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Filename : trapezoidal.hpp
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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)
 *
 * Use the adaptive trapezoidal rule to estimate the integral of periodic functions over a period,
 * or to integrate a function whose derivative vanishes at the endpoints.
 *
 * If your function does not satisfy these conditions, and instead is simply continuous and bounded
 * over the whole interval, then this routine will still converge, albeit slowly. However, there
 * are much more efficient methods in this case, including Romberg, Simpson, and double exponential quadrature.
 */

#ifndef BOOST_MATH_QUADRATURE_TRAPEZOIDAL_HPP
#define BOOST_MATH_QUADRATURE_TRAPEZOIDAL_HPP

#include <cmath>
#include <limits>
#include <stdexcept>
#include <boost/math/constants/constants.hpp>
#include <boost/math/special_functions/fpclassify.hpp>
#include <boost/math/policies/error_handling.hpp>
#include <boost/math/tools/cxx03_warn.hpp>

namespace boost{ namespace math{ namespace quadrature {

template<class F, class Real, class Policy>
auto trapezoidal(F f, Real a, Real b, Real tol, std::size_t max_refinements, Real* error_estimate, Real* L1, const Policy& pol)->decltype(std::declval<F>()(std::declval<Real>()))
{
    static const char* function = "boost::math::quadrature::trapezoidal<%1%>(F, %1%, %1%, %1%)";
    using std::abs;
    using boost::math::constants::half;
    // In many math texts, K represents the field of real or complex numbers.
    // Too bad we can't put blackboard bold into C++ source!
    typedef decltype(f(a)) K;
    if (!(boost::math::isfinite)(a))
    {
       return static_cast<K>(boost::math::policies::raise_domain_error(function, "Left endpoint of integration must be finite for adaptive trapezoidal integration but got a = %1%.\n", a, pol));
    }
    if (!(boost::math::isfinite)(b))
    {
       return static_cast<K>(boost::math::policies::raise_domain_error(function, "Right endpoint of integration must be finite for adaptive trapezoidal integration but got b = %1%.\n", b, pol));
    }

    if (a == b)
    {
        return static_cast<K>(0);
    }
    if(a > b)
    {
        return -trapezoidal(f, b, a, tol, max_refinements, error_estimate, L1, pol);
    }


    K ya = f(a);
    K yb = f(b);
    Real h = (b - a)*half<Real>();
    K I0 = (ya + yb)*h;
    Real IL0 = (abs(ya) + abs(yb))*h;

    K yh = f(a + h);
    K I1;
    I1 = I0*half<Real>() + yh*h;
    Real IL1 = IL0*half<Real>() + abs(yh)*h;

    // The recursion is:
    // I_k = 1/2 I_{k-1} + 1/2^k \sum_{j=1; j odd, j < 2^k} f(a + j(b-a)/2^k)
    std::size_t k = 2;
    // We want to go through at least 5 levels so we have sampled the function at least 20 times.
    // Otherwise, we could terminate prematurely and miss essential features.
    // This is of course possible anyway, but 20 samples seems to be a reasonable compromise.
    Real error = abs(I0 - I1);
    // I take k < 5, rather than k < 4, or some other smaller minimum number,
    // because I hit a truly exceptional bug where the k = 2 and k =3 refinement were bitwise equal,
    // but the quadrature had not yet converged.
    while (k < 5 || (k < max_refinements && error > tol*IL1) )
    {
        I0 = I1;
        IL0 = IL1;

        I1 = I0*half<Real>();
        IL1 = IL0*half<Real>();
        std::size_t p = static_cast<std::size_t>(1u) << k;
        h *= half<Real>();
        K sum = 0;
        Real absum = 0;

        for(std::size_t j = 1; j < p; j += 2)
        {
            K y = f(a + j*h);
            sum += y;
            absum += abs(y);
        }

        I1 += sum*h;
        IL1 += absum*h;
        ++k;
        error = abs(I0 - I1);
    }

    if (error_estimate)
    {
        *error_estimate = error;
    }

    if (L1)
    {
        *L1 = IL1;
    }

    return static_cast<K>(I1);
}
#if BOOST_WORKAROUND(BOOST_MSVC, < 1800)
// Template argument deduction failure otherwise:
template<class F, class Real>
auto trapezoidal(F f, Real a, Real b, Real tol = 0, std::size_t max_refinements = 12, Real* error_estimate = 0, Real* L1 = 0)->decltype(std::declval<F>()(std::declval<Real>()))
#elif !defined(BOOST_NO_CXX11_NULLPTR)
template<class F, class Real>
auto trapezoidal(F f, Real a, Real b, Real tol = boost::math::tools::root_epsilon<Real>(), std::size_t max_refinements = 12, Real* error_estimate = nullptr, Real* L1 = nullptr)->decltype(std::declval<F>()(std::declval<Real>()))
#else
template<class F, class Real>
auto trapezoidal(F f, Real a, Real b, Real tol = boost::math::tools::root_epsilon<Real>(), std::size_t max_refinements = 12, Real* error_estimate = 0, Real* L1 = 0)->decltype(std::declval<F>()(std::declval<Real>()))
#endif
{
#if BOOST_WORKAROUND(BOOST_MSVC, <= 1600)
   if (tol == 0)
      tol = boost::math::tools::root_epsilon<Real>();
#endif
   return trapezoidal(f, a, b, tol, max_refinements, error_estimate, L1, boost::math::policies::policy<>());
}

}}}
#endif