/*
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* Copyright Nick Thompson, 2017
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* Use, modification and distribution are subject to the
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* Boost Software License, Version 1.0. (See accompanying file
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* LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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*
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* Use the adaptive trapezoidal rule to estimate the integral of periodic functions over a period,
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* or to integrate a function whose derivative vanishes at the endpoints.
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*
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* If your function does not satisfy these conditions, and instead is simply continuous and bounded
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* over the whole interval, then this routine will still converge, albeit slowly. However, there
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* are much more efficient methods in this case, including Romberg, Simpson, and double exponential quadrature.
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*/
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#ifndef BOOST_MATH_QUADRATURE_TRAPEZOIDAL_HPP
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#define BOOST_MATH_QUADRATURE_TRAPEZOIDAL_HPP
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#include <cmath>
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#include <limits>
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#include <stdexcept>
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#include <boost/math/constants/constants.hpp>
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#include <boost/math/special_functions/fpclassify.hpp>
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#include <boost/math/policies/error_handling.hpp>
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#include <boost/math/tools/cxx03_warn.hpp>
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namespace boost{ namespace math{ namespace quadrature {
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template<class F, class Real, class Policy>
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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>()))
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{
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static const char* function = "boost::math::quadrature::trapezoidal<%1%>(F, %1%, %1%, %1%)";
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using std::abs;
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using boost::math::constants::half;
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// In many math texts, K represents the field of real or complex numbers.
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// Too bad we can't put blackboard bold into C++ source!
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typedef decltype(f(a)) K;
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if (!(boost::math::isfinite)(a))
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{
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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));
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}
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if (!(boost::math::isfinite)(b))
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{
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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));
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}
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if (a == b)
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{
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return static_cast<K>(0);
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}
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if(a > b)
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{
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return -trapezoidal(f, b, a, tol, max_refinements, error_estimate, L1, pol);
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}
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K ya = f(a);
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K yb = f(b);
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Real h = (b - a)*half<Real>();
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K I0 = (ya + yb)*h;
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Real IL0 = (abs(ya) + abs(yb))*h;
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K yh = f(a + h);
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K I1;
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I1 = I0*half<Real>() + yh*h;
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Real IL1 = IL0*half<Real>() + abs(yh)*h;
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// The recursion is:
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// 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)
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std::size_t k = 2;
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// We want to go through at least 5 levels so we have sampled the function at least 20 times.
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// Otherwise, we could terminate prematurely and miss essential features.
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// This is of course possible anyway, but 20 samples seems to be a reasonable compromise.
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Real error = abs(I0 - I1);
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// I take k < 5, rather than k < 4, or some other smaller minimum number,
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// because I hit a truly exceptional bug where the k = 2 and k =3 refinement were bitwise equal,
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// but the quadrature had not yet converged.
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while (k < 5 || (k < max_refinements && error > tol*IL1) )
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{
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I0 = I1;
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IL0 = IL1;
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I1 = I0*half<Real>();
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IL1 = IL0*half<Real>();
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std::size_t p = static_cast<std::size_t>(1u) << k;
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h *= half<Real>();
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K sum = 0;
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Real absum = 0;
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for(std::size_t j = 1; j < p; j += 2)
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{
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K y = f(a + j*h);
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sum += y;
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absum += abs(y);
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}
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I1 += sum*h;
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IL1 += absum*h;
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++k;
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error = abs(I0 - I1);
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}
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if (error_estimate)
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{
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*error_estimate = error;
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}
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if (L1)
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{
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*L1 = IL1;
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}
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return static_cast<K>(I1);
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}
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#if BOOST_WORKAROUND(BOOST_MSVC, < 1800)
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// Template argument deduction failure otherwise:
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template<class F, class Real>
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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>()))
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#elif !defined(BOOST_NO_CXX11_NULLPTR)
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template<class F, class Real>
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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>()))
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#else
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template<class F, class Real>
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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>()))
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#endif
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{
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#if BOOST_WORKAROUND(BOOST_MSVC, <= 1600)
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if (tol == 0)
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tol = boost::math::tools::root_epsilon<Real>();
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#endif
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return trapezoidal(f, a, b, tol, max_refinements, error_estimate, L1, boost::math::policies::policy<>());
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}
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}}}
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#endif
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