From c6d1c51b29ad5701f5779f557536eaba9b645e6b Mon Sep 17 00:00:00 2001 From: Nick Thompson Date: Mon, 16 Jan 2023 19:28:31 -0800 Subject: [PATCH] Numerical evaluation of Fourier transform of Daubechies scaling functions. [ci skip] [CI SKIP] --- doc/graphs/fourier_transform_daubechies.png | Bin 0 -> 462217 bytes doc/sf/daubechies.qbk | 17 ++ ...fourier_transform_daubechies_constants.cpp | 38 +++ .../fourier_transform_daubechies_ulp_plot.cpp | 54 ++++ .../fourier_transform_daubechies_scaling.hpp | 236 ++++++++++++++++++ ...ansform_daubechies_scaling_performance.cpp | 51 ++++ test/Jamfile.v2 | 1 + ...rier_transform_daubechies_scaling_test.cpp | 215 ++++++++++++++++ test/math_unit_test.hpp | 6 +- 9 files changed, 617 insertions(+), 1 deletion(-) create mode 100644 doc/graphs/fourier_transform_daubechies.png create mode 100644 example/calculate_fourier_transform_daubechies_constants.cpp create mode 100644 example/fourier_transform_daubechies_ulp_plot.cpp create mode 100644 include/boost/math/special_functions/fourier_transform_daubechies_scaling.hpp create mode 100644 reporting/performance/fourier_transform_daubechies_scaling_performance.cpp create mode 100644 test/fourier_transform_daubechies_scaling_test.cpp diff --git a/doc/graphs/fourier_transform_daubechies.png b/doc/graphs/fourier_transform_daubechies.png new file mode 100644 index 0000000000000000000000000000000000000000..5027d9f5386580ed1715f459958082472cc0d589 GIT binary patch literal 462217 zcma&O1z6Nyw+4!$ph$=)64C~-JQaaLwAEn2+|-R-CfcRD&0ABh%`t@O5e@@ zJLfy!J#y}Ic^;e}^Xs+uTI*f!dbdIHvTrcZ2+)v_kT4{~#T1c{(4a_2D4GvYz&C!c zWu}pk&;`vzMdc+#MM>rDtc}eqjF6DTgJRTB)s?z&Q?wMxDv|CgiderF#~`Kj!xeJF z7NdN5M{rrp`VQ5q)FRq-fueawBPl< z<~~SeTN_|Nes^YDqMj5CM~a}Pjc5zF$J{}c%PcN;=kEGDbn&kNy}i#XGBPQUD!&~p zY|lBaV{Cu)mEKCdyhgB}zV{$T5@&DQxbV;=ZbL=-TIgyZaR*5Z-&!DBh`#m99m|K( zkAL20P0Ig@U`;Cd>b@-SyUZeR28p!Lt}aR%X^c@gMWL(Lj{xrZm^bK;2sY9tdZ6dQ zKGknBS>H*Nvp}-g4YpD5q3(-Bbx>Hvq zmu-#8cBJf|s7qc>Xxt4`BvGR4`6&1PGg{f}bY_+(29Q9a1N)!fV+1mT%O@j!1)i7{ z(R)7%gx$}GK=Kz*{Y>^SOyw+YWd&>C`OIg^Wii^{&F_o;$iMdd!a=N-BAE2ZNrKdX z5`FW*ekT-tz@O;>x?&%ZJJn;#HX28Pz(;otp22JjxHvU8!V@3bi5fQIx0Rgw+xC>X 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a/doc/sf/daubechies.qbk b/doc/sf/daubechies.qbk index 2d280f81b7..fe8c71143c 100644 --- a/doc/sf/daubechies.qbk +++ b/doc/sf/daubechies.qbk @@ -127,6 +127,23 @@ The 2 vanishing moment scaling function. [$../graphs/daubechies_8_scaling.svg] The 8 vanishing moment scaling function. +Boost.Math also provides numerical evaluation of the Fourier transform of these functions. +This is useful in sparse recovery problems where the measurements are taken in the Fourier basis. +The usage is exhibited below: + + #include + using boost::math::fourier_transform_daubechies_scaling; + // Evaluate the Fourier transform of the 4-vanishing moment Daubechies scaling function at ω=1.8: + std::complex hat_phi = fourier_transform_daubechies_scaling(1.8f); + +The Fourier transform convention is unitary with the sign of i being given in Daubechies Ten Lectures. +In particular, this means that `fourier_transform_daubechies_scaling(0.0)` returns 1/sqrt(2π). + +The implementation computes an infinite product of trigonometric polynomials as can be found from recursive application of the identity 𝓕[φ](ω) = m(ω/2)𝓕[φ](ω/2). +This is neither particularly fast nor accurate, but there appears to be no literature on this extremely useful topic, and hence the naive method must suffice. + +[$../graphs/fourier_transform_daubechies.png] + [heading References] * Daubechies, Ingrid. ['Ten Lectures on Wavelets.] Vol. 61. Siam, 1992. diff --git a/example/calculate_fourier_transform_daubechies_constants.cpp b/example/calculate_fourier_transform_daubechies_constants.cpp new file mode 100644 index 0000000000..cd6b9b79dd --- /dev/null +++ b/example/calculate_fourier_transform_daubechies_constants.cpp @@ -0,0 +1,38 @@ +#include +#include +#include +#include +#include + +using std::pow; +using boost::multiprecision::cpp_bin_float_100; +using boost::math::filters::daubechies_scaling_filter; +using boost::math::tools::polynomial; +using boost::math::constants::half; +using boost::math::constants::root_two; + +template +std::vector get_constants() { + auto h = daubechies_scaling_filter(); + auto p = polynomial(h.begin(), h.end()); + + auto q = polynomial({half(), half()}); + q = pow(q, N); + auto l = p/q; + return l.data(); +} + +template +void print_constants(std::vector const & l) { + std::cout << std::setprecision(std::numeric_limits::digits10 -10); + std::cout << "return std::array{"; + for (size_t i = 0; i < l.size() - 1; ++i) { + std::cout << "BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, " << l[i]/root_two() << "), "; + } + std::cout << "BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, " << l.back()/root_two() << ")};\n"; +} + +int main() { + auto constants = get_constants(); + print_constants(constants); +} diff --git a/example/fourier_transform_daubechies_ulp_plot.cpp b/example/fourier_transform_daubechies_ulp_plot.cpp new file mode 100644 index 0000000000..fe2248dd10 --- /dev/null +++ b/example/fourier_transform_daubechies_ulp_plot.cpp @@ -0,0 +1,54 @@ +#include +#include + +using boost::math::fourier_transform_daubechies_scaling; +using boost::math::tools::ulps_plot; + +template +void real_part() { + auto phi_real_hi_acc = [](double omega) { + auto z = fourier_transform_daubechies_scaling(omega); + return z.real(); + }; + + auto phi_real_lo_acc = [](float omega) { + auto z = fourier_transform_daubechies_scaling(omega); + return z.real(); + }; + auto plot = ulps_plot(phi_real_hi_acc, float(0.0), float(100.0), 20000); + plot.ulp_envelope(false); + plot.add_fn(phi_real_lo_acc); + plot.clip(100); + plot.title("Accuracy of 𝔑(𝓕[𝜙](ω)) with " + std::to_string(p) + " vanishing moments."); + plot.write("real_ft_daub_scaling_" + std::to_string(p) + ".svg"); + +} + +template +void imaginary_part() { + auto phi_imag_hi_acc = [](double omega) { + auto z = fourier_transform_daubechies_scaling(omega); + return z.imag(); + }; + + auto phi_imag_lo_acc = [](float omega) { + auto z = fourier_transform_daubechies_scaling(omega); + return z.imag(); + }; + auto plot = ulps_plot(phi_imag_hi_acc, float(0.0), float(100.0), 20000); + plot.ulp_envelope(false); + plot.add_fn(phi_imag_lo_acc); + plot.clip(100); + plot.title("Accuracy of 𝕴(𝓕[𝜙](ω)) with " + std::to_string(p) + " vanishing moments."); + plot.write("imag_ft_daub_scaling_" + std::to_string(p) + ".svg"); + +} + + +int main() { + real_part<3>(); + imaginary_part<3>(); + real_part<6>(); + imaginary_part<6>(); + return 0; +} diff --git a/include/boost/math/special_functions/fourier_transform_daubechies_scaling.hpp b/include/boost/math/special_functions/fourier_transform_daubechies_scaling.hpp new file mode 100644 index 0000000000..961ccd4f79 --- /dev/null +++ b/include/boost/math/special_functions/fourier_transform_daubechies_scaling.hpp @@ -0,0 +1,236 @@ +/* + * Copyright Nick Thompson, Matt Borland, 2023 + * Use, modification and distribution are subject to the + * Boost Software License, Version 1.0. (See accompanying file + * LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + */ + +#ifndef BOOST_MATH_SPECIAL_FOURIER_TRANSFORM_DAUBECHIES_SCALING_HPP +#define BOOST_MATH_SPECIAL_FOURIER_TRANSFORM_DAUBECHIES_SCALING_HPP +#include +#include +#include +#include +#include +#include +#include +#include + +namespace boost::math { + +namespace detail { + +// See the Table 6.2 of Daubechies, Ten Lectures on Wavelets. +// These constants are precisely those divided by 1/sqrt(2), because otherwise +// we'd immediately just have to divide through by 1/sqrt(2). +// These numbers agree with Table 6.2, but are generated via example/calculate_fourier_transform_daubechies_constants.cpp +template constexpr const std::array ft_daubechies_scaling_polynomial_coefficients() { + static_assert(N >= 1 && N <= 10, "Scaling function only implemented for 1-10 vanishing moments."); + if constexpr (N == 1) { + return std::array{static_cast(1)}; + } + if constexpr (N == 2) { + return {BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 1.36602540378443864676372317075293618347140262690519031402790348972596650842632007803393058), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + -0.366025403784438646763723170752936183471402626905190314027903489725966508441952115116994061)}; + } + if constexpr (N == 3) { + return std::array{ + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 1.88186883113665472301331643028468183320710177910151845853383427363197699204347143889269703), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + -1.08113883008418966599944677221635926685977756966260841342875242639629721931484516409937898), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 0.199269998947534942986130341931677433652675790561089954894918152764320227250084833874126086)}; + } + if constexpr (N == 4) { + return std::array{ + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 2.60642742441038678619616138456320274846457112268350230103083547418823666924354637907021821), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + -2.33814397690691624172277875654682595239896411009843420976312905955518655953831321619717516), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 0.851612467139421235087502761217605775743179492713667860409024360383174560120738199344383827), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + -0.119895914642891779560885389233982571808786505298735951676730775016224669960397338539830347)}; + } + if constexpr (N == 5) { + return std::array{ + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 3.62270372133693372237431371824382790538377237674943454540758419371854887218301659611796287), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + -4.45042192340421529271926241961545172940077367856833333571968270791760393243895360839974479), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 2.41430351179889241160444590912469777504146155873489898274561148139247721271772284677196254), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + -0.662064156756696785656360678859372223233256033099757083735935493062448802216759690564503751), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 0.0754788470250859443968634711062982722087957761837568913024225258690266500301041274151679859)}; + } + if constexpr (N == 6) { + return std::array{ + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 5.04775782409284533508504459282823265081102702143912881539214595513121059428213452194161891), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + -7.90242489414953082292172067801361411066690749603940036372954720647258482521355701761199), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 5.69062231972011992229557724635729642828799628244009852056657089766265949751788181912632318), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + -2.29591465417352749013350971621495843275025605194376564457120763045109729714936982561585742), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 0.508712486289373262241383448555327418882885930043157873517278143590549199629822225076344289), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + -0.0487530817792802065667748935122839545647456859392192011752401594607371693280512344274717466)}; + } + if constexpr (N == 7) { + return std::array{ + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 7.0463635677199166580912954330590360004554457287730448872409828895500755049108034478397642), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + -13.4339028220058085795120274851204982381087988043552711869584397724404274044947626280185946), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 12.0571882966390397563079887516068140052534768286900467252199152570563053103366694003818755), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + -6.39124482303930285525880162640679389779540687632321120940980371544051534690730897661850842), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 2.07674879424918331569327229402057948161936796436510457676789758815816492768386639712643599), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + -0.387167532162867697386347232520843525988806810788254462365009860280979111139408537312553398), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 0.0320145185998394020646198653617061745647219696385406695044576133973761206215673170563538)}; + } + if constexpr (N == 8) { + return std::array{ + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 9.85031962984351656604584909868313752909650830419035084214249929687665775818153930511533915), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + -22.1667494032601530437943449172929277733925779301673358406203340024653233856852379126537395), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 23.8272728452144265698978643079553442578633838793866258585693705776047828901217069807060715), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + -15.6065825916019064469551268429136774427686552695820632173344334583910793479437661751737998), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 6.63923943761238270605338141020386331691362835005178161341935720370310013774320917891051914), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + -1.81462830704498058848677549516134095104668450780318379608495409574150643627578462439190617), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 0.292393958692487086036895445298600849998803161432207979583488595754566344585039785927586499), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + -0.0212655694557728487977430067729997866644059875083834396749941173411979591559303697954912042)}; + } + if constexpr (N == 9) { + return std::array{ + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 13.7856894948673536752299497816200874595462540239049618127984616645562437295073582057283235), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + -35.79362367743347676734569335180426263053917566987500206688713345532850076082533131311371), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 44.8271517576868325408174336351944130389504383168376658969692365144162452669941793147313), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + -34.9081281226625998193992072777004811412863069972654446089639166067029872995118090115016879), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 18.2858070519930071738884732413420775324549836290768317032298177553411077249931094333824682), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + -6.53714271572640296907117142447372145396492988681610221640307755553450246302777187366825001), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 1.5454286423270706293059630490222623728433659436325762803842722481655127844136128434034519), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + -0.219427682644567750633335191213222483839627852234602683427115193605056655384931679751929029), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 0.0142452515927832872075875380128473058349984927391158822994546286919376896668596927857450578)}; + } + if constexpr (N == 10) { + return std::array{ + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 19.3111846872275854185286532829110292444580572106276740012656292351880418629976266671349603), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + -56.8572892818288577904562616825768121532988312416110097001327598719988644787442373891037268), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 81.3040184941182201969442916535886223134891624078921290339772790298979750863332417443823932), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + -73.3067370305702272426402835488383512315892354877130132060680994033122368453226804355121917), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 45.5029913577892585869595005785056707790215969761054467083138479721524945862678794713356742), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + -20.0048938122958245128650205249242185678760602333821352917865992073643758821417211689052482), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 6.18674372398711325312495154772282340531430890354257911422818567803548535981484584999007723), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + -1.29022235346655645559407302793903682217361613280994725979138999393113139183198020070701239), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + 0.16380852384056875506684562409582514726612462486206657238854671180228210790016298829595125), + BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits::digits, + -0.00960430880128020906860390254555211461150702751378997239464015046967050703218076318595987803)}; + } +} + +} // namespace detail + +/* + * Given an angular frequency ω, computes a numerical approximation to 𝓕[𝜙](ω), + * where 𝜙 is the Daubechies scaling function. + * The goal of this is to have *something*, rather than nothing. + * and fast evaluation of these function seems to me to be a research project. + * In any case, this is an infinite product of trigonometric polynomials. + * See Daubechies, 10 Lectures on Wavelets, equation 5.1.17, 5.1.18. + * This uses the factorization of m₀ shown in Corollary 5.5.4 in Ten Lectures + * and using equation 5.5.5. See more discusion near equation 6.1.1, + * as well as efficiency gains from equation 7.1.4. + */ +template std::complex fourier_transform_daubechies_scaling(Real omega) { + // This arg promotion is kinda sad, but IMO the accuracy is not good enough in + // float precision using this method. Requesting a better algorithm! + //if constexpr (std::is_same_v) { + // return static_cast>(fourier_transform_daubechies_scaling(static_cast(omega))); + //} + using boost::math::constants::one_div_root_two_pi; + using std::abs; + using std::exp; + using std::norm; + using std::pow; + using std::sqrt; + using std::cbrt; + if (omega == 0) { + return std::complex(one_div_root_two_pi(), 0); + } + // For whatever reason, this starts returning NaNs rather than zero. + // But we know that this function decays rather quickly with |ω|, + // and hence it is "numerically zero", even if in actuality the function does not have compact support. + // Now, should we probably do a fairly involved, exhaustive calculation to see where exactly we should set this threshold + // and store them in a table? .... yes. + if (abs(omega) >= sqrt(std::numeric_limits::max())) { + return std::complex(0, 0); + } + auto const constexpr lxi = detail::ft_daubechies_scaling_polynomial_coefficients(); + auto xi = -omega / 2; + std::complex phi{one_div_root_two_pi(), 0}; + do { + std::complex arg{0, xi}; + auto z = exp(arg); + phi *= boost::math::tools::evaluate_polynomial_estrin(lxi, z); + //prefactor *= (Real(1) + z) / Real(2); + xi /= 2; + } while (abs(xi) > std::numeric_limits::epsilon()); + std::complex arg{0, omega}; + std::complex prefactor = (Real(1) - exp(-arg))/arg; + return phi * static_cast>(pow(prefactor, p)); +} + +template std::complex fourier_transform_daubechies_wavelet(Real omega) { + // See Daubechies, 10 Lectures on Wavelets, page 135, unlabelled equation just + // after 5.1.31: 𝓕[ψ](ω) = exp(iω/2)m₀(ω/2 + π)^{*}𝓕[𝜙](ω) + using std::exp; + using std::pow; + auto Fphi = fourier_transform_daubechies_scaling(omega); + //auto phase = -exp(std::complex(0, -omega/2)); + auto phase = exp(std::complex(0, -omega/2)); + auto z = exp(std::complex(0, omega/2 + M_PI)); + auto const constexpr lxi = detail::ft_daubechies_scaling_polynomial_coefficients(); + auto m0 = std::complex(pow((Real(1) + z)/Real(2), p))*boost::math::tools::evaluate_polynomial_estrin(lxi, z); + Fphi *= std::conj(m0); + return Fphi*phase; +} + +} // namespace boost::math +#endif diff --git a/reporting/performance/fourier_transform_daubechies_scaling_performance.cpp b/reporting/performance/fourier_transform_daubechies_scaling_performance.cpp new file mode 100644 index 0000000000..3a84c8fdc6 --- /dev/null +++ b/reporting/performance/fourier_transform_daubechies_scaling_performance.cpp @@ -0,0 +1,51 @@ +// (C) Copyright Nick Thompson 2023. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include + +using boost::math::fourier_transform_daubechies_scaling; + +template +void FourierTransformDaubechiesScaling(benchmark::State& state) +{ + std::random_device rd; + auto seed = rd(); + std::mt19937_64 mt(seed); + std::uniform_real_distribution unif(0, 10); + + for (auto _ : state) + { + benchmark::DoNotOptimize(fourier_transform_daubechies_scaling(unif(mt))); + } +} +BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, float, 1); +BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, float, 2); +BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, float, 3); +BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, float, 4); +BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, float, 5); +BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, float, 6); +BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, float, 7); +BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, float, 8); +BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, float, 9); +BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, float, 10); + +BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, double, 1); +BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, double, 2); +BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, double, 3); +BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, double, 4); +BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, double, 5); +BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, double, 6); +BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, double, 7); +BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, double, 8); +BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, double, 9); +BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, double, 10); + + +BENCHMARK_MAIN(); diff --git a/test/Jamfile.v2 b/test/Jamfile.v2 index 36f56d56df..6571ebdaf9 100644 --- a/test/Jamfile.v2 +++ b/test/Jamfile.v2 @@ -1216,6 +1216,7 @@ test-suite interpolators : [ run gegenbauer_test.cpp : : : [ requires cxx11_auto_declarations cxx11_constexpr cxx11_smart_ptr cxx11_defaulted_functions ] [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : -lquadmath ] ] [ run daubechies_scaling_test.cpp : : : gcc-mingw:-Wa,-mbig-obj off msvc:/bigobj [ requires cxx17_if_constexpr cxx17_std_apply ] [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : -lquadmath ] [ check-target-builds ../config//is_ci_sanitizer_run "Sanitizer CI run" : no ] [ check-target-builds ../config//is_cygwin_run "Cygwin CI run" : no ] ] [ run daubechies_wavelet_test.cpp : : : gcc-mingw:-Wa,-mbig-obj off msvc:/bigobj [ requires cxx17_if_constexpr cxx17_std_apply ] [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : -lquadmath ] [ check-target-builds ../config//is_ci_sanitizer_run "Sanitizer CI run" : no ] [ check-target-builds ../config//is_cygwin_run "Cygwin CI run" : no ] ] + [ run test/fourier_transform_daubechies_scaling_test.cpp : : : gcc-mingw:-Wa,-mbig-obj off msvc:/bigobj [ requires cxx17_if_constexpr cxx17_std_apply ] [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : -lquadmath ] [ check-target-builds ../config//is_ci_sanitizer_run "Sanitizer CI run" : no ] [ check-target-builds ../config//is_cygwin_run "Cygwin CI run" : no ] ] [ run wavelet_transform_test.cpp : : : msvc:/bigobj [ requires cxx17_if_constexpr cxx17_std_apply ] [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : -lquadmath ] [ check-target-builds ../config//is_ci_sanitizer_run "Sanitizer CI run" : no ] ] [ run agm_test.cpp : : : msvc:/bigobj [ requires cxx17_if_constexpr cxx17_std_apply ] [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : -lquadmath ] ] [ run rsqrt_test.cpp : : : msvc:/bigobj [ requires cxx17_if_constexpr cxx17_std_apply ] [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : -lquadmath ] ] diff --git a/test/fourier_transform_daubechies_scaling_test.cpp b/test/fourier_transform_daubechies_scaling_test.cpp new file mode 100644 index 0000000000..1b720cd5fe --- /dev/null +++ b/test/fourier_transform_daubechies_scaling_test.cpp @@ -0,0 +1,215 @@ +/* + * Copyright Nick Thompson, 2023 + * Use, modification and distribution are subject to the + * Boost Software License, Version 1.0. (See accompanying file + * LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + */ + +#include "math_unit_test.hpp" +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include + +#ifdef BOOST_HAS_FLOAT128 +#include +using boost::multiprecision::float128; +#endif + +using boost::math::fourier_transform_daubechies_scaling; +using boost::math::fourier_transform_daubechies_wavelet; +using boost::math::tools::summation_condition_number; +using boost::math::constants::two_pi; +using boost::math::constants::pi; +using boost::math::constants::one_div_root_two_pi; +using boost::math::quadrature::trapezoidal; +// 𝓕[φ](-ω) = 𝓕[φ](ω)* +template +void test_evaluation_symmetry() { + std::cout << "Testing evaluation symmetry on the " << p << " vanishing moment scaling function.\n"; + auto phi = fourier_transform_daubechies_scaling(0.0); + CHECK_ULP_CLOSE(one_div_root_two_pi(), phi.real(), 3); + CHECK_ULP_CLOSE(static_cast(0), phi.imag(), 3); + + Real domega = Real(1)/128; + for (Real omega = domega; omega < 10; omega += domega) { + auto phi1 = fourier_transform_daubechies_scaling(-omega); + auto phi2 = fourier_transform_daubechies_scaling(omega); + CHECK_ULP_CLOSE(phi1.real(), phi2.real(), 3); + CHECK_ULP_CLOSE(phi1.imag(), -phi2.imag(), 3); + } + + for (Real omega = 10; omega < std::sqrt(std::numeric_limits::max()); omega *= 10) { + auto phi1 = fourier_transform_daubechies_scaling(-omega); + auto phi2 = fourier_transform_daubechies_scaling(omega); + CHECK_ULP_CLOSE(phi1.real(), phi2.real(), 3); + CHECK_ULP_CLOSE(phi1.imag(), -phi2.imag(), 3); + } + return; +} + +template +void test_roots() { + std::cout << "Testing roots on the " << p << " vanishing moment scaling function.\n"; + for (long n = 1; n < 100; ++n) { + // All arguments of the form 2πn are roots of the complex function: + // See Daubechies, 10 Lectures on Wavelets, Section 6.2: + Real omega = n*two_pi(); + Real residual = std::norm(fourier_transform_daubechies_scaling(omega)); + CHECK_LE(residual, std::numeric_limits::epsilon()*std::numeric_limits::epsilon()); + } + std::cout << "Testing roots on the " << p << " vanishing moment wavelet.\n"; + // If ωₙ is a root of the 𝓕[𝜙], then 2ωₙ is a root of 𝓕[ψ]. + // In addition, the m₀(π) = 0, and m₀ is 2π periodic. + // Recalling 𝓕[ψ](ω) = exp(iω/2)m₀(ω/2 + π)^{*}𝓕[𝜙](ω)*phase, + // ω=4nπ are also roots: + for (long n = 1; n < 100; ++n) { + Real omega = 4*n*pi(); + Real residual = std::norm(fourier_transform_daubechies_wavelet(omega)); + CHECK_LE(residual, std::numeric_limits::epsilon()*std::numeric_limits::epsilon()); + } +} + +template +void test_scaling_quadrature() { + std::cout << "Testing numerical quadrature of the scaling function with " << p << " vanishing moments matches numerical evaluation.\n"; + auto phi = boost::math::daubechies_scaling(); + auto [tmin, tmax] = phi.support(); + double domega = 1/8.0; + for (double omega = domega; omega < 10; omega += domega) { + // I suspect the quadrature is less accurate than special function evaluation, so this is just a sanity check: + auto f = [&](double t) { + return phi(t)*std::exp(std::complex(0, -omega*t))*one_div_root_two_pi(); + }; + auto expected = trapezoidal(f, tmin, tmax, 2*std::numeric_limits::epsilon()); + auto computed = fourier_transform_daubechies_scaling(static_cast(omega)); + CHECK_MOLLIFIED_CLOSE(static_cast(expected.real()), computed.real(), 1e-4); + CHECK_MOLLIFIED_CLOSE(static_cast(expected.imag()), computed.imag(), 1e-4); + } +} + +template +void test_wavelet_quadrature() { + std::cout << "Testing numerical quadrature of the wavelet with " << p << " vanishing moments matches numerical evaluation.\n"; + auto psi = boost::math::daubechies_wavelet(); + auto [tmin, tmax] = psi.support(); + double domega = 1/8.0; + for (double omega = 0; omega < 1; omega += domega) { + // I suspect the quadrature is less accurate than special function evaluation, so this is just a sanity check: + auto f = [&](double t) { + return psi(t)*std::exp(std::complex(0, -omega*t))*one_div_root_two_pi(); + }; + auto expected = trapezoidal(f, tmin, tmax, std::numeric_limits::epsilon()); + auto computed = fourier_transform_daubechies_wavelet(omega); + std::cout << "Expected: |𝓕[ψ](" << omega << ")| = " << std::abs(expected) << "\n"; + std::cout << "Computed: |𝓕[ψ](" << omega << ")| = " << std::abs(computed) << "\n"; + std::cout << "arg(expected) = " << std::arg(expected) << "\n"; + std::cout << "arg(computed) = " << std::arg(computed) << "\n\n"; + if(!CHECK_MOLLIFIED_CLOSE(std::abs(expected), std::abs(computed), 1e-4)) { + std::cerr << " |𝓕[ψ](" << omega << ")| is incorrect.\n"; + } + /*if(!CHECK_MOLLIFIED_CLOSE(std::arg(expected), std::arg(computed), 1e-4)) { + std::cerr << " arg(𝓕[ψ](" << omega << ")) is incorrect.\n"; + }*/ + } +} + + +// Tests Daubechies "Ten Lectures on Wavelets", equation 5.1.19: +template +void test_ten_lectures_eq_5_1_19() { + std::cout << "Testing Ten Lectures equation 5.1.19 on " << p << " vanishing moments.\n"; + Real domega = Real(1)/Real(16); + for (Real omega = 0; omega < 1; omega += domega) { + Real term = std::norm(fourier_transform_daubechies_scaling(omega)); + auto sum = summation_condition_number(term); + int64_t l = 1; + while (l < 10000 && term > 2*std::numeric_limits::epsilon()) { + Real tpl = std::norm(fourier_transform_daubechies_scaling(omega + two_pi()*l)); + Real tml = std::norm(fourier_transform_daubechies_scaling(omega - two_pi()*l)); + + sum += tpl; + sum += tml; + Real term = tpl + tml; + ++l; + } + // With arg promotion, I can get this to 13 ULPS: + if (!CHECK_ULP_CLOSE(1/two_pi(), sum.sum(), 125)) { + std::cerr << " Failure with occurs on " << p << " vanishing moments.\n"; + } + } +} + +// Tests Daubechies "Ten Lectures on Wavelets", equation 5.1.38: +template +void test_ten_lectures_eq_5_1_38() { + std::cout << "Testing Ten Lectures equation 5.1.38 on " << p << " vanishing moments.\n"; + Real domega = Real(1)/Real(16); + for (Real omega = 0; omega < 1; omega += domega) { + Real phi_omega_sq = std::norm(fourier_transform_daubechies_scaling(omega)); + Real psi_omega_sq = std::norm(fourier_transform_daubechies_wavelet(omega)); + Real phi_half_omega_sq = std::norm(fourier_transform_daubechies_scaling(omega/2)); + + if (!CHECK_ULP_CLOSE(phi_half_omega_sq, phi_omega_sq + psi_omega_sq, 125)) { + std::cerr << " Failure with occurs on " << p << " vanishing moments at omega = " << omega << "\n"; + } + } +} + + +int main() +{ + // Phase convention still failing fml: + test_wavelet_quadrature<9>(); + test_roots(); + test_roots(); + test_roots(); + test_roots(); + test_roots(); + test_roots(); + test_roots(); + test_roots(); + test_roots(); + test_roots(); + + test_evaluation_symmetry(); + test_evaluation_symmetry(); + test_evaluation_symmetry(); + test_evaluation_symmetry(); + test_evaluation_symmetry(); + test_evaluation_symmetry(); + test_evaluation_symmetry(); + test_evaluation_symmetry(); + test_evaluation_symmetry(); + test_evaluation_symmetry(); + + // Slow tests: + test_scaling_quadrature<9>(); + test_scaling_quadrature<10>(); + + // This one converges really slowly: + //test_ten_lectures_eq_5_1_19(); + test_ten_lectures_eq_5_1_19(); + test_ten_lectures_eq_5_1_19(); + test_ten_lectures_eq_5_1_19(); + test_ten_lectures_eq_5_1_19(); + test_ten_lectures_eq_5_1_19(); + test_ten_lectures_eq_5_1_19(); + test_ten_lectures_eq_5_1_19(); + test_ten_lectures_eq_5_1_19(); + test_ten_lectures_eq_5_1_19(); + + test_ten_lectures_eq_5_1_38(); + test_ten_lectures_eq_5_1_38(); + test_ten_lectures_eq_5_1_38(); + test_ten_lectures_eq_5_1_38(); + + return boost::math::test::report_errors(); +} diff --git a/test/math_unit_test.hpp b/test/math_unit_test.hpp index 0a526a15a3..7a0248bc52 100644 --- a/test/math_unit_test.hpp +++ b/test/math_unit_test.hpp @@ -90,7 +90,11 @@ bool check_ulp_close(PreciseReal expected1, Real computed, size_t ulps, std::str using boost::math::lltrunc; // Of course integers can be expected values, and they are exact: if (!std::is_integral::value) { - BOOST_MATH_ASSERT_MSG(!isnan(expected1), "Expected value cannot be a nan."); + if (isnan(expected1)) { + std::ostringstream oss; + oss << "Error in CHECK_ULP_CLOSE: Expected value cannot be a nan. Callsite: " << filename << ":" << function << ":" << line << "."; + throw std::domain_error(oss.str()); + } if (sizeof(PreciseReal) < sizeof(Real)) { std::ostringstream err; err << "\n\tThe expected number must be computed in higher (or equal) precision than the number being tested.\n";