/*
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* Copyright Nick Thompson, 2020
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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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#ifndef BOOST_MATH_SPECIAL_DAUBECHIES_WAVELET_HPP
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#define BOOST_MATH_SPECIAL_DAUBECHIES_WAVELET_HPP
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#include <vector>
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#include <array>
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#include <cmath>
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#include <thread>
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#include <future>
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#include <iostream>
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#include <boost/math/constants/constants.hpp>
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#include <boost/math/special_functions/detail/daubechies_scaling_integer_grid.hpp>
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#include <boost/math/special_functions/daubechies_scaling.hpp>
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#include <boost/math/filters/daubechies.hpp>
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#include <boost/math/interpolators/detail/cubic_hermite_detail.hpp>
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#include <boost/math/interpolators/detail/quintic_hermite_detail.hpp>
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#include <boost/math/interpolators/detail/septic_hermite_detail.hpp>
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namespace boost::math {
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template<class Real, int p, int order>
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std::vector<Real> daubechies_wavelet_dyadic_grid(int64_t j_max)
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{
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if (j_max == 0)
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{
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throw std::domain_error("The wavelet dyadic grid is refined from the scaling integer grid, so its minimum amount of data is half integer widths.");
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}
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auto phijk = daubechies_scaling_dyadic_grid<Real, p, order>(j_max - 1);
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//psi_j[l] = psi(-p+1 + l/2^j) = \sum_{k=0}^{2p-1} (-1)^k c_k \phi(1-2p+k + l/2^{j-1})
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//For derivatives just map c_k -> 2^order c_k.
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auto d = boost::math::filters::daubechies_scaling_filter<Real, p>();
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Real scale = boost::math::constants::root_two<Real>() * (1 << order);
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for (size_t i = 0; i < d.size(); ++i)
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{
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d[i] *= scale;
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if (!(i & 1))
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{
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d[i] = -d[i];
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}
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}
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std::vector<Real> v(2 * p + (2 * p - 1) * ((int64_t(1) << j_max) - 1), std::numeric_limits<Real>::quiet_NaN());
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v[0] = 0;
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v[v.size() - 1] = 0;
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for (int64_t l = 1; l < static_cast<int64_t>(v.size() - 1); ++l)
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{
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Real term = 0;
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for (int64_t k = 0; k < static_cast<int64_t>(d.size()); ++k)
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{
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int64_t idx = (int64_t(1) << (j_max - 1)) * (1 - 2 * p + k) + l;
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if (idx < 0 || idx >= static_cast<int64_t>(phijk.size()))
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{
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continue;
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}
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term += d[k] * phijk[idx];
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}
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v[l] = term;
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}
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return v;
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}
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template<class Real, int p>
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class daubechies_wavelet {
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//
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// Some type manipulation so we know the type of the interpolator, and the vector type it requires:
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//
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typedef std::vector < std::array < Real, p < 6 ? 2 : p < 10 ? 3 : 4>> vector_type;
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//
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// List our interpolators:
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//
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typedef std::tuple<
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detail::null_interpolator, detail::matched_holder_aos<vector_type>, detail::linear_interpolation_aos<vector_type>,
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interpolators::detail::cardinal_cubic_hermite_detail_aos<vector_type>, interpolators::detail::cardinal_quintic_hermite_detail_aos<vector_type>,
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interpolators::detail::cardinal_septic_hermite_detail_aos<vector_type> > interpolator_list;
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//
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// Select the one we need:
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//
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typedef std::tuple_element_t<
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p == 1 ? 0 :
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p == 2 ? 1 :
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p == 3 ? 2 :
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p <= 5 ? 3 :
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p <= 9 ? 4 : 5, interpolator_list> interpolator_type;
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public:
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daubechies_wavelet(int grid_refinements = -1)
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{
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static_assert(p < 20, "Daubechies wavelets are only implemented for p < 20.");
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static_assert(p > 0, "Daubechies wavelets must have at least 1 vanishing moment.");
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if (grid_refinements == 0)
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{
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throw std::domain_error("The wavelet requires at least 1 grid refinement.");
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}
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if constexpr (p == 1)
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{
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return;
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}
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else
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{
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if (grid_refinements < 0)
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{
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if (std::is_same_v<Real, float>)
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{
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if (grid_refinements == -2)
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{
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// Control absolute error:
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// p= 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19
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std::array<int, 20> r{ -1, -1, 18, 19, 16, 11, 8, 7, 7, 7, 5, 5, 4, 4, 4, 4, 3, 3, 3, 3 };
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grid_refinements = r[p];
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}
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else
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{
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// Control relative error:
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// p= 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19
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std::array<int, 20> r{ -1, -1, 21, 21, 21, 17, 16, 15, 14, 13, 12, 11, 11, 11, 11, 11, 11, 11, 11, 11 };
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grid_refinements = r[p];
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}
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}
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else if (std::is_same_v<Real, double>)
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{
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// p= 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19
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std::array<int, 20> r{ -1, -1, 21, 21, 21, 21, 21, 21, 21, 21, 20, 20, 19, 18, 18, 18, 18, 18, 18, 18 };
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grid_refinements = r[p];
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}
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else
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{
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grid_refinements = 21;
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}
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}
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// Compute the refined grid:
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// In fact for float precision I know the grid must be computed in double precision and then cast back down, or else parts of the support are systematically inaccurate.
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std::future<std::vector<Real>> t0 = std::async(std::launch::async, [&grid_refinements]() {
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// Computing in higher precision and downcasting is essential for 1ULP evaluation in float precision:
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auto v = daubechies_wavelet_dyadic_grid<typename detail::daubechies_eval_type<Real>::type, p, 0>(grid_refinements);
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return detail::daubechies_eval_type<Real>::vector_cast(v);
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});
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// Compute the derivative of the refined grid:
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std::future<std::vector<Real>> t1 = std::async(std::launch::async, [&grid_refinements]() {
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auto v = daubechies_wavelet_dyadic_grid<typename detail::daubechies_eval_type<Real>::type, p, 1>(grid_refinements);
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return detail::daubechies_eval_type<Real>::vector_cast(v);
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});
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// if necessary, compute the second and third derivative:
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std::vector<Real> d2ydx2;
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std::vector<Real> d3ydx3;
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if constexpr (p >= 6) {
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std::future<std::vector<Real>> t3 = std::async(std::launch::async, [&grid_refinements]() {
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auto v = daubechies_wavelet_dyadic_grid<typename detail::daubechies_eval_type<Real>::type, p, 2>(grid_refinements);
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return detail::daubechies_eval_type<Real>::vector_cast(v);
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});
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if constexpr (p >= 10) {
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std::future<std::vector<Real>> t4 = std::async(std::launch::async, [&grid_refinements]() {
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auto v = daubechies_wavelet_dyadic_grid<typename detail::daubechies_eval_type<Real>::type, p, 3>(grid_refinements);
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return detail::daubechies_eval_type<Real>::vector_cast(v);
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});
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d3ydx3 = t4.get();
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}
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d2ydx2 = t3.get();
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}
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auto y = t0.get();
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auto dydx = t1.get();
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if constexpr (p >= 2)
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{
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vector_type data(y.size());
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for (size_t i = 0; i < y.size(); ++i)
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{
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data[i][0] = y[i];
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data[i][1] = dydx[i];
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if constexpr (p >= 6)
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data[i][2] = d2ydx2[i];
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if constexpr (p >= 10)
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data[i][3] = d3ydx3[i];
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}
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if constexpr (p <= 3)
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m_interpolator = std::make_shared<interpolator_type>(std::move(data), grid_refinements, Real(-p + 1));
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else
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m_interpolator = std::make_shared<interpolator_type>(std::move(data), Real(-p + 1), Real(1) / (1 << grid_refinements));
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}
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else
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m_interpolator = std::make_shared<detail::null_interpolator>();
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}
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}
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inline Real operator()(Real x) const
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{
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if (x <= -p + 1 || x >= p)
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{
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return 0;
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}
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if constexpr (p == 1)
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{
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if (x < Real(1) / Real(2))
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{
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return 1;
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}
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else if (x == Real(1) / Real(2))
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{
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return 0;
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}
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return -1;
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}
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return (*m_interpolator)(x);
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}
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inline Real prime(Real x) const
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{
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static_assert(p > 2, "The 3-vanishing moment Daubechies wavelet is the first which is continuously differentiable.");
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if (x <= -p + 1 || x >= p)
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{
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return 0;
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}
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return m_interpolator->prime(x);
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}
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inline Real double_prime(Real x) const
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{
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static_assert(p >= 6, "Second derivatives of Daubechies wavelets require at least 6 vanishing moments.");
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if (x <= -p + 1 || x >= p)
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{
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return Real(0);
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}
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return m_interpolator->double_prime(x);
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}
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std::pair<Real, Real> support() const
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{
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return { Real(-p + 1), Real(p) };
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}
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int64_t bytes() const
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{
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return m_interpolator->bytes() + sizeof(*this);
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}
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private:
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std::shared_ptr<interpolator_type> m_interpolator;
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};
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}
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#endif
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