# Continuous Fourier integrals by Ooura's method

I have a PR implementing Ooura and Mori's method for continuous Fourier integrals. I used it here to compute an oscillatory integral that Mathematica got completely wrong, and then I thought "well maybe this method is pretty good!" and figured I'd finish it after letting it languish for over a year.

Here are a few concerns:

• I agonized over computing the nodes and weights accurately. But in the end, I had to precompute the nodes and weights in higher accuracy and cast them back down. Is there any way to avoid this that I'm missing? (If I don't do this, then the error goes down after a few levels, then starts increasing. Use -DBOOST_MATH_INSTRUMENT_OOURA to see it if you're interested.)

• There is also some code duplication that I can't really figure out how to get rid of. For example, for the sine integral, f(0) = inf is allowed, but for the cosine integral, f(0) must be finite. So you have this very small difference that disallows extracting the generic code into a single location. But maybe I'm just not being creative enough here.

• I'm also interested in whether the comments are well-written and informative. I couldn't understand my comments from when I stopped working on this last year, and now I'm worried I won't be able to understand my current comments next year!

So here's the code:

// Copyright Nick Thompson, 2019
// Use, modification and distribution are subject to the
// Boost Software License, Version 1.0.

/*
* References:
* Ooura, Takuya, and Masatake Mori. "A robust double exponential formula for Fourier-type integrals." Journal of computational and applied mathematics 112.1-2 (1999): 229-241.
* http://www.kurims.kyoto-u.ac.jp/~ooura/intde.html
*/
#include <memory>

namespace boost { namespace math { namespace quadrature {

template<class Real>
class ooura_fourier_sin {
public:
ooura_fourier_sin(const Real relative_error_tolerance = tools::root_epsilon<Real>(), size_t levels = sizeof(Real)) : impl_(std::make_shared<detail::ooura_fourier_sin_detail<Real>>(relative_error_tolerance, levels))
{}

template<class F>
std::pair<Real, Real> integrate(F const & f, Real omega) {
return impl_->integrate(f, omega);
}

std::vector<std::vector<Real>> const & big_nodes() const {
return impl_->big_nodes();
}

std::vector<std::vector<Real>> const & weights_for_big_nodes() const {
return impl_->weights_for_big_nodes();
}

std::vector<std::vector<Real>> const & little_nodes() const {
return impl_->little_nodes();
}

std::vector<std::vector<Real>> const & weights_for_little_nodes() const {
return impl_->weights_for_little_nodes();
}

private:
std::shared_ptr<detail::ooura_fourier_sin_detail<Real>> impl_;
};

template<class Real>
class ooura_fourier_cos {
public:
ooura_fourier_cos(const Real relative_error_tolerance = tools::root_epsilon<Real>(), size_t levels = sizeof(Real)) : impl_(std::make_shared<detail::ooura_fourier_cos_detail<Real>>(relative_error_tolerance, levels))
{}

template<class F>
std::pair<Real, Real> integrate(F const & f, Real omega) {
return impl_->integrate(f, omega);
}
private:
std::shared_ptr<detail::ooura_fourier_cos_detail<Real>> impl_;
};

}}}
#endif


And the detail (which contains the real meat):

// Copyright Nick Thompson, 2019
// Use, modification and distribution are subject to the
// Boost Software License, Version 1.0.
#include <utility> // for std::pair.
#include <mutex>
#include <atomic>
#include <vector>
#include <iostream>
#include <boost/math/special_functions/expm1.hpp>
#include <boost/math/special_functions/sin_pi.hpp>
#include <boost/math/special_functions/cos_pi.hpp>
#include <boost/math/constants/constants.hpp>

namespace boost { namespace math { namespace quadrature { namespace detail {

// Ooura and Mori, A robust double exponential formula for Fourier-type integrals,
// eta is the argument to the exponential in equation 3.3:
template<class Real>
std::pair<Real, Real> ooura_eta(Real x, Real alpha) {
using std::expm1;
using std::exp;
using std::abs;
Real expx = exp(x);
Real eta_prime = 2 + alpha/expx + expx/4;
Real eta;
// This is the fast branch:
if (abs(x) > 0.125) {
eta = 2*x - alpha*(1/expx - 1) + (expx - 1)/4;
}
else {// this is the slow branch using expm1 for small x:
eta = 2*x - alpha*expm1(-x) + expm1(x)/4;
}
return {eta, eta_prime};
}

// Ooura and Mori, A robust double exponential formula for Fourier-type integrals,
// equation 3.6:
template<class Real>
Real calculate_ooura_alpha(Real h)
{
using boost::math::constants::pi;
using std::log1p;
using std::sqrt;
Real x = sqrt(16 + 4*log1p(pi<Real>()/h)/h);
return 1/x;
}

template<class Real>
std::pair<Real, Real> ooura_sin_node_and_weight(long n, Real h, Real alpha)
{
using std::expm1;
using std::exp;
using std::abs;
using boost::math::constants::pi;
using std::isnan;

if (n == 0) {
// Equation 44 of https://arxiv.org/pdf/0911.4796.pdf
Real eta_prime_0 = Real(2) + alpha + Real(1)/Real(4);
Real node = pi<Real>()/(eta_prime_0*h);
Real weight = pi<Real>()*boost::math::sin_pi(1/(eta_prime_0*h));
Real eta_dbl_prime = -alpha + Real(1)/Real(4);
Real phi_prime_0 = (1 - eta_dbl_prime/(eta_prime_0*eta_prime_0))/2;
weight *= phi_prime_0;
return {node, weight};
}
Real x = n*h;
auto p = ooura_eta(x, alpha);
auto eta = p.first;
auto eta_prime = p.second;

Real expm1_meta = expm1(-eta);
Real exp_meta = exp(-eta);
Real node = -n*pi<Real>()/expm1_meta;

// I have verified that this is not a significant source of inaccuracy in the weight computation:
Real phi_prime = -(expm1_meta + x*exp_meta*eta_prime)/(expm1_meta*expm1_meta);

// The main source of inaccuracy is in computation of sin_pi.
// But I've agonized over this, and I think it's as good as it can get:
Real s = pi<Real>();
Real arg;
if(eta > 1) {
arg = n/( 1/exp_meta - 1 );
s *= boost::math::sin_pi(arg);
if (n&1) {
s *= -1;
}
}
else if (eta < -1) {
arg = n/(1-exp_meta);
s *= boost::math::sin_pi(arg);
}
else {
arg = -n*exp_meta/expm1_meta;
s *= boost::math::sin_pi(arg);
if (n&1) {
s *= -1;
}
}

Real weight = s*phi_prime;
return {node, weight};
}

#ifdef BOOST_MATH_INSTRUMENT_OOURA
template<class Real>
void print_ooura_estimate(size_t i, Real I0, Real I1, Real omega) {
using std::abs;
std::cout << std::defaultfloat
<< std::setprecision(std::numeric_limits<Real>::digits10)
<< std::fixed;
std::cout << "h = " << Real(1)/Real(1<<i) << ", I_h = " << I0/omega
<< " = " << std::hexfloat << I0/omega << ", absolute error est = "
<< std::defaultfloat << std::scientific << abs(I0-I1)  << "\n";
}
#endif

template<class Real>
std::pair<Real, Real> ooura_cos_node_and_weight(long n, Real h, Real alpha)
{
using std::expm1;
using std::exp;
using std::abs;
using boost::math::constants::pi;

Real x = h*(n-Real(1)/Real(2));
auto p = ooura_eta(x, alpha);
auto eta = p.first;
auto eta_prime = p.second;
Real expm1_meta = expm1(-eta);
Real exp_meta = exp(-eta);
Real node = pi<Real>()*(Real(1)/Real(2)-n)/expm1_meta;

Real phi_prime = -(expm1_meta + x*exp_meta*eta_prime)/(expm1_meta*expm1_meta);

// Equation 4.6 of A robust double exponential formula for Fourier-type integrals
Real s = pi<Real>();
Real arg;
if (eta < -1) {
arg = -(n-Real(1)/Real(2))/expm1_meta;
s *= boost::math::cos_pi(arg);
}
else {
arg = -(n-Real(1)/Real(2))*exp_meta/expm1_meta;
s *= boost::math::sin_pi(arg);
if (n&1) {
s *= -1;
}
}

Real weight = s*phi_prime;
return {node, weight};
}

template<class Real>
class ooura_fourier_sin_detail {
public:
ooura_fourier_sin_detail(const Real relative_error_goal, size_t levels) {
if (relative_error_goal <= std::numeric_limits<Real>::epsilon()/2) {
throw std::domain_error("The relative error goal cannot be smaller than the unit roundoff.");
}
using std::abs;
requested_levels_ = levels;
starting_level_ = 0;
rel_err_goal_ = relative_error_goal;
big_nodes_.reserve(levels);
bweights_.reserve(levels);
little_nodes_.reserve(levels);
lweights_.reserve(levels);

for (size_t i = 0; i < levels; ++i) {
if (std::is_same<Real, float>::value) {
}
else if (std::is_same<Real, double>::value) {
}
else {
}
}
}

std::vector<std::vector<Real>> const & big_nodes() const {
return big_nodes_;
}

std::vector<std::vector<Real>> const & weights_for_big_nodes() const {
return bweights_;
}

std::vector<std::vector<Real>> const & little_nodes() const {
return little_nodes_;
}

std::vector<std::vector<Real>> const & weights_for_little_nodes() const {
return lweights_;
}

template<class F>
std::pair<Real,Real> integrate(F const & f, Real omega) {
using std::abs;
using std::max;
using boost::math::constants::pi;

if (omega == 0) {
return {Real(0), Real(0)};
}
if (omega < 0) {
auto p = this->integrate(f, -omega);
return {-p.first, p.second};
}

Real I1 = std::numeric_limits<Real>::quiet_NaN();
Real relative_error_estimate = std::numeric_limits<Real>::quiet_NaN();
// As we compute integrals, we learn about their structure.
// Assuming we compute f(t)sin(wt) for many different omega, this gives some
// a posteriori ability to choose a refinement level that is roughly appropriate.
size_t i = starting_level_;
do {
Real I0 = estimate_integral(f, omega, i);
#ifdef BOOST_MATH_INSTRUMENT_OOURA
print_ooura_estimate(i, I0, I1, omega);
#endif
Real absolute_error_estimate = abs(I0-I1);
Real scale = max(abs(I0), abs(I1));
if (!isnan(I1) && absolute_error_estimate <= rel_err_goal_*scale) {
starting_level_ = std::max(long(i) - 1, long(0));
return {I0/omega, absolute_error_estimate/scale};
}
I1 = I0;
} while(++i < big_nodes_.size());

// We've used up all our precomputed levels.
// Now we need to add more.
// It might seems reasonable to just keep adding levels indefinitely, if that's what the user wants.
// But in fact the nodes and weights just merge into each other and the error gets worse after a certain number.
// This value for max_additional_levels was chosen by observation of a slowly converging oscillatory integral:
// f(x) := cos(7cos(x))sin(x)/x
while (big_nodes_.size() < requested_levels_ + max_additional_levels) {
size_t i = big_nodes_.size();
if (std::is_same<Real, float>::value) {
}
else if (std::is_same<Real, double>::value) {
}
else {
}
Real I0 = estimate_integral(f, omega, i);
Real absolute_error_estimate = abs(I0-I1);
Real scale = max(abs(I0), abs(I1));
#ifdef BOOST_MATH_INSTRUMENT_OOURA
print_ooura_estimate(i, I0, I1, omega);
#endif
if (absolute_error_estimate <= rel_err_goal_*scale) {
starting_level_ = std::max(long(i) - 1, long(0));
return {I0/omega, absolute_error_estimate/scale};
}
I1 = I0;
++i;
}

starting_level_ = big_nodes_.size() - 2;
return {I1/omega, relative_error_estimate};
}

private:

template<class PreciseReal>
size_t current_num_levels = big_nodes_.size();
Real unit_roundoff = std::numeric_limits<Real>::epsilon()/2;
// h0 = 1. Then all further levels have h_i = 1/2^i.
// Since the nodes don't nest, we could conceivably divide h by (say) 1.5, or 3.
// It's not clear how much benefit (or loss) would be obtained from this.
PreciseReal h = PreciseReal(1)/PreciseReal(1<<i);

std::vector<Real> bnode_row;
std::vector<Real> bweight_row;
// Definitely could use a more sophisticated heuristic for how many elements
// will be placed in the vector. This is a pretty huge overestimate:
bnode_row.reserve((1<<i)*sizeof(Real));
bweight_row.reserve((1<<i)*sizeof(Real));

std::vector<Real> lnode_row;
std::vector<Real> lweight_row;

lnode_row.reserve((1<<i)*sizeof(Real));
lweight_row.reserve((1<<i)*sizeof(Real));

Real max_weight = 1;
auto alpha = calculate_ooura_alpha(h);
long n = 0;
Real w;
do {
auto precise_nw = ooura_sin_node_and_weight(n, h, alpha);
Real node = static_cast<Real>(precise_nw.first);
Real weight = static_cast<Real>(precise_nw.second);
w = weight;
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);

// This class tends to consume a lot of memory; shrink the vectors back down to size:
bnode_row.shrink_to_fit();
bweight_row.shrink_to_fit();
// Why we are splitting the nodes into regimes where t_n >> 1 and t_n << 1?
// It will create the opportunity to sensibly truncate the quadrature sum to significant terms.
n = -1;
do {
auto precise_nw = ooura_sin_node_and_weight(n, h, alpha);
Real node = static_cast<Real>(precise_nw.first);
if (node <= 0) {
break;
}
Real weight = static_cast<Real>(precise_nw.second);
w = weight;
using std::isnan;
if (isnan(node)) {
// This occurs at n = -11 in quad precision:
break;
}
if (lnode_row.size() > 0) {
if (lnode_row[lnode_row.size()-1] == node) {
// The nodes have fused into each other:
break;
}
}
lnode_row.push_back(node);
lweight_row.push_back(weight);
if (abs(weight) > max_weight) {
max_weight = abs(weight);
}
--n;
// f(t)->infty is possible as t->0, hence compute up to the min.
} while(abs(w) > std::numeric_limits<Real>::min()*max_weight);

lnode_row.shrink_to_fit();
lweight_row.shrink_to_fit();

// std::scoped_lock once C++17 is more common?
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) {
// Because so few function evaluations are required to get high accuracy on the integrals in the tests,
// Kahan summation doesn't really help.
//auto cond = boost::math::tools::summation_condition_number<Real, true>(0);
Real I0 = 0;
auto const & b_nodes = big_nodes_[i];
auto const & b_weights = bweights_[i];
// Will benchmark if this is helpful:
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];
// If f decays rapidly as |t|->infty, not all of these calls are necessary.
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_;
// Nodes for n >= 0, giving t_n = pi*phi(nh)/h. Generally t_n >> 1.
std::vector<std::vector<Real>> big_nodes_;
// The term bweights_ will indicate that these are weights corresponding
// to the big nodes:
std::vector<std::vector<Real>> bweights_;

// Nodes for n < 0: Generally t_n << 1, and an invariant is that t_n > 0.
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_;
};

template<class Real>
class ooura_fourier_cos_detail {
public:
ooura_fourier_cos_detail(const Real relative_error_goal, size_t levels) {
if (relative_error_goal <= std::numeric_limits<Real>::epsilon()/2) {
throw std::domain_error("The relative error goal cannot be smaller than the unit roundoff.");
}
using std::abs;
requested_levels_ = levels;
starting_level_ = 0;
rel_err_goal_ = relative_error_goal;
big_nodes_.reserve(levels);
bweights_.reserve(levels);
little_nodes_.reserve(levels);
lweights_.reserve(levels);

for (size_t i = 0; i < levels; ++i) {
if (std::is_same<Real, float>::value) {
}
else if (std::is_same<Real, double>::value) {
}
else {
}
}

}

template<class F>
std::pair<Real,Real> integrate(F const & f, Real omega) {
using std::abs;
using std::max;
using boost::math::constants::pi;

if (omega == 0) {
throw std::domain_error("At omega = 0, the integral is not oscillatory. The user must choose an appropriate method for this case.\n");
}

if (omega < 0) {
return this->integrate(f, -omega);
}

Real I1 = std::numeric_limits<Real>::quiet_NaN();
Real absolute_error_estimate = std::numeric_limits<Real>::quiet_NaN();
Real scale = std::numeric_limits<Real>::quiet_NaN();
size_t i = starting_level_;
do {
Real I0 = estimate_integral(f, omega, i);
#ifdef BOOST_MATH_INSTRUMENT_OOURA
print_ooura_estimate(i, I0, I1, omega);
#endif
absolute_error_estimate = abs(I0-I1);
scale = max(abs(I0), abs(I1));
if (!isnan(I1) && absolute_error_estimate <= rel_err_goal_*scale) {
starting_level_ = std::max(long(i) - 1, long(0));
return {I0/omega, absolute_error_estimate/scale};
}
I1 = I0;
} while(++i < big_nodes_.size());

while (big_nodes_.size() < requested_levels_ + max_additional_levels) {
size_t i = big_nodes_.size();
if (std::is_same<Real, float>::value) {
}
else if (std::is_same<Real, double>::value) {
}
else {
}
Real I0 = estimate_integral(f, omega, i);
#ifdef BOOST_MATH_INSTRUMENT_OOURA
print_ooura_estimate(i, I0, I1, omega);
#endif
absolute_error_estimate = abs(I0-I1);
scale = max(abs(I0), abs(I1));
if (absolute_error_estimate <= rel_err_goal_*scale) {
starting_level_ = std::max(long(i) - 1, long(0));
return {I0/omega, absolute_error_estimate/scale};
}
I1 = I0;
++i;
}

starting_level_ = big_nodes_.size() - 2;
return {I1/omega, absolute_error_estimate/scale};
}

private:

template<class PreciseReal>
size_t current_num_levels = big_nodes_.size();
Real unit_roundoff = std::numeric_limits<Real>::epsilon()/2;
PreciseReal h = PreciseReal(1)/PreciseReal(1<<i);

std::vector<Real> bnode_row;
std::vector<Real> bweight_row;
bnode_row.reserve((1<<i)*sizeof(Real));
bweight_row.reserve((1<<i)*sizeof(Real));

std::vector<Real> lnode_row;
std::vector<Real> lweight_row;

lnode_row.reserve((1<<i)*sizeof(Real));
lweight_row.reserve((1<<i)*sizeof(Real));

Real max_weight = 1;
auto alpha = calculate_ooura_alpha(h);
long n = 0;
Real w;
do {
auto precise_nw = ooura_cos_node_and_weight(n, h, alpha);
Real node = static_cast<Real>(precise_nw.first);
Real weight = static_cast<Real>(precise_nw.second);
w = weight;
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;
}
}
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
$$$$


# Accuracy of floating point calculations

Indeed, when doing a large number of mathematical operations, you have to be careful to ensure proper accuracy of the results. Luckily, a lot of operations are mostly fine. The main thing that causes loss of precision is when adding together numbers that differ greatly in size. For example, adding 1 and 1e-100 will just result in 1. However, multiplying them is perfectly fine, and will result in the expected 1e-100.

The most accurate way to do repeated summing (which is what numerical integration is all about) is to calculate the values of all the individual points, put them into an array, sort that array, and then sum adjacent pairs of array elements into another array that is half the size, and repeat that until you have one value left. The only issue with this of course is that it can use a lot of memory. However, since you already have arrays for nodes and weights, I think it could be an option here.

Using a higher precision and converting it back may work in some cases. However, if the difference in magnitude between the smallest and the largest value that you are summing together is still bigger than what the higher precision variables can represent, you are out of luck.

I would try to implement the sorting + repeated summing approach, and check the results you get with that against naive integration, and against integration with higher precision variables, and see whether any of this matters. It also wouldn't hurt to do some performance benchmarks to see the cost of each of these approaches.

# Avoiding code duplication

Indeed there is a lot of code duplication between oora_fourier_sin_detail and oora_fourier_cos_detail. You could try to create a generic class that takes a (template) parameter that chooses between the sine and cosine variant, and then check that parameter to specialize some parts of the code, similar to how you used if (std::is_same<Real, ...>::value).

The references to papers and equations are very good to have. However, most of the other comments are unfortunately bad in my opinion.

First of all, it would help to understand what the code is doing without having to find the equations myself. Especially for someone not having a university level maths education, it can be hard to track down references when they are not used to how math papers are published. So instead of just saying "Ooura and Mori, equation 3.3", describe what it actually is that oora_eta() is calculating.

In oora_sin_node_and_weight(), why is n == 0 a special case?

A comment like "I have verified that this is not a signifant source of inaccuracy" is a bit weird to see in code. Who is "I"? And how insignificant is it? And why would this line of code not be significant to begin with? It just adds more questions. If the way you calculate rhi_prime is an approximation instead of a more exact formula, then mention that this is an approximation and why it is valid to use here. If it is the exact formula, then I would not add any comment at all here. The fact that floating point operations are not perfect should be understood by all programmers, and the formula in that line doesn't look like there is any reason to worry about its accuracy.

The comments in oora_sin_node_and_weight() and oora_cos_node_and_weight() are quite different, even though the functions are almost the same.

A lot of comments say things like "we could", "it's not clear", "this is a huge overestimate". That doesn't inspire a lot of confidence in this code. Why are these comments there? If it's to remind you of something you have to fix later, mark it with "TODO", so it's clear that this is something for the future. It also makes it easy to grep for. An example:

// TODO: the size of the vectors is bigger than necessary, use a better heuristic


It's hard to write comments, especially for code you've just written. I would wait a week or two and then reread the code. Try to imagine a colleague or a student having to read your code.

# Use unique_ptr<> instead of shared_ptr<>

Your impl_ variable is never shared with anything else, so there is no reason for it to be a shared pointer. Declare it as std::unique<...> impl_ instead, and use std::make_unique<>() in the constructor.

# Use a proper two-dimensional array class

Nesting vectors is not an efficient way of storing multi-dimensional arrays. Why not use boost::multi_array instead?

# Why use mutexes and atomics if there are no threads?

You have a mutex to guard node weight vectors, and an atomic variable for the starting level. However, I see no mention of threads or asynchronous execution in your code, unless I am missing something. So it seems to me these things are useless, and should be removed from the code.

# Avoid useless using statements

I see a lot of using statements in the code in functions that don't even use the functions or constants mentioned in these statement. For example, using std::abs in oora_sin_node_and_weight() is not used at all.

In many cases, you only use these things once anyway, so it just increases the number of lines of code, for little gain in readability. I recommend that you just avoid using using altogether, and just write out all namespaces explicitly.

# Use consistent whitespace

Sometimes you use spaces around operators, sometimes not. Be consistent in how you format your code. I suggest you use spaces around all binary operator. Conversely, don't use a space after a '(' and before a ')'.

You don't have to fix all this by hand, I recommend you use a code formatting tool to do this for you. Check the Boost style guide for the recommended coding style.

# Use std::function<> to pass function pointers

I see you are using template<class F> to allow passing function pointers as arguments. The problem is that F is allowed to be anything, even things that are not functions, and this will cause hard to read compiler errors. Instead, you can use std::function<> to specify that a function takes another function as an argument. For example:

Real estimate_integral(std::function<Real(Real)> f, Real omega, size_t i) {
...
I0 += f(b_nodes[j] * inv_omega) * b_weights[j];
...
}

• The integrator does a lot of precompute which occupies a lot of memory and is independent of the function. Hence the documentation say to share this object between threads if you need the performance while (for example) assembling a stiffness matrix. Hence the std::shared_ptr, and hence the atomics. Commented Dec 21, 2019 at 1:43
• The using statements are required for ADL, since std::cos is only defined on float, double, and long double, but this quadrature works in arbitrary precision provided definitions of cos, sin`, so on are available. I'll double check that all of them are in fact necessary. Commented Dec 21, 2019 at 1:45