249 lines
16 KiB
C++
249 lines
16 KiB
C++
// boost-no-inspect
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/*
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* Copyright Nick Thompson, Matt Borland, 2023
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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_FOURIER_TRANSFORM_DAUBECHIES_HPP
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#define BOOST_MATH_SPECIAL_FOURIER_TRANSFORM_DAUBECHIES_HPP
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#include <array>
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#include <cmath>
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#include <complex>
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#include <iostream>
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#include <limits>
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#include <boost/math/constants/constants.hpp>
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#include <boost/math/tools/big_constant.hpp>
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#include <boost/math/tools/estrin.hpp>
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namespace boost::math {
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namespace detail {
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// See the Table 6.2 of Daubechies, Ten Lectures on Wavelets.
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// These constants are precisely those divided by 1/sqrt(2), because otherwise
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// we'd immediately just have to divide through by 1/sqrt(2).
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// These numbers agree with Table 6.2, but are generated via example/calculate_fourier_transform_daubechies_constants.cpp
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template <typename Real, unsigned N> constexpr std::array<Real, N> ft_daubechies_scaling_polynomial_coefficients() {
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static_assert(N >= 1 && N <= 10, "Scaling function only implemented for 1-10 vanishing moments.");
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if constexpr (N == 1) {
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return std::array<Real, 1>{static_cast<Real>(1)};
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}
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if constexpr (N == 2) {
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return {BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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1.36602540378443864676372317075293618347140262690519031402790348972596650842632007803393058),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-0.366025403784438646763723170752936183471402626905190314027903489725966508441952115116994061)};
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}
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if constexpr (N == 3) {
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return std::array<Real, 3>{
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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1.88186883113665472301331643028468183320710177910151845853383427363197699204347143889269703),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-1.08113883008418966599944677221635926685977756966260841342875242639629721931484516409937898),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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0.199269998947534942986130341931677433652675790561089954894918152764320227250084833874126086)};
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}
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if constexpr (N == 4) {
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return std::array<Real, 4>{
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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2.60642742441038678619616138456320274846457112268350230103083547418823666924354637907021821),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-2.33814397690691624172277875654682595239896411009843420976312905955518655953831321619717516),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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0.851612467139421235087502761217605775743179492713667860409024360383174560120738199344383827),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-0.119895914642891779560885389233982571808786505298735951676730775016224669960397338539830347)};
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}
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if constexpr (N == 5) {
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return std::array<Real, 5>{
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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3.62270372133693372237431371824382790538377237674943454540758419371854887218301659611796287),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-4.45042192340421529271926241961545172940077367856833333571968270791760393243895360839974479),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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2.41430351179889241160444590912469777504146155873489898274561148139247721271772284677196254),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-0.662064156756696785656360678859372223233256033099757083735935493062448802216759690564503751),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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0.0754788470250859443968634711062982722087957761837568913024225258690266500301041274151679859)};
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}
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if constexpr (N == 6) {
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return std::array<Real, 6>{
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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5.04775782409284533508504459282823265081102702143912881539214595513121059428213452194161891),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-7.90242489414953082292172067801361411066690749603940036372954720647258482521355701761199),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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5.69062231972011992229557724635729642828799628244009852056657089766265949751788181912632318),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-2.29591465417352749013350971621495843275025605194376564457120763045109729714936982561585742),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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0.508712486289373262241383448555327418882885930043157873517278143590549199629822225076344289),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-0.0487530817792802065667748935122839545647456859392192011752401594607371693280512344274717466)};
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}
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if constexpr (N == 7) {
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return std::array<Real, 7>{
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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7.0463635677199166580912954330590360004554457287730448872409828895500755049108034478397642),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-13.4339028220058085795120274851204982381087988043552711869584397724404274044947626280185946),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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12.0571882966390397563079887516068140052534768286900467252199152570563053103366694003818755),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-6.39124482303930285525880162640679389779540687632321120940980371544051534690730897661850842),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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2.07674879424918331569327229402057948161936796436510457676789758815816492768386639712643599),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-0.387167532162867697386347232520843525988806810788254462365009860280979111139408537312553398),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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0.0320145185998394020646198653617061745647219696385406695044576133973761206215673170563538)};
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}
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if constexpr (N == 8) {
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return std::array<Real, 8>{
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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9.85031962984351656604584909868313752909650830419035084214249929687665775818153930511533915),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-22.1667494032601530437943449172929277733925779301673358406203340024653233856852379126537395),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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23.8272728452144265698978643079553442578633838793866258585693705776047828901217069807060715),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-15.6065825916019064469551268429136774427686552695820632173344334583910793479437661751737998),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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6.63923943761238270605338141020386331691362835005178161341935720370310013774320917891051914),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-1.81462830704498058848677549516134095104668450780318379608495409574150643627578462439190617),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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0.292393958692487086036895445298600849998803161432207979583488595754566344585039785927586499),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-0.0212655694557728487977430067729997866644059875083834396749941173411979591559303697954912042)};
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}
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if constexpr (N == 9) {
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return std::array<Real, 9>{
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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13.7856894948673536752299497816200874595462540239049618127984616645562437295073582057283235),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-35.79362367743347676734569335180426263053917566987500206688713345532850076082533131311371),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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44.8271517576868325408174336351944130389504383168376658969692365144162452669941793147313),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-34.9081281226625998193992072777004811412863069972654446089639166067029872995118090115016879),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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18.2858070519930071738884732413420775324549836290768317032298177553411077249931094333824682),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-6.53714271572640296907117142447372145396492988681610221640307755553450246302777187366825001),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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1.5454286423270706293059630490222623728433659436325762803842722481655127844136128434034519),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-0.219427682644567750633335191213222483839627852234602683427115193605056655384931679751929029),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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0.0142452515927832872075875380128473058349984927391158822994546286919376896668596927857450578)};
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}
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if constexpr (N == 10) {
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return std::array<Real, 10>{
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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19.3111846872275854185286532829110292444580572106276740012656292351880418629976266671349603),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-56.8572892818288577904562616825768121532988312416110097001327598719988644787442373891037268),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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81.3040184941182201969442916535886223134891624078921290339772790298979750863332417443823932),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-73.3067370305702272426402835488383512315892354877130132060680994033122368453226804355121917),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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45.5029913577892585869595005785056707790215969761054467083138479721524945862678794713356742),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-20.0048938122958245128650205249242185678760602333821352917865992073643758821417211689052482),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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6.18674372398711325312495154772282340531430890354257911422818567803548535981484584999007723),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-1.29022235346655645559407302793903682217361613280994725979138999393113139183198020070701239),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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0.16380852384056875506684562409582514726612462486206657238854671180228210790016298829595125),
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-0.00960430880128020906860390254555211461150702751378997239464015046967050703218076318595987803)};
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}
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}
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} // namespace detail
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/*
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* Given ω∈ℝ, computes a numerical approximation to 𝓕[𝜙](ω),
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* where 𝜙 is the Daubechies scaling function.
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* Fast and accurate evaluation of these function seems to me to be a rather involved research project,
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* which I have not endeavored to complete.
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* In particular, recovering ~1ULP evaluation is not possible using the techniques
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* employed here-you should use this with the understanding it is good enough for almost
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* all uses with empirical data, but probably doesn't recover enough accuracy
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* for pure mathematical uses (other than graphing-in which case it's fine).
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* The implementation uses an infinite product of trigonometric polynomials.
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* See Daubechies, 10 Lectures on Wavelets, equation 5.1.17, 5.1.18.
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* It uses the factorization of m₀ shown in Corollary 5.5.4 and equation 5.5.5.
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* See more discusion near equation 6.1.1,
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* as well as efficiency gains from equation 7.1.4.
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*/
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template <class Real, unsigned p> std::complex<Real> fourier_transform_daubechies_scaling(Real omega) {
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// This arg promotion is kinda sad, but IMO the accuracy is not good enough in
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// float precision using this method. Requesting a better algorithm!
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if constexpr (std::is_same_v<Real, float>) {
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return static_cast<std::complex<float>>(fourier_transform_daubechies_scaling<double, p>(static_cast<double>(omega)));
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}
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using boost::math::constants::one_div_root_two_pi;
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using std::abs;
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using std::exp;
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using std::norm;
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using std::pow;
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using std::sqrt;
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using std::cbrt;
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// Equation 7.1.4 of 10 Lectures on Wavelets is singular at ω=0:
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if (omega == 0) {
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return std::complex<Real>(one_div_root_two_pi<Real>(), 0);
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}
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// For whatever reason, this starts returning NaNs rather than zero for |ω|≫1.
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// But we know that this function decays rather quickly with |ω|,
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// and hence it is "numerically zero", even if in actuality the function does not have compact support.
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// Now, should we probably do a fairly involved, exhaustive calculation to see where exactly we should set this threshold
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// and store them in a table? .... yes.
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if (abs(omega) >= sqrt(std::numeric_limits<Real>::max())) {
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return std::complex<Real>(0, 0);
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}
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auto const constexpr lxi = detail::ft_daubechies_scaling_polynomial_coefficients<Real, p>();
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auto xi = -omega / 2;
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std::complex<Real> phi{one_div_root_two_pi<Real>(), 0};
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do {
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std::complex<Real> arg{0, xi};
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auto z = exp(arg);
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phi *= boost::math::tools::evaluate_polynomial_estrin(lxi, z);
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xi /= 2;
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} while (abs(xi) > std::numeric_limits<Real>::epsilon());
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std::complex<Real> arg{0, omega};
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// There is no std::expm1 for complex numbers.
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// We may therefore be leaving accuracy gains on the table for small |ω|:
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std::complex<Real> prefactor = (Real(1) - exp(-arg))/arg;
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return phi * static_cast<std::complex<Real>>(pow(prefactor, p));
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}
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template <class Real, unsigned p> std::complex<Real> fourier_transform_daubechies_wavelet(Real omega) {
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// See Daubechies, 10 Lectures on Wavelets, page 193, unlabelled equation in Theorem 6.3.6:
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// 𝓕[ψ](ω) = -exp(-iω/2)m₀(ω/2 + π)^{*}𝓕[𝜙](ω/2)
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if constexpr (std::is_same_v<Real, float>) {
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return static_cast<std::complex<float>>(fourier_transform_daubechies_wavelet<double, p>(static_cast<double>(omega)));
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}
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using std::exp;
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using std::pow;
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auto Fphi = fourier_transform_daubechies_scaling<Real, p>(omega/2);
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auto phase = -exp(std::complex<Real>(0, -omega/2));
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// See Section 6.4 for the sign convention on the argument,
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// as well as Table 6.2:
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auto z = phase; // strange coincidence.
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//auto z = exp(std::complex<Real>(0, -omega/2 - boost::math::constants::pi<Real>()));
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auto constexpr lxi = detail::ft_daubechies_scaling_polynomial_coefficients<Real, p>();
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auto m0 = std::complex<Real>(pow((Real(1) + z)/Real(2), p))*boost::math::tools::evaluate_polynomial_estrin(lxi, z);
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return Fphi*std::conj(m0)*phase;
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}
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} // namespace boost::math
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#endif
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