// 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)
|
|
/*
|
* This class performs tanh-sinh quadrature on the real line.
|
* Tanh-sinh quadrature is exponentially convergent for integrands in Hardy spaces,
|
* (see https://en.wikipedia.org/wiki/Hardy_space for a formal definition), and is optimal for a random function from that class.
|
*
|
* The tanh-sinh quadrature is one of a class of so called "double exponential quadratures"-there is a large family of them,
|
* but this one seems to be the most commonly used.
|
*
|
* As always, there are caveats: For instance, if the function you want to integrate is not holomorphic on the unit disk,
|
* then the rapid convergence will be spoiled. In this case, a more appropriate quadrature is (say) Romberg, which does not
|
* require the function to be holomorphic, only differentiable up to some order.
|
*
|
* In addition, if you are integrating a periodic function over a period, the trapezoidal rule is better.
|
*
|
* References:
|
*
|
* 1) Mori, Masatake. "Quadrature formulas obtained by variable transformation and the DE-rule." Journal of Computational and Applied Mathematics 12 (1985): 119-130.
|
* 2) Bailey, David H., Karthik Jeyabalan, and Xiaoye S. Li. "A comparison of three high-precision quadrature schemes." Experimental Mathematics 14.3 (2005): 317-329.
|
* 3) Press, William H., et al. "Numerical recipes third edition: the art of scientific computing." Cambridge University Press 32 (2007): 10013-2473.
|
*
|
*/
|
|
#ifndef BOOST_MATH_QUADRATURE_TANH_SINH_HPP
|
#define BOOST_MATH_QUADRATURE_TANH_SINH_HPP
|
|
#include <cmath>
|
#include <limits>
|
#include <memory>
|
#include <boost/math/quadrature/detail/tanh_sinh_detail.hpp>
|
|
namespace boost{ namespace math{ namespace quadrature {
|
|
template<class Real, class Policy = policies::policy<> >
|
class tanh_sinh
|
{
|
public:
|
tanh_sinh(size_t max_refinements = 15, const Real& min_complement = tools::min_value<Real>() * 4)
|
: m_imp(std::make_shared<detail::tanh_sinh_detail<Real, Policy>>(max_refinements, min_complement)) {}
|
|
template<class F>
|
auto integrate(const F f, Real a, Real b, Real tolerance = tools::root_epsilon<Real>(), Real* error = nullptr, Real* L1 = nullptr, std::size_t* levels = nullptr) ->decltype(std::declval<F>()(std::declval<Real>())) const;
|
template<class F>
|
auto integrate(const F f, Real a, Real b, Real tolerance = tools::root_epsilon<Real>(), Real* error = nullptr, Real* L1 = nullptr, std::size_t* levels = nullptr) ->decltype(std::declval<F>()(std::declval<Real>(), std::declval<Real>())) const;
|
|
template<class F>
|
auto integrate(const F f, Real tolerance = tools::root_epsilon<Real>(), Real* error = nullptr, Real* L1 = nullptr, std::size_t* levels = nullptr) ->decltype(std::declval<F>()(std::declval<Real>())) const;
|
template<class F>
|
auto integrate(const F f, Real tolerance = tools::root_epsilon<Real>(), Real* error = nullptr, Real* L1 = nullptr, std::size_t* levels = nullptr) ->decltype(std::declval<F>()(std::declval<Real>(), std::declval<Real>())) const;
|
|
private:
|
std::shared_ptr<detail::tanh_sinh_detail<Real, Policy>> m_imp;
|
};
|
|
template<class Real, class Policy>
|
template<class F>
|
auto tanh_sinh<Real, Policy>::integrate(const F f, Real a, Real b, Real tolerance, Real* error, Real* L1, std::size_t* levels) ->decltype(std::declval<F>()(std::declval<Real>())) const
|
{
|
BOOST_MATH_STD_USING
|
using boost::math::constants::half;
|
using boost::math::quadrature::detail::tanh_sinh_detail;
|
|
static const char* function = "tanh_sinh<%1%>::integrate";
|
|
typedef decltype(std::declval<F>()(std::declval<Real>())) result_type;
|
|
if (!(boost::math::isnan)(a) && !(boost::math::isnan)(b))
|
{
|
|
// Infinite limits:
|
if ((a <= -tools::max_value<Real>()) && (b >= tools::max_value<Real>()))
|
{
|
auto u = [&](const Real& t, const Real& tc)->result_type
|
{
|
Real t_sq = t*t;
|
Real inv;
|
if (t > 0.5f)
|
inv = 1 / ((2 - tc) * tc);
|
else if(t < -0.5)
|
inv = 1 / ((2 + tc) * -tc);
|
else
|
inv = 1 / (1 - t_sq);
|
return f(t*inv)*(1 + t_sq)*inv*inv;
|
};
|
Real limit = sqrt(tools::min_value<Real>()) * 4;
|
return m_imp->integrate(u, error, L1, function, limit, limit, tolerance, levels);
|
}
|
|
// Right limit is infinite:
|
if ((boost::math::isfinite)(a) && (b >= tools::max_value<Real>()))
|
{
|
auto u = [&](const Real& t, const Real& tc)->result_type
|
{
|
Real z, arg;
|
if (t > -0.5f)
|
z = 1 / (t + 1);
|
else
|
z = -1 / tc;
|
if (t < 0.5)
|
arg = 2 * z + a - 1;
|
else
|
arg = a + tc / (2 - tc);
|
return f(arg)*z*z;
|
};
|
Real left_limit = sqrt(tools::min_value<Real>()) * 4;
|
result_type Q = Real(2) * m_imp->integrate(u, error, L1, function, left_limit, tools::min_value<Real>(), tolerance, levels);
|
if (L1)
|
{
|
*L1 *= 2;
|
}
|
if (error)
|
{
|
*error *= 2;
|
}
|
|
return Q;
|
}
|
|
if ((boost::math::isfinite)(b) && (a <= -tools::max_value<Real>()))
|
{
|
auto v = [&](const Real& t, const Real& tc)->result_type
|
{
|
Real z;
|
if (t > -0.5)
|
z = 1 / (t + 1);
|
else
|
z = -1 / tc;
|
Real arg;
|
if (t < 0.5)
|
arg = 2 * z - 1;
|
else
|
arg = tc / (2 - tc);
|
return f(b - arg) * z * z;
|
};
|
|
Real left_limit = sqrt(tools::min_value<Real>()) * 4;
|
result_type Q = Real(2) * m_imp->integrate(v, error, L1, function, left_limit, tools::min_value<Real>(), tolerance, levels);
|
if (L1)
|
{
|
*L1 *= 2;
|
}
|
if (error)
|
{
|
*error *= 2;
|
}
|
return Q;
|
}
|
|
if ((boost::math::isfinite)(a) && (boost::math::isfinite)(b))
|
{
|
if (a == b)
|
{
|
return result_type(0);
|
}
|
if (b < a)
|
{
|
return -this->integrate(f, b, a, tolerance, error, L1, levels);
|
}
|
Real avg = (a + b)*half<Real>();
|
Real diff = (b - a)*half<Real>();
|
Real avg_over_diff_m1 = a / diff;
|
Real avg_over_diff_p1 = b / diff;
|
bool have_small_left = fabs(a) < 0.5f;
|
bool have_small_right = fabs(b) < 0.5f;
|
Real left_min_complement = float_next(avg_over_diff_m1) - avg_over_diff_m1;
|
Real min_complement_limit = (std::max)(tools::min_value<Real>(), Real(tools::min_value<Real>() / diff));
|
if (left_min_complement < min_complement_limit)
|
left_min_complement = min_complement_limit;
|
Real right_min_complement = avg_over_diff_p1 - float_prior(avg_over_diff_p1);
|
if (right_min_complement < min_complement_limit)
|
right_min_complement = min_complement_limit;
|
//
|
// These asserts will fail only if rounding errors on
|
// type Real have accumulated so much error that it's
|
// broken our internal logic. Should that prove to be
|
// a persistent issue, we might need to add a bit of fudge
|
// factor to move left_min_complement and right_min_complement
|
// further from the end points of the range.
|
//
|
BOOST_ASSERT((left_min_complement * diff + a) > a);
|
BOOST_ASSERT((b - right_min_complement * diff) < b);
|
auto u = [&](Real z, Real zc)->result_type
|
{
|
Real position;
|
if (z < -0.5)
|
{
|
if(have_small_left)
|
return f(diff * (avg_over_diff_m1 - zc));
|
position = a - diff * zc;
|
}
|
else if (z > 0.5)
|
{
|
if(have_small_right)
|
return f(diff * (avg_over_diff_p1 - zc));
|
position = b - diff * zc;
|
}
|
else
|
position = avg + diff*z;
|
BOOST_ASSERT(position != a);
|
BOOST_ASSERT(position != b);
|
return f(position);
|
};
|
result_type Q = diff*m_imp->integrate(u, error, L1, function, left_min_complement, right_min_complement, tolerance, levels);
|
|
if (L1)
|
{
|
*L1 *= diff;
|
}
|
if (error)
|
{
|
*error *= diff;
|
}
|
return Q;
|
}
|
}
|
return policies::raise_domain_error(function, "The domain of integration is not sensible; please check the bounds.", a, Policy());
|
}
|
|
template<class Real, class Policy>
|
template<class F>
|
auto tanh_sinh<Real, Policy>::integrate(const F f, Real a, Real b, Real tolerance, Real* error, Real* L1, std::size_t* levels) ->decltype(std::declval<F>()(std::declval<Real>(), std::declval<Real>())) const
|
{
|
BOOST_MATH_STD_USING
|
using boost::math::constants::half;
|
using boost::math::quadrature::detail::tanh_sinh_detail;
|
|
static const char* function = "tanh_sinh<%1%>::integrate";
|
|
if ((boost::math::isfinite)(a) && (boost::math::isfinite)(b))
|
{
|
if (b <= a)
|
{
|
return policies::raise_domain_error(function, "Arguments to integrate are in wrong order; integration over [a,b] must have b > a.", a, Policy());
|
}
|
auto u = [&](Real z, Real zc)->Real
|
{
|
if (z < 0)
|
return f((a - b) * zc / 2 + a, (b - a) * zc / 2);
|
else
|
return f((a - b) * zc / 2 + b, (b - a) * zc / 2);
|
};
|
Real diff = (b - a)*half<Real>();
|
Real left_min_complement = tools::min_value<Real>() * 4;
|
Real right_min_complement = tools::min_value<Real>() * 4;
|
Real Q = diff*m_imp->integrate(u, error, L1, function, left_min_complement, right_min_complement, tolerance, levels);
|
|
if (L1)
|
{
|
*L1 *= diff;
|
}
|
if (error)
|
{
|
*error *= diff;
|
}
|
return Q;
|
}
|
return policies::raise_domain_error(function, "The domain of integration is not sensible; please check the bounds.", a, Policy());
|
}
|
|
template<class Real, class Policy>
|
template<class F>
|
auto tanh_sinh<Real, Policy>::integrate(const F f, Real tolerance, Real* error, Real* L1, std::size_t* levels) ->decltype(std::declval<F>()(std::declval<Real>())) const
|
{
|
using boost::math::quadrature::detail::tanh_sinh_detail;
|
static const char* function = "tanh_sinh<%1%>::integrate";
|
Real min_complement = tools::epsilon<Real>();
|
return m_imp->integrate([&](const Real& arg, const Real&) { return f(arg); }, error, L1, function, min_complement, min_complement, tolerance, levels);
|
}
|
|
template<class Real, class Policy>
|
template<class F>
|
auto tanh_sinh<Real, Policy>::integrate(const F f, Real tolerance, Real* error, Real* L1, std::size_t* levels) ->decltype(std::declval<F>()(std::declval<Real>(), std::declval<Real>())) const
|
{
|
using boost::math::quadrature::detail::tanh_sinh_detail;
|
static const char* function = "tanh_sinh<%1%>::integrate";
|
Real min_complement = tools::min_value<Real>() * 4;
|
return m_imp->integrate(f, error, L1, function, min_complement, min_complement, tolerance, levels);
|
}
|
|
}
|
}
|
}
|
#endif
|
