::epsilon<Real>();
Real Q;
Real Q_expected;
Real error;
Real L1;
auto integrator = get_integrator<Real>();
auto f1 = [](const Real& t) { return 1/(1+t*t);};
Q = integrator.integrate(f1, std::numeric_limits<Real>::has_infinity ? -std::numeric_limits<Real>::infinity() : -boost::math::tools::max_value<Real>(), std::numeric_limits<Real>::has_infinity ? std::numeric_limits<Real>::infinity() : boost::math::tools::max_value<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
Q_expected = pi<Real>();
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol);
auto f2 = [](const Real& t) { return exp(-t*t*half<Real>()); };
Q = integrator.integrate(f2, std::numeric_limits<Real>::has_infinity ? -std::numeric_limits<Real>::infinity() : -boost::math::tools::max_value<Real>(), std::numeric_limits<Real>::has_infinity ? std::numeric_limits<Real>::infinity() : boost::math::tools::max_value<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
Q_expected = root_two_pi<Real>();
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol * 2);
BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol * 2);
// This test shows how oscillatory integrals are approximated very poorly by this method:
//std::cout << "Testing sinc integral: \n";
//Q = integrator.integrate(boost::math::sinc_pi<Real>, -std::numeric_limits<Real>::infinity(), std::numeric_limits<Real>::infinity(), &error, &L1);
//std::cout << "Error estimate of sinc integral: " << error << std::endl;
//std::cout << "L1 norm of sinc integral " << L1 << std::endl;
//Q_expected = pi<Real>();
//BOOST_CHECK_CLOSE(Q, Q_expected, 100*tol);
auto f4 = [](const Real& t) { return 1/cosh(t);};
Q = integrator.integrate(f4, std::numeric_limits<Real>::has_infinity ? -std::numeric_limits<Real>::infinity() : -boost::math::tools::max_value<Real>(), std::numeric_limits<Real>::has_infinity ? std::numeric_limits<Real>::infinity() : boost::math::tools::max_value<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
Q_expected = pi<Real>();
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol);
}
template<class Real>
void test_right_limit_infinite()
{
std::cout << "Testing right limit infinite for tanh_sinh in 'A Comparison of Three High Precision Quadrature Schemes' on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
Real tol = 10 * boost::math::tools::epsilon<Real>();
Real Q;
Real Q_expected;
Real error;
Real L1;
auto integrator = get_integrator<Real>();
// Example 11:
auto f1 = [](const Real& t) { return 1/(1+t*t);};
Q = integrator.integrate(f1, 0, std::numeric_limits<Real>::has_infinity ? std::numeric_limits<Real>::infinity() : boost::math::tools::max_value<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
Q_expected = half_pi<Real>();
BOOST_CHECK_CLOSE(Q, Q_expected, 100*tol);
// Example 12
auto f2 = [](const Real& t) { return exp(-t)/sqrt(t); };
Q = integrator.integrate(f2, 0, std::numeric_limits<Real>::has_infinity ? std::numeric_limits<Real>::infinity() : boost::math::tools::max_value<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
Q_expected = root_pi<Real>();
BOOST_CHECK_CLOSE(Q, Q_expected, 1000*tol);
auto f3 = [](const Real& t) { return exp(-t)*cos(t); };
Q = integrator.integrate(f3, 0, std::numeric_limits<Real>::has_infinity ? std::numeric_limits<Real>::infinity() : boost::math::tools::max_value<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
Q_expected = half<Real>();
BOOST_CHECK_CLOSE(Q, Q_expected, 100*tol);
auto f4 = [](const Real& t) { return 1/(1+t*t); };
Q = integrator.integrate(f4, 1, std::numeric_limits<Real>::has_infinity ? std::numeric_limits<Real>::infinity() : boost::math::tools::max_value<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
Q_expected = pi<Real>()/4;
BOOST_CHECK_CLOSE(Q, Q_expected, 100*tol);
}
template<class Real>
void test_left_limit_infinite()
{
std::cout << "Testing left limit infinite for tanh_sinh in 'A Comparison of Three High Precision Quadrature Schemes' on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
Real tol = 10 * boost::math::tools::epsilon<Real>();
Real Q;
Real Q_expected;
auto integrator = get_integrator<Real>();
// Example 11:
auto f1 = [](const Real& t) { return 1/(1+t*t);};
Q = integrator.integrate(f1, std::numeric_limits<Real>::has_infinity ? -std::numeric_limits<Real>::infinity() : -boost::math::tools::max_value<Real>(), Real(0));
Q_expected = half_pi<Real>();
BOOST_CHECK_CLOSE(Q, Q_expected, 100*tol);
}
// A horrible function taken from
// http://www.chebfun.org/examples/quad/GaussClenCurt.html
template<class Real>
void test_horrible()
{
std::cout << "Testing Trefenthen's horrible integral on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
// We only know the integral to double precision, so requesting a higher tolerance doesn't make sense.
Real tol = 10 * std::numeric_limits<float>::epsilon();
Real Q;
Real Q_expected;
Real error;
Real L1;
auto integrator = get_integrator<Real>();
auto f = [](Real x)->Real { return x*sin(2*exp(2*sin(2*exp(2*x) ) ) ); };
Q = integrator.integrate(f, (Real) -1, (Real) 1, get_convergence_tolerance<Real>(), &error, &L1);
// NIntegrate[x*Sin[2*Exp[2*Sin[2*Exp[2*x]]]], {x, -1, 1}, WorkingPrecision -> 130, MaxRecursion -> 100]
Q_expected = boost::lexical_cast<Real>("0.33673283478172753598559003181355241139806404130031017259552729882281");
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
// Over again without specifying the bounds:
Q = integrator.integrate(f, get_convergence_tolerance<Real>(), &error, &L1);
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
}
// Some examples of tough integrals from NR, section 4.5.4:
template<class Real>
void test_nr_examples()
{
std::cout << "Testing singular integrals from NR 4.5.4 on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
Real tol = 10 * boost::math::tools::epsilon<Real>();
Real Q;
Real Q_expected;
Real error;
Real L1;
auto integrator = get_integrator<Real>();
auto f1 = [](Real x)->Real
{
return (sin(x * half<Real>()) * exp(-x) / x) / sqrt(x);
};
Q = integrator.integrate(f1, 0, std::numeric_limits<Real>::has_infinity ? std::numeric_limits<Real>::infinity() : boost::math::tools::max_value<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
Q_expected = sqrt(pi<Real>()*(sqrt((Real) 5) - 2));
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, 25*tol);
auto f2 = [](Real x)->Real { return pow(x, -(Real) 2/(Real) 7)*exp(-x*x); };
Q = integrator.integrate(f2, 0, std::numeric_limits<Real>::has_infinity ? std::numeric_limits<Real>::infinity() : boost::math::tools::max_value<Real>());
Q_expected = half<Real>()*boost::math::tgamma((Real) 5/ (Real) 14);
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol * 6);
}
// Test integrand known to fool some termination schemes:
template<class Real>
void test_early_termination()
{
std::cout << "Testing Clenshaw & Curtis's example of integrand which fools termination schemes on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
Real tol = 10 * boost::math::tools::epsilon<Real>();
Real Q;
Real Q_expected;
Real error;
Real L1;
auto integrator = get_integrator<Real>();
auto f1 = [](Real x)->Real { return 23*cosh(x)/ (Real) 25 - cos(x) ; };
Q = integrator.integrate(f1, (Real) -1, (Real) 1, get_convergence_tolerance<Real>(), &error, &L1);
Q_expected = 46*sinh((Real) 1)/(Real) 25 - 2*sin((Real) 1);
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
// Over again with no bounds:
Q = integrator.integrate(f1);
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
}
// Test some definite integrals from the CRC handbook, 32nd edition:
template<class Real>
void test_crc()
{
std::cout << "Testing CRC formulas on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
Real tol = 10 * boost::math::tools::epsilon<Real>();
Real Q;
Real Q_expected;
Real error;
Real L1;
auto integrator = get_integrator<Real>();
// CRC Definite integral 585
auto f1 = [](Real x)->Real { Real t = log(1/x); return x*x*t*t*t; };
Q = integrator.integrate(f1, (Real) 0, (Real) 1, get_convergence_tolerance<Real>(), &error, &L1);
Q_expected = (Real) 2/ (Real) 27;
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
// CRC 636:
auto f2 = [](Real x)->Real { return sqrt(cos(x)); };
Q = integrator.integrate(f2, (Real) 0, (Real) half_pi<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
//Q_expected = pow(two_pi<Real>(), 3*half<Real>())/pow(boost::math::tgamma((Real) 1/ (Real) 4), 2);
Q_expected = boost::lexical_cast<Real>("1.198140234735592207439922492280323878227212663215651558263674952946405214143915670835885556489793389375907225");
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
// CRC Section 5.5, integral 585:
for (int n = 0; n < 3; ++n) {
for (int m = 0; m < 3; ++m) {
auto f = [&](Real x)->Real { return pow(x, Real(m))*pow(log(1/x), Real(n)); };
Q = integrator.integrate(f, (Real) 0, (Real) 1, get_convergence_tolerance<Real>(), &error, &L1);
// Calculation of the tgamma function is not exact, giving spurious failures.
// Casting to cpp_bin_float_100 beforehand fixes most of them.
cpp_bin_float_100 np1 = n + 1;
cpp_bin_float_100 mp1 = m + 1;
Q_expected = boost::lexical_cast<Real>((tgamma(np1)/pow(mp1, np1)).str());
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
}
}
// CRC Section 5.5, integral 591
// The parameter p allows us to control the strength of the singularity.
// Rapid convergence is not guaranteed for this function, as the branch cut makes it non-analytic on a disk.
// This converges only when our test type has an extended exponent range as all the area of the integral
// occurs so close to 0 (or 1) that we need abscissa values exceptionally small to find it.
// "There's a lot of room at the bottom".
// We also use a 2 argument functor so that 1-x is evaluated accurately:
if (std::numeric_limits<Real>::max_exponent > std::numeric_limits<double>::max_exponent)
{
for (Real p = Real (-0.99); p < 1; p += Real(0.1)) {
auto f = [&](Real x, Real cx)->Real
{
//return pow(x, p) / pow(1 - x, p);
return cx < 0 ? exp(p * (log(x) - boost::math::log1p(-x))) : pow(x, p) / pow(cx, p);
};
Q = integrator.integrate(f, (Real)0, (Real)1, get_convergence_tolerance<Real>(), &error, &L1);
Q_expected = 1 / sinc_pi(p*pi<Real>());
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, 10 * tol);
}
}
// There is an alternative way to evaluate the above integral: by noticing that all the area of the integral
// is near zero for p < 0 and near 1 for p > 0 we can substitute exp(-x) for x and remap the integral to the
// domain (0, INF). Internally we need to expand out the terms and evaluate using logs to avoid spurous overflow,
// this gives us
// for p > 0:
for (Real p = Real(0.99); p > 0; p -= Real(0.1)) {
auto f = [&](Real x)->Real
{
return exp(-x * (1 - p) + p * log(-boost::math::expm1(-x)));
};
Q = integrator.integrate(f, 0, boost::math::tools::max_value<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
Q_expected = 1 / sinc_pi(p*pi<Real>());
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, 10 * tol);
}
// and for p < 1:
for (Real p = Real (-0.99); p < 0; p += Real(0.1)) {
auto f = [&](Real x)->Real
{
return exp(-p * log(-boost::math::expm1(-x)) - (1 + p) * x);
};
Q = integrator.integrate(f, 0, boost::math::tools::max_value<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
Q_expected = 1 / sinc_pi(p*pi<Real>());
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, 10 * tol);
}
// CRC Section 5.5, integral 635
for (int m = 0; m < 10; ++m) {
auto f = [&](Real x)->Real { return 1/(1 + pow(tan(x), m)); };
Q = integrator.integrate(f, (Real) 0, half_pi<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
Q_expected = half_pi<Real>()/2;
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
}
// CRC Section 5.5, integral 637:
//
// When h gets very close to 1, the strength of the singularity gradually increases until we
// no longer have enough exponent range to evaluate the integral....
// We also have to use the 2-argument functor version of the integrator to avoid
// cancellation error, since the singularity is near PI/2.
//
Real limit = std::numeric_limits<Real>::max_exponent > std::numeric_limits<double>::max_exponent
? .95f : .85f;
for (Real h = Real(0.01); h < limit; h += Real(0.1)) {
auto f = [&](Real x, Real xc)->Real { return xc > 0 ? pow(1/tan(xc), h) : pow(tan(x), h); };
Q = integrator.integrate(f, (Real) 0, half_pi<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
Q_expected = half_pi<Real>()/cos(h*half_pi<Real>());
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
}
// CRC Section 5.5, integral 637:
//
// Over again, but with argument transformation, we want:
//
// Integral of tan(x)^h over (0, PI/2)
//
// Note that the bulk of the area is next to the singularity at PI/2,
// so we'll start by replacing x by PI/2 - x, and that tan(PI/2 - x) == 1/tan(x)
// so we now have:
//
// Integral of 1/tan(x)^h over (0, PI/2)
//
// Which is almost the ideal form, except that when h is very close to 1
// we run out of exponent range in evaluating the integral arbitrarily close to 0.
// So now we substitute exp(-x) for x: this stretches out the range of the
// integral to (-log(PI/2), +INF) with the singularity at +INF giving:
//
// Integral of exp(-x)/tan(exp(-x))^h over (-log(PI/2), +INF)
//
// We just need a way to evaluate the function without spurious under/overflow
// in the exp terms. Note that for small x: tan(x) ~= x, so making this
// substitution and evaluating by logs we have:
//
// exp(-x)/tan(exp(-x))^h ~= exp((h - 1) * x) for x > -log(epsion);
//
// Here's how that looks in code:
//
for (Real i = 80; i < 100; ++i) {
Real h = i / 100;
auto f = [&](Real x)->Real
{
if (x > -log(boost::math::tools::epsilon<Real>()))
return exp((h - 1) * x);
else
{
Real et = exp(-x);
// Need to deal with numeric instability causing et to be greater than PI/2:
return et >= boost::math::constants::half_pi<Real>() ? 0 : et * pow(1 / tan(et), h);
}
};
Q = integrator.integrate(f, -log(half_pi<Real>()), boost::math::tools::max_value<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
Q_expected = half_pi<Real>() / cos(h*half_pi<Real>());
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, 5 * tol);
}
// CRC Section 5.5, integral 670:
auto f3 = [](Real x)->Real { return sqrt(log(1/x)); };
Q = integrator.integrate(f3, (Real) 0, (Real) 1, get_convergence_tolerance<Real>(), &error, &L1);
Q_expected = root_pi<Real>()/2;
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
}
template <class Real>
void test_sf()
{
using std::sqrt;
// Test some special functions that we already know how to evaluate:
std::cout << "Testing special functions on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
Real tol = 10 * boost::math::tools::epsilon<Real>();
auto integrator = get_integrator<Real>();
// incomplete beta:
if (std::numeric_limits<Real>::digits10 < 37) // Otherwise too slow
{
Real a(100), b(15);
auto f = [&](Real x)->Real { return boost::math::ibeta_derivative(a, b, x); };
BOOST_CHECK_CLOSE_FRACTION(integrator.integrate(f, 0, Real(0.25)), boost::math::ibeta(100, 15, Real(0.25)), tol * 10);
// Check some really extreme versions:
a = 1000;
b = 500;
BOOST_CHECK_CLOSE_FRACTION(integrator.integrate(f, 0, 1), Real(1), tol * 15);
//
// This is as extreme as we can get in this domain: otherwise the function has all it's
// area so close to zero we never get in there no matter how many levels we go down:
//
a = Real(1) / 15;
b = 30;
BOOST_CHECK_CLOSE_FRACTION(integrator.integrate(f, 0, 1), Real(1), tol * 25);
}
Real x = 2, y = 3, z = 0.5, p = 0.25;
// Elliptic integral RC:
Real Q = integrator.integrate([&](const Real& t)->Real { return 1 / (sqrt(t + x) * (t + y)); }, 0, std::numeric_limits<Real>::has_infinity ? std::numeric_limits<Real>::infinity() : boost::math::tools::max_value<Real>()) / 2;
BOOST_CHECK_CLOSE_FRACTION(Q, boost::math::ellint_rc(x, y), tol);
// Elliptic Integral RJ:
BOOST_CHECK_CLOSE_FRACTION(Real((Real(3) / 2) * integrator.integrate([&](Real t)->Real { return 1 / (sqrt((t + x) * (t + y) * (t + z)) * (t + p)); }, 0, std::numeric_limits<Real>::has_infinity ? std::numeric_limits<Real>::infinity() : boost::math::tools::max_value<Real>())), boost::math::ellint_rj(x, y, z, p), tol);
z = 5.5;
if (std::numeric_limits<Real>::digits10 > 40)
tol *= 200;
else if (!std::numeric_limits<Real>::is_specialized)
tol *= 3;
// tgamma expressed as an integral:
BOOST_CHECK_CLOSE_FRACTION(integrator.integrate([&](Real t)->Real { using std::pow; using std::exp; return t > 10000 ? Real(0) : Real(pow(t, z - 1) * exp(-t)); }, 0, std::numeric_limits<Real>::has_infinity ? std::numeric_limits<Real>::infinity() : boost::math::tools::max_value<Real>()),
boost::math::tgamma(z), tol);
BOOST_CHECK_CLOSE_FRACTION(integrator.integrate([](const Real& t)->Real { using std::exp; return exp(-t*t); }, std::numeric_limits<Real>::has_infinity ? -std::numeric_limits<Real>::infinity() : -boost::math::tools::max_value<Real>(), std::numeric_limits<Real>::has_infinity ? std::numeric_limits<Real>::infinity() : boost::math::tools::max_value<Real>()),
boost::math::constants::root_pi<Real>(), tol);
}
template <class Real>
void test_2_arg()
{
BOOST_MATH_STD_USING
std::cout << "Testing 2 argument functors on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
Real tol = 10 * boost::math::tools::epsilon<Real>();
auto integrator = get_integrator<Real>();
//
// There are a whole family of integrals of the general form
// x(1-x)^-N ; N < 1
// which have all the interesting behaviour near the 2 singularities
// and all converge, see: http://www.wolframalpha.com/input/?i=integrate+(x+*+(1-x))%5E-1%2FN+from+0+to+1
//
Real Q = integrator.integrate([&](const Real& t, const Real & tc)->Real
{
return tc < 0 ? 1 / sqrt(t * (1-t)) : 1 / sqrt(t * tc);
}, 0, 1);
BOOST_CHECK_CLOSE_FRACTION(Q, boost::math::constants::pi<Real>(), tol);
Q = integrator.integrate([&](const Real& t, const Real & tc)->Real
{
return tc < 0 ? 1 / boost::math::cbrt(t * (1-t)) : 1 / boost::math::cbrt(t * tc);
}, 0, 1);
BOOST_CHECK_CLOSE_FRACTION(Q, boost::math::pow<2>(boost::math::tgamma(Real(2) / 3)) / boost::math::tgamma(Real(4) / 3), tol * 3);
//
// We can do the same thing with ((1+x)(1-x))^-N ; N < 1
//
Q = integrator.integrate([&](const Real& t, const Real & tc)->Real
{
return t < 0 ? 1 / sqrt(-tc * (1-t)) : 1 / sqrt((t + 1) * tc);
}, -1, 1);
BOOST_CHECK_CLOSE_FRACTION(Q, boost::math::constants::pi<Real>(), tol);
Q = integrator.integrate([&](const Real& t, const Real & tc)->Real
{
return t < 0 ? 1 / sqrt(-tc * (1-t)) : 1 / sqrt((t + 1) * tc);
});
BOOST_CHECK_CLOSE_FRACTION(Q, boost::math::constants::pi<Real>(), tol);
Q = integrator.integrate([&](const Real& t, const Real & tc)->Real
{
return t < 0 ? 1 / boost::math::cbrt(-tc * (1-t)) : 1 / boost::math::cbrt((t + 1) * tc);
}, -1, 1);
BOOST_CHECK_CLOSE_FRACTION(Q, sqrt(boost::math::constants::pi<Real>()) * boost::math::tgamma(Real(2) / 3) / boost::math::tgamma(Real(7) / 6), tol * 10);
Q = integrator.integrate([&](const Real& t, const Real & tc)->Real
{
return t < 0 ? 1 / boost::math::cbrt(-tc * (1-t)) : 1 / boost::math::cbrt((t + 1) * tc);
});
BOOST_CHECK_CLOSE_FRACTION(Q, sqrt(boost::math::constants::pi<Real>()) * boost::math::tgamma(Real(2) / 3) / boost::math::tgamma(Real(7) / 6), tol * 10);
//
// These are taken from above, and do not get to full precision via the single arg functor:
//
auto f7 = [](const Real& t, const Real& tc) { return t < 1 ? sqrt(tan(t)) : sqrt(1/tan(tc)); };
Real error, L1;
Q = integrator.integrate(f7, (Real)0, (Real)half_pi<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
Real Q_expected = pi<Real>() / root_two<Real>();
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol);
auto f8 = [](const Real& t, const Real& tc) { return t < 1 ? log(cos(t)) : log(sin(tc)); };
Q = integrator.integrate(f8, (Real)0, half_pi<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
Q_expected = -pi<Real>()*ln_two<Real>()*half<Real>();
BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
BOOST_CHECK_CLOSE_FRACTION(L1, -Q_expected, tol);
}
template <class Complex>
void test_complex()
{
typedef typename Complex::value_type value_type;
//
// Integral version of the confluent hypergeometric function:
// https://dlmf.nist.gov/13.4#E1
//
value_type tol = 10 * boost::math::tools::epsilon<value_type>();
Complex a(2, 3), b(3, 4), z(0.5, -2);
auto f = [&](value_type t)
{
return exp(z * t) * pow(t, a - value_type(1)) * pow(value_type(1) - t, b - a - value_type(1));
};
auto integrator = get_integrator<value_type>();
auto Q = integrator.integrate(f, value_type(0), value_type(1), get_convergence_tolerance<value_type>());
//
// Expected result computed from http://www.wolframalpha.com/input/?i=1F1%5B(2%2B3i),+(3%2B4i);+(0.5-2i)%5D+*+gamma(2%2B3i)+*+gamma(1%2Bi)+%2F+gamma(3%2B4i)
//
Complex expected(boost::lexical_cast<value_type>("-0.2911081612888249710582867318081776512805281815037891183828405999609246645054069649838607112484426042883371996"),
boost::lexical_cast<value_type>("0.4507983563969959578849120188097153649211346293694903758252662015991543519595834937475296809912196906074655385"));
value_type error = abs(expected - Q);
BOOST_CHECK_LE(error, tol);
//
// Sin Integral https://dlmf.nist.gov/6.2#E9
//
auto f2 = [z](value_type t)
{
return -exp(-z * cos(t)) * cos(z * sin(t));
};
Q = integrator.integrate(f2, value_type(0), boost::math::constants::half_pi<value_type>(), get_convergence_tolerance<value_type>());
expected = Complex(boost::lexical_cast<value_type>("0.8893822921008980697856313681734926564752476188106405688951257340480164694708337246829840859633322683740376134733"),
-boost::lexical_cast<value_type>("2.381380802906111364088958767973164614925936185337231718483495612539455538280372745733208000514737758457795502168"));
expected -= boost::math::constants::half_pi<value_type>();
error = abs(expected - Q);
BOOST_CHECK_LE(error, tol);
}
BOOST_AUTO_TEST_CASE(tanh_sinh_quadrature_test)
{
#ifdef GENERATE_CONSTANTS
//
// Generate pre-computed coefficients:
std::cout << std::setprecision(35);
print_levels(generate_constants<cpp_bin_float_100>(10, 8), "L");
#else
#ifdef TEST1
test_right_limit_infinite<float>();
test_left_limit_infinite<float>();
test_linear<float>();
test_quadratic<float>();
test_singular<float>();
test_ca<float>();
test_three_quadrature_schemes_examples<float>();
test_horrible<float>();
test_integration_over_real_line<float>();
test_nr_examples<float>();
#endif
#ifdef TEST1A
test_early_termination<float>();
test_2_arg<float>();
#endif
#ifdef TEST1B
test_crc<float>();
#endif
#ifdef TEST2
test_right_limit_infinite<double>();
test_left_limit_infinite<double>();
test_linear<double>();
test_quadratic<double>();
test_singular<double>();
test_ca<double>();
test_three_quadrature_schemes_examples<double>();
test_horrible<double>();
test_integration_over_real_line<double>();
test_nr_examples<double>();
test_early_termination<double>();
test_sf<double>();
test_2_arg<double>();
#endif
#ifdef TEST2A
test_crc<double>();
#endif
#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
#ifdef TEST3
test_right_limit_infinite<long double>();
test_left_limit_infinite<long double>();
test_linear<long double>();
test_quadratic<long double>();
test_singular<long double>();
test_ca<long double>();
test_three_quadrature_schemes_examples<long double>();
test_horrible<long double>();
test_integration_over_real_line<long double>();
test_nr_examples<long double>();
test_early_termination<long double>();
test_sf<long double>();
test_2_arg<long double>();
#endif
#ifdef TEST3A
test_crc<long double>();
#endif
#endif
#ifdef TEST4
test_right_limit_infinite<cpp_bin_float_quad>();
test_left_limit_infinite<cpp_bin_float_quad>();
test_linear<cpp_bin_float_quad>();
test_quadratic<cpp_bin_float_quad>();
test_singular<cpp_bin_float_quad>();
test_ca<cpp_bin_float_quad>();
test_three_quadrature_schemes_examples<cpp_bin_float_quad>();
test_horrible<cpp_bin_float_quad>();
test_nr_examples<cpp_bin_float_quad>();
test_early_termination<cpp_bin_float_quad>();
test_crc<cpp_bin_float_quad>();
test_sf<cpp_bin_float_quad>();
test_2_arg<cpp_bin_float_quad>();
#endif
#ifdef TEST5
test_sf<cpp_bin_float_50>();
test_sf<cpp_bin_float_100>();
test_sf<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<150> > >();
#endif
#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
#ifdef TEST6
test_right_limit_infinite<boost::math::concepts::real_concept>();
test_left_limit_infinite<boost::math::concepts::real_concept>();
test_linear<boost::math::concepts::real_concept>();
test_quadratic<boost::math::concepts::real_concept>();
test_singular<boost::math::concepts::real_concept>();
test_ca<boost::math::concepts::real_concept>();
test_three_quadrature_schemes_examples<boost::math::concepts::real_concept>();
test_horrible<boost::math::concepts::real_concept>();
test_integration_over_real_line<boost::math::concepts::real_concept>();
test_nr_examples<boost::math::concepts::real_concept>();
test_early_termination<boost::math::concepts::real_concept>();
test_sf<boost::math::concepts::real_concept>();
test_2_arg<boost::math::concepts::real_concept>();
#endif
#ifdef TEST6A
test_crc<boost::math::concepts::real_concept>();
#endif
#endif
#ifdef TEST7
test_sf<cpp_dec_float_50>();
#endif
#if defined(TEST8) && defined(BOOST_HAS_FLOAT128)
test_right_limit_infinite<boost::multiprecision::float128>();
test_left_limit_infinite<boost::multiprecision::float128>();
test_linear<boost::multiprecision::float128>();
test_quadratic<boost::multiprecision::float128>();
test_singular<boost::multiprecision::float128>();
test_ca<boost::multiprecision::float128>();
test_three_quadrature_schemes_examples<boost::multiprecision::float128>();
test_horrible<boost::multiprecision::float128>();
test_integration_over_real_line<boost::multiprecision::float128>();
test_nr_examples<boost::multiprecision::float128>();
test_early_termination<boost::multiprecision::float128>();
test_crc<boost::multiprecision::float128>();
test_sf<boost::multiprecision::float128>();
test_2_arg<boost::multiprecision::float128>();
#endif
#ifdef TEST9
test_complex<std::complex<double> >();
test_complex<std::complex<float> >();
#endif
#endif
}
#else
int main() { return 0; }
#endif
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