// Copyright Nick Thompson, 2019
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0.
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// (See accompanying file LICENSE_1_0.txt
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// or copy at http://www.boost.org/LICENSE_1_0.txt)
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#ifndef BOOST_MATH_QUADRATURE_DETAIL_OOURA_FOURIER_INTEGRALS_DETAIL_HPP
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#define BOOST_MATH_QUADRATURE_DETAIL_OOURA_FOURIER_INTEGRALS_DETAIL_HPP
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#include <utility> // for std::pair.
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#include <mutex>
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#include <atomic>
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#include <vector>
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#include <iostream>
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#include <boost/math/special_functions/expm1.hpp>
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#include <boost/math/special_functions/sin_pi.hpp>
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#include <boost/math/special_functions/cos_pi.hpp>
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#include <boost/math/constants/constants.hpp>
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namespace boost { namespace math { namespace quadrature { namespace detail {
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// Ooura and Mori, A robust double exponential formula for Fourier-type integrals,
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// eta is the argument to the exponential in equation 3.3:
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template<class Real>
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std::pair<Real, Real> ooura_eta(Real x, Real alpha) {
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using std::expm1;
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using std::exp;
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using std::abs;
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Real expx = exp(x);
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Real eta_prime = 2 + alpha/expx + expx/4;
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Real eta;
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// This is the fast branch:
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if (abs(x) > 0.125) {
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eta = 2*x - alpha*(1/expx - 1) + (expx - 1)/4;
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}
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else {// this is the slow branch using expm1 for small x:
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eta = 2*x - alpha*expm1(-x) + expm1(x)/4;
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}
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return {eta, eta_prime};
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}
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// Ooura and Mori, A robust double exponential formula for Fourier-type integrals,
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// equation 3.6:
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template<class Real>
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Real calculate_ooura_alpha(Real h)
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{
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using boost::math::constants::pi;
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using std::log1p;
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using std::sqrt;
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Real x = sqrt(16 + 4*log1p(pi<Real>()/h)/h);
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return 1/x;
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}
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template<class Real>
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std::pair<Real, Real> ooura_sin_node_and_weight(long n, Real h, Real alpha)
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{
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using std::expm1;
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using std::exp;
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using std::abs;
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using boost::math::constants::pi;
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using std::isnan;
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if (n == 0) {
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// Equation 44 of https://arxiv.org/pdf/0911.4796.pdf
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// Fourier Transform of the Stretched Exponential Function: Analytic Error Bounds,
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// Double Exponential Transform, and Open-Source Implementation,
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// Joachim Wuttke,
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// The C library libkww provides functions to compute the Kohlrausch-Williams-Watts function,
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// the Laplace-Fourier transform of the stretched (or compressed) exponential function exp(-t^beta)
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// for exponent beta between 0.1 and 1.9 with sixteen decimal digits accuracy.
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Real eta_prime_0 = Real(2) + alpha + Real(1)/Real(4);
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Real node = pi<Real>()/(eta_prime_0*h);
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Real weight = pi<Real>()*boost::math::sin_pi(1/(eta_prime_0*h));
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Real eta_dbl_prime = -alpha + Real(1)/Real(4);
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Real phi_prime_0 = (1 - eta_dbl_prime/(eta_prime_0*eta_prime_0))/2;
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weight *= phi_prime_0;
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return {node, weight};
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}
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Real x = n*h;
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auto p = ooura_eta(x, alpha);
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auto eta = p.first;
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auto eta_prime = p.second;
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Real expm1_meta = expm1(-eta);
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Real exp_meta = exp(-eta);
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Real node = -n*pi<Real>()/expm1_meta;
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// I have verified that this is not a significant source of inaccuracy in the weight computation:
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Real phi_prime = -(expm1_meta + x*exp_meta*eta_prime)/(expm1_meta*expm1_meta);
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// The main source of inaccuracy is in computation of sin_pi.
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// But I've agonized over this, and I think it's as good as it can get:
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Real s = pi<Real>();
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Real arg;
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if(eta > 1) {
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arg = n/( 1/exp_meta - 1 );
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s *= boost::math::sin_pi(arg);
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if (n&1) {
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s *= -1;
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}
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}
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else if (eta < -1) {
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arg = n/(1-exp_meta);
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s *= boost::math::sin_pi(arg);
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}
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else {
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arg = -n*exp_meta/expm1_meta;
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s *= boost::math::sin_pi(arg);
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if (n&1) {
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s *= -1;
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}
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}
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Real weight = s*phi_prime;
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return {node, weight};
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}
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#ifdef BOOST_MATH_INSTRUMENT_OOURA
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template<class Real>
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void print_ooura_estimate(size_t i, Real I0, Real I1, Real omega) {
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using std::abs;
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std::cout << std::defaultfloat
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<< std::setprecision(std::numeric_limits<Real>::digits10)
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<< std::fixed;
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std::cout << "h = " << Real(1)/Real(1<<i) << ", I_h = " << I0/omega
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<< " = " << std::hexfloat << I0/omega << ", absolute error estimate = "
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<< std::defaultfloat << std::scientific << abs(I0-I1) << std::endl;
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}
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#endif
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template<class Real>
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std::pair<Real, Real> ooura_cos_node_and_weight(long n, Real h, Real alpha)
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{
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using std::expm1;
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using std::exp;
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using std::abs;
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using boost::math::constants::pi;
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Real x = h*(n-Real(1)/Real(2));
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auto p = ooura_eta(x, alpha);
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auto eta = p.first;
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auto eta_prime = p.second;
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Real expm1_meta = expm1(-eta);
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Real exp_meta = exp(-eta);
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Real node = pi<Real>()*(Real(1)/Real(2)-n)/expm1_meta;
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Real phi_prime = -(expm1_meta + x*exp_meta*eta_prime)/(expm1_meta*expm1_meta);
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// Takuya Ooura and Masatake Mori,
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// Journal of Computational and Applied Mathematics, 112 (1999) 229-241.
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// A robust double exponential formula for Fourier-type integrals.
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// Equation 4.6
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Real s = pi<Real>();
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Real arg;
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if (eta < -1) {
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arg = -(n-Real(1)/Real(2))/expm1_meta;
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s *= boost::math::cos_pi(arg);
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}
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else {
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arg = -(n-Real(1)/Real(2))*exp_meta/expm1_meta;
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s *= boost::math::sin_pi(arg);
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if (n&1) {
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s *= -1;
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}
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}
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Real weight = s*phi_prime;
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return {node, weight};
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}
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template<class Real>
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class ooura_fourier_sin_detail {
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public:
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ooura_fourier_sin_detail(const Real relative_error_goal, size_t levels) {
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#ifdef BOOST_MATH_INSTRUMENT_OOURA
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std::cout << "ooura_fourier_sin with relative error goal " << relative_error_goal
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<< " & " << levels << " levels." << std::endl;
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#endif // BOOST_MATH_INSTRUMENT_OOURA
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if (relative_error_goal < std::numeric_limits<Real>::epsilon() * 2) {
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throw std::domain_error("The relative error goal cannot be smaller than the unit roundoff.");
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}
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using std::abs;
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requested_levels_ = levels;
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starting_level_ = 0;
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rel_err_goal_ = relative_error_goal;
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big_nodes_.reserve(levels);
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bweights_.reserve(levels);
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little_nodes_.reserve(levels);
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lweights_.reserve(levels);
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for (size_t i = 0; i < levels; ++i) {
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if (std::is_same<Real, float>::value) {
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add_level<double>(i);
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}
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else if (std::is_same<Real, double>::value) {
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add_level<long double>(i);
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}
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else {
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add_level<Real>(i);
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}
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}
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}
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std::vector<std::vector<Real>> const & big_nodes() const {
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return big_nodes_;
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}
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std::vector<std::vector<Real>> const & weights_for_big_nodes() const {
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return bweights_;
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}
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std::vector<std::vector<Real>> const & little_nodes() const {
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return little_nodes_;
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}
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std::vector<std::vector<Real>> const & weights_for_little_nodes() const {
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return lweights_;
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}
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template<class F>
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std::pair<Real,Real> integrate(F const & f, Real omega) {
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using std::abs;
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using std::max;
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using boost::math::constants::pi;
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if (omega == 0) {
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return {Real(0), Real(0)};
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}
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if (omega < 0) {
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auto p = this->integrate(f, -omega);
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return {-p.first, p.second};
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}
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Real I1 = std::numeric_limits<Real>::quiet_NaN();
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Real relative_error_estimate = std::numeric_limits<Real>::quiet_NaN();
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// As we compute integrals, we learn about their structure.
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// Assuming we compute f(t)sin(wt) for many different omega, this gives some
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// a posteriori ability to choose a refinement level that is roughly appropriate.
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size_t i = starting_level_;
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do {
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Real I0 = estimate_integral(f, omega, i);
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#ifdef BOOST_MATH_INSTRUMENT_OOURA
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print_ooura_estimate(i, I0, I1, omega);
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#endif
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Real absolute_error_estimate = abs(I0-I1);
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Real scale = (max)(abs(I0), abs(I1));
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if (!isnan(I1) && absolute_error_estimate <= rel_err_goal_*scale) {
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starting_level_ = (max)(long(i) - 1, long(0));
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return {I0/omega, absolute_error_estimate/scale};
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}
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I1 = I0;
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} while(++i < big_nodes_.size());
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// We've used up all our precomputed levels.
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// Now we need to add more.
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// It might seems reasonable to just keep adding levels indefinitely, if that's what the user wants.
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// But in fact the nodes and weights just merge into each other and the error gets worse after a certain number.
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// This value for max_additional_levels was chosen by observation of a slowly converging oscillatory integral:
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// f(x) := cos(7cos(x))sin(x)/x
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size_t max_additional_levels = 4;
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while (big_nodes_.size() < requested_levels_ + max_additional_levels) {
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size_t ii = big_nodes_.size();
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if (std::is_same<Real, float>::value) {
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add_level<double>(ii);
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}
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else if (std::is_same<Real, double>::value) {
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add_level<long double>(ii);
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}
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else {
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add_level<Real>(ii);
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}
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Real I0 = estimate_integral(f, omega, ii);
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Real absolute_error_estimate = abs(I0-I1);
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Real scale = (max)(abs(I0), abs(I1));
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#ifdef BOOST_MATH_INSTRUMENT_OOURA
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print_ooura_estimate(ii, I0, I1, omega);
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#endif
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if (absolute_error_estimate <= rel_err_goal_*scale) {
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starting_level_ = (max)(long(ii) - 1, long(0));
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return {I0/omega, absolute_error_estimate/scale};
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}
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I1 = I0;
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++ii;
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}
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starting_level_ = static_cast<long>(big_nodes_.size() - 2);
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return {I1/omega, relative_error_estimate};
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}
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private:
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template<class PreciseReal>
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void add_level(size_t i) {
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using std::abs;
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size_t current_num_levels = big_nodes_.size();
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Real unit_roundoff = std::numeric_limits<Real>::epsilon()/2;
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// h0 = 1. Then all further levels have h_i = 1/2^i.
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// Since the nodes don't nest, we could conceivably divide h by (say) 1.5, or 3.
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// It's not clear how much benefit (or loss) would be obtained from this.
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PreciseReal h = PreciseReal(1)/PreciseReal(1<<i);
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std::vector<Real> bnode_row;
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std::vector<Real> bweight_row;
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// This is a pretty good estimate for how many elements will be placed in the vector:
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bnode_row.reserve((static_cast<size_t>(1)<<i)*sizeof(Real));
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bweight_row.reserve((static_cast<size_t>(1)<<i)*sizeof(Real));
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std::vector<Real> lnode_row;
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std::vector<Real> lweight_row;
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lnode_row.reserve((static_cast<size_t>(1)<<i)*sizeof(Real));
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lweight_row.reserve((static_cast<size_t>(1)<<i)*sizeof(Real));
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Real max_weight = 1;
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auto alpha = calculate_ooura_alpha(h);
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long n = 0;
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Real w;
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do {
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auto precise_nw = ooura_sin_node_and_weight(n, h, alpha);
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Real node = static_cast<Real>(precise_nw.first);
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Real weight = static_cast<Real>(precise_nw.second);
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w = weight;
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if (bnode_row.size() == bnode_row.capacity()) {
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bnode_row.reserve(2*bnode_row.size());
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bweight_row.reserve(2*bnode_row.size());
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}
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bnode_row.push_back(node);
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bweight_row.push_back(weight);
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if (abs(weight) > max_weight) {
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max_weight = abs(weight);
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}
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++n;
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// f(t)->0 as t->infty, which is why the weights are computed up to the unit roundoff.
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} while(abs(w) > unit_roundoff*max_weight);
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// This class tends to consume a lot of memory; shrink the vectors back down to size:
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bnode_row.shrink_to_fit();
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bweight_row.shrink_to_fit();
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// Why we are splitting the nodes into regimes where t_n >> 1 and t_n << 1?
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// It will create the opportunity to sensibly truncate the quadrature sum to significant terms.
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n = -1;
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do {
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auto precise_nw = ooura_sin_node_and_weight(n, h, alpha);
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Real node = static_cast<Real>(precise_nw.first);
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if (node <= 0) {
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break;
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}
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Real weight = static_cast<Real>(precise_nw.second);
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w = weight;
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using std::isnan;
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if (isnan(node)) {
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// This occurs at n = -11 in quad precision:
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break;
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}
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if (lnode_row.size() > 0) {
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if (lnode_row[lnode_row.size()-1] == node) {
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// The nodes have fused into each other:
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break;
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}
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}
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if (lnode_row.size() == lnode_row.capacity()) {
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lnode_row.reserve(2*lnode_row.size());
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lweight_row.reserve(2*lnode_row.size());
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}
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lnode_row.push_back(node);
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lweight_row.push_back(weight);
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if (abs(weight) > max_weight) {
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max_weight = abs(weight);
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}
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--n;
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// f(t)->infty is possible as t->0, hence compute up to the min.
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} while(abs(w) > (std::numeric_limits<Real>::min)()*max_weight);
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lnode_row.shrink_to_fit();
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lweight_row.shrink_to_fit();
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// std::scoped_lock once C++17 is more common?
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std::lock_guard<std::mutex> lock(node_weight_mutex_);
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// Another thread might have already finished this calculation and appended it to the nodes/weights:
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if (current_num_levels == big_nodes_.size()) {
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big_nodes_.push_back(bnode_row);
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bweights_.push_back(bweight_row);
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little_nodes_.push_back(lnode_row);
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lweights_.push_back(lweight_row);
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}
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}
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template<class F>
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Real estimate_integral(F const & f, Real omega, size_t i) {
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// Because so few function evaluations are required to get high accuracy on the integrals in the tests,
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// Kahan summation doesn't really help.
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//auto cond = boost::math::tools::summation_condition_number<Real, true>(0);
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Real I0 = 0;
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auto const & b_nodes = big_nodes_[i];
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auto const & b_weights = bweights_[i];
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// Will benchmark if this is helpful:
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Real inv_omega = 1/omega;
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for(size_t j = 0 ; j < b_nodes.size(); ++j) {
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I0 += f(b_nodes[j]*inv_omega)*b_weights[j];
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}
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auto const & l_nodes = little_nodes_[i];
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auto const & l_weights = lweights_[i];
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// If f decays rapidly as |t|->infty, not all of these calls are necessary.
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for (size_t j = 0; j < l_nodes.size(); ++j) {
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I0 += f(l_nodes[j]*inv_omega)*l_weights[j];
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}
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return I0;
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}
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std::mutex node_weight_mutex_;
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// Nodes for n >= 0, giving t_n = pi*phi(nh)/h. Generally t_n >> 1.
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std::vector<std::vector<Real>> big_nodes_;
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// The term bweights_ will indicate that these are weights corresponding
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// to the big nodes:
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std::vector<std::vector<Real>> bweights_;
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// Nodes for n < 0: Generally t_n << 1, and an invariant is that t_n > 0.
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std::vector<std::vector<Real>> little_nodes_;
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std::vector<std::vector<Real>> lweights_;
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Real rel_err_goal_;
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std::atomic<long> starting_level_;
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size_t requested_levels_;
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};
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template<class Real>
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class ooura_fourier_cos_detail {
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public:
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ooura_fourier_cos_detail(const Real relative_error_goal, size_t levels) {
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#ifdef BOOST_MATH_INSTRUMENT_OOURA
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std::cout << "ooura_fourier_cos with relative error goal " << relative_error_goal
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<< " & " << levels << " levels." << std::endl;
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std::cout << "epsilon for type = " << std::numeric_limits<Real>::epsilon() << std::endl;
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#endif // BOOST_MATH_INSTRUMENT_OOURA
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if (relative_error_goal < std::numeric_limits<Real>::epsilon() * 2) {
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throw std::domain_error("The relative error goal cannot be smaller than the unit roundoff!");
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}
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using std::abs;
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requested_levels_ = levels;
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starting_level_ = 0;
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rel_err_goal_ = relative_error_goal;
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big_nodes_.reserve(levels);
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bweights_.reserve(levels);
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little_nodes_.reserve(levels);
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lweights_.reserve(levels);
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for (size_t i = 0; i < levels; ++i) {
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if (std::is_same<Real, float>::value) {
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add_level<double>(i);
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}
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else if (std::is_same<Real, double>::value) {
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add_level<long double>(i);
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}
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else {
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add_level<Real>(i);
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}
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}
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}
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template<class F>
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std::pair<Real,Real> integrate(F const & f, Real omega) {
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using std::abs;
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using std::max;
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using boost::math::constants::pi;
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if (omega == 0) {
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throw std::domain_error("At omega = 0, the integral is not oscillatory. The user must choose an appropriate method for this case.\n");
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}
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if (omega < 0) {
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return this->integrate(f, -omega);
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}
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Real I1 = std::numeric_limits<Real>::quiet_NaN();
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Real absolute_error_estimate = std::numeric_limits<Real>::quiet_NaN();
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Real scale = std::numeric_limits<Real>::quiet_NaN();
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size_t i = starting_level_;
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do {
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Real I0 = estimate_integral(f, omega, i);
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#ifdef BOOST_MATH_INSTRUMENT_OOURA
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print_ooura_estimate(i, I0, I1, omega);
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#endif
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absolute_error_estimate = abs(I0-I1);
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scale = (max)(abs(I0), abs(I1));
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if (!isnan(I1) && absolute_error_estimate <= rel_err_goal_*scale) {
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starting_level_ = (max)(long(i) - 1, long(0));
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return {I0/omega, absolute_error_estimate/scale};
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}
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I1 = I0;
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} while(++i < big_nodes_.size());
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size_t max_additional_levels = 4;
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while (big_nodes_.size() < requested_levels_ + max_additional_levels) {
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size_t ii = big_nodes_.size();
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if (std::is_same<Real, float>::value) {
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add_level<double>(ii);
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}
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else if (std::is_same<Real, double>::value) {
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add_level<long double>(ii);
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}
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else {
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add_level<Real>(ii);
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}
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Real I0 = estimate_integral(f, omega, ii);
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#ifdef BOOST_MATH_INSTRUMENT_OOURA
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print_ooura_estimate(ii, I0, I1, omega);
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#endif
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absolute_error_estimate = abs(I0-I1);
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scale = (max)(abs(I0), abs(I1));
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if (absolute_error_estimate <= rel_err_goal_*scale) {
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starting_level_ = (max)(long(ii) - 1, long(0));
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return {I0/omega, absolute_error_estimate/scale};
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}
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I1 = I0;
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++ii;
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}
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starting_level_ = static_cast<long>(big_nodes_.size() - 2);
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return {I1/omega, absolute_error_estimate/scale};
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}
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private:
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template<class PreciseReal>
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void add_level(size_t i) {
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using std::abs;
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size_t current_num_levels = big_nodes_.size();
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Real unit_roundoff = std::numeric_limits<Real>::epsilon()/2;
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PreciseReal h = PreciseReal(1)/PreciseReal(1<<i);
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std::vector<Real> bnode_row;
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std::vector<Real> bweight_row;
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bnode_row.reserve((static_cast<size_t>(1)<<i)*sizeof(Real));
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bweight_row.reserve((static_cast<size_t>(1)<<i)*sizeof(Real));
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std::vector<Real> lnode_row;
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std::vector<Real> lweight_row;
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lnode_row.reserve((static_cast<size_t>(1)<<i)*sizeof(Real));
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lweight_row.reserve((static_cast<size_t>(1)<<i)*sizeof(Real));
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Real max_weight = 1;
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auto alpha = calculate_ooura_alpha(h);
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long n = 0;
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Real w;
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do {
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auto precise_nw = ooura_cos_node_and_weight(n, h, alpha);
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Real node = static_cast<Real>(precise_nw.first);
|
Real weight = static_cast<Real>(precise_nw.second);
|
w = weight;
|
if (bnode_row.size() == bnode_row.capacity()) {
|
bnode_row.reserve(2*bnode_row.size());
|
bweight_row.reserve(2*bnode_row.size());
|
}
|
|
bnode_row.push_back(node);
|
bweight_row.push_back(weight);
|
if (abs(weight) > max_weight) {
|
max_weight = abs(weight);
|
}
|
++n;
|
// f(t)->0 as t->infty, which is why the weights are computed up to the unit roundoff.
|
} while(abs(w) > unit_roundoff*max_weight);
|
|
bnode_row.shrink_to_fit();
|
bweight_row.shrink_to_fit();
|
n = -1;
|
do {
|
auto precise_nw = ooura_cos_node_and_weight(n, h, alpha);
|
Real node = static_cast<Real>(precise_nw.first);
|
// The function cannot be singular at zero,
|
// so zero is not a unreasonable node,
|
// unlike in the case of the Fourier Sine.
|
// Hence only break if the node is negative.
|
if (node < 0) {
|
break;
|
}
|
Real weight = static_cast<Real>(precise_nw.second);
|
w = weight;
|
if (lnode_row.size() > 0) {
|
if (lnode_row.back() == node) {
|
// The nodes have fused into each other:
|
break;
|
}
|
}
|
if (lnode_row.size() == lnode_row.capacity()) {
|
lnode_row.reserve(2*lnode_row.size());
|
lweight_row.reserve(2*lnode_row.size());
|
}
|
|
lnode_row.push_back(node);
|
lweight_row.push_back(weight);
|
if (abs(weight) > max_weight) {
|
max_weight = abs(weight);
|
}
|
--n;
|
} while(abs(w) > (std::numeric_limits<Real>::min)()*max_weight);
|
|
lnode_row.shrink_to_fit();
|
lweight_row.shrink_to_fit();
|
|
std::lock_guard<std::mutex> lock(node_weight_mutex_);
|
// Another thread might have already finished this calculation and appended it to the nodes/weights:
|
if (current_num_levels == big_nodes_.size()) {
|
big_nodes_.push_back(bnode_row);
|
bweights_.push_back(bweight_row);
|
|
little_nodes_.push_back(lnode_row);
|
lweights_.push_back(lweight_row);
|
}
|
}
|
|
template<class F>
|
Real estimate_integral(F const & f, Real omega, size_t i) {
|
Real I0 = 0;
|
auto const & b_nodes = big_nodes_[i];
|
auto const & b_weights = bweights_[i];
|
Real inv_omega = 1/omega;
|
for(size_t j = 0 ; j < b_nodes.size(); ++j) {
|
I0 += f(b_nodes[j]*inv_omega)*b_weights[j];
|
}
|
|
auto const & l_nodes = little_nodes_[i];
|
auto const & l_weights = lweights_[i];
|
for (size_t j = 0; j < l_nodes.size(); ++j) {
|
I0 += f(l_nodes[j]*inv_omega)*l_weights[j];
|
}
|
return I0;
|
}
|
|
std::mutex node_weight_mutex_;
|
std::vector<std::vector<Real>> big_nodes_;
|
std::vector<std::vector<Real>> bweights_;
|
|
std::vector<std::vector<Real>> little_nodes_;
|
std::vector<std::vector<Real>> lweights_;
|
Real rel_err_goal_;
|
std::atomic<long> starting_level_;
|
size_t requested_levels_;
|
};
|
|
|
}}}}
|
#endif
|
