From 18c53aa93d535c1d39c01860ce7f1b17173eadfa Mon Sep 17 00:00:00 2001 From: Kshitij Surjuse Date: Fri, 23 Feb 2024 10:30:09 -0500 Subject: [PATCH] factorials and related quantities are computed in a safer manner, with graceful switching to inexact arithmetic for large argument values ... this allows to compute SolidHarmonicsCoefficients for arbitrary (in practice) L --- include/libint2/shell.h | 614 ++++++++++++++++++++++++++++++- include/libint2/solidharmonics.h | 40 +- tests/unit/test-core.cc | 12 + 3 files changed, 639 insertions(+), 27 deletions(-) diff --git a/include/libint2/shell.h b/include/libint2/shell.h index 44c705b82..e0a3df5cf 100644 --- a/include/libint2/shell.h +++ b/include/libint2/shell.h @@ -33,12 +33,14 @@ #include #include #include +#include #include namespace libint2 { namespace math { -/// fac[k] = k! + +/// @brief fac[k] = k!, `0<=k<=20` static constexpr std::array fac = {{1LL, 1LL, 2LL, @@ -60,8 +62,224 @@ static constexpr std::array fac = {{1LL, 6402373705728000LL, 121645100408832000LL, 2432902008176640000LL}}; -/// df_Kminus1[k] = (k-1)!! -static constexpr std::array df_Kminus1 = {{1LL, + +/// @brief fac_ld[k] = static_cast(k!), `0<=k<=170` +/// @note This is an inexact representation for most values, so should only be +/// used when `fac` does not +/// suffice, i.e. for `k>=21`. +/// @note This should be sufficient for computing with `double` reals +static constexpr std::array fac_ld = { + {1.l, + 1.l, + 2.l, + 6.l, + 24.l, + 120.l, + 720.l, + 5040.l, + 40320.l, + 362880.l, + 3.6288e6l, + 3.99168e7l, + 4.790016e8l, + 6.2270208e9l, + 8.71782912e10l, + 1.307674368e12l, + 2.0922789888e13l, + 3.55687428096e14l, + 6.402373705728e15l, + 1.21645100408832e17l, + 2.43290200817664e18l, + 5.109094217170944e19l, + 1.12400072777760768e21l, + 2.585201673888497664e22l, + 6.2044840173323943936e23l, + 1.5511210043330985984e25l, + 4.03291461126605635584e26l, + 1.0888869450418352160768e28l, + 3.04888344611713860501504e29l, + 8.841761993739701954543616e30l, + 2.6525285981219105863630848e32l, + 8.22283865417792281772556288e33l, + 2.6313083693369353016721801216e35l, + 8.68331761881188649551819440128e36l, + 2.9523279903960414084761860964352e38l, + 1.03331479663861449296666513375232e40l, + 3.719933267899012174679994481508352e41l, + 1.37637530912263450463159795815809024e43l, + 5.23022617466601111760007224100074291e44l, + 2.03978820811974433586402817399028974e46l, + 8.15915283247897734345611269596115894e47l, + 3.34525266131638071081700620534407517e49l, + 1.40500611775287989854314260624451157e51l, + 6.04152630633738356373551320685139975e52l, + 2.65827157478844876804362581101461589e54l, + 1.19622220865480194561963161495657715e56l, + 5.50262215981208894985030542880025489e57l, + 2.5862324151116818064296435515361198e59l, + 1.2413915592536072670862289047373375e61l, + 6.08281864034267560872252163321295377e62l, + 3.04140932017133780436126081660647688e64l, + 1.55111875328738228022424301646930321e66l, + 8.0658175170943878571660636856403767e67l, + 4.27488328406002556429801375338939965e69l, + 2.30843697339241380472092742683027581e71l, + 1.2696403353658275925965100847566517e73l, + 7.1099858780486345185404564746372495e74l, + 4.05269195048772167556806019054323221e76l, + 2.35056133128287857182947491051507468e78l, + 1.38683118545689835737939019720389406e80l, + 8.32098711274139014427634118322336438e81l, + 5.07580213877224798800856812176625227e83l, + 3.14699732603879375256531223549507641e85l, + 1.98260831540444006411614670836189814e87l, + 1.26886932185884164103433389335161481e89l, + 8.24765059208247066672317030678549625e90l, + 5.44344939077443064003729240247842753e92l, + 3.64711109181886852882498590966054644e94l, + 2.48003554243683059960099041856917158e96l, + 1.71122452428141311372468338881272839e98l, + 1.19785716699698917960727837216890987e100l, + 8.5047858856786231752116764423992601e101l, + 6.12344583768860868615240703852746727e103l, + 4.47011546151268434089125713812505111e105l, + 3.30788544151938641225953028221253782e107l, + 2.48091408113953980919464771165940337e109l, + 1.88549470166605025498793226086114656e111l, + 1.45183092028285869634070784086308285e113l, + 1.13242811782062978314575211587320462e115l, + 8.94618213078297528685144171539831652e116l, + 7.15694570462638022948115337231865322e118l, + 5.79712602074736798587973423157810911e120l, + 4.75364333701284174842138206989404947e122l, + 3.94552396972065865118974711801206106e124l, + 3.31424013456535326699938757913013129e126l, + 2.81710411438055027694947944226061159e128l, + 2.42270953836727323817655232034412597e130l, + 2.1077572983795277172136005186993896e132l, + 1.85482642257398439114796845645546284e134l, + 1.65079551609084610812169192624536193e136l, + 1.48571596448176149730952273362082574e138l, + 1.35200152767840296255166568759495142e140l, + 1.24384140546413072554753243258735531e142l, + 1.15677250708164157475920516230624044e144l, + 1.08736615665674308027365285256786601e146l, + 1.03299784882390592625997020993947271e148l, + 9.91677934870949689209571401541893801e149l, + 9.61927596824821198533284259495636987e151l, + 9.42689044888324774562618574305724247e153l, + 9.33262154439441526816992388562667005e155l, + 9.33262154439441526816992388562667005e157l, + 9.42594775983835942085162312448293675e159l, + 9.61446671503512660926865558697259548e161l, + 9.90290071648618040754671525458177335e163l, + 1.02990167451456276238485838647650443e166l, + 1.08139675824029090050410130580032965e168l, + 1.14628056373470835453434738414834943e170l, + 1.22652020319613793935175170103873389e172l, + 1.3246418194518289744998918371218326e174l, + 1.44385958320249358220488210246279753e176l, + 1.58824554152274294042537031270907729e178l, + 1.76295255109024466387216104710707579e180l, + 1.97450685722107402353682037275992488e182l, + 2.23119274865981364659660702121871512e184l, + 2.54355973347218755712013200418933523e186l, + 2.92509369349301569068815180481773552e188l, + 3.3931086844518982011982560935885732e190l, + 3.96993716080872089540195962949863065e192l, + 4.68452584975429065657431236280838416e194l, + 5.57458576120760588132343171174197716e196l, + 6.68950291344912705758811805409037259e198l, + 8.09429852527344373968162284544935083e200l, + 9.87504420083360136241157987144820801e202l, + 1.21463043670253296757662432418812959e205l, + 1.50614174151114087979501416199328069e207l, + 1.88267717688892609974376770249160086e209l, + 2.37217324288004688567714730513941708e211l, + 3.01266001845765954480997707752705969e213l, + 3.85620482362580421735677065923463641e215l, + 4.97450422247728744039023415041268096e217l, + 6.46685548922047367250730439553648525e219l, + 8.47158069087882051098456875815279568e221l, + 1.11824865119600430744996307607616903e224l, + 1.48727070609068572890845089118130481e226l, + 1.99294274616151887673732419418294845e228l, + 2.6904727073180504835953876621469804e230l, + 3.65904288195254865768972722051989335e232l, + 5.01288874827499166103492629211225388e234l, + 6.91778647261948849222819828311491036e236l, + 9.6157231969410890041971956135297254e238l, + 1.34620124757175246058760738589416156e241l, + 1.89814375907617096942852641411076779e243l, + 2.69536413788816277658850750803729027e245l, + 3.85437071718007277052156573649332508e247l, + 5.55029383273930478955105466055038812e249l, + 8.04792605747199194484902925779806277e251l, + 1.17499720439091082394795827163851716e254l, + 1.72724589045463891120349865930862023e256l, + 2.55632391787286558858117801577675794e258l, + 3.80892263763056972698595524350736934e260l, + 5.713383956445854590478932865261054e262l, + 8.62720977423324043162318862654419154e264l, + 1.31133588568345254560672467123471711e267l, + 2.00634390509568239477828874698911719e269l, + 3.08976961384735088795856467036324047e271l, + 4.78914290146339387633577523906302272e273l, + 7.47106292628289444708380937293831545e275l, + 1.17295687942641442819215807155131553e278l, + 1.85327186949373479654360975305107853e280l, + 2.94670227249503832650433950735121486e282l, + 4.71472363599206132240694321176194378e284l, + 7.59070505394721872907517857093672949e286l, + 1.22969421873944943411017892849175018e289l, + 2.00440157654530257759959165344155279e291l, + 3.28721858553429622726333031164414657e293l, + 5.42391066613158877498449501421284184e295l, + 9.00369170577843736647426172359331746e297l, + 1.50361651486499904020120170784008402e300l, + 2.52607574497319838753801886917134115e302l, + 4.26906800900470527493925188889956654e304l, + 7.25741561530799896739672821112926311e306l}}; + +/// @brief computes factorial as an integer + +/// @param k non-negative integer +/// @pre `k::value>::type> +constexpr int64_t fac_int(Int k) { +#if LIBINT2_CPLUSPLUS_STD >= 2014 + assert(!std::is_signed::value || k >= 0); + assert(k < fac.size()); +#endif + return fac[k]; +} + +/// @brief computes factorial as a real number + +/// @param k non-negative integer +/// @pre `k::value>::type> +constexpr Real fac_real(Int k) { +#if LIBINT2_CPLUSPLUS_STD >= 2014 + assert(!std::is_signed::value || k >= 0); + assert(k < fac.size() || k < fac_ld.size()); + // for extended-precision types should do better ... + assert(k < fac.size() || + (std::is_same_v || std::is_same_v)); +#endif + + // use exact representation when available + return (k < fac.size()) ? static_cast(fac[k]) + : static_cast(fac_ld[k]); +} + +/// `df_Kminus1[k] = (k-1)!!`, `0<=k<=34` +static constexpr std::array df_Kminus1 = {{1LL, 1LL, 1LL, 2LL, @@ -91,15 +309,391 @@ static constexpr std::array df_Kminus1 = {{1LL, 51011754393600LL, 213458046676875LL, 1428329123020800LL, - 6190283353629375LL}}; -/// bc(i,j) = binomial coefficient, i! / (j! (i-j)!) -template -int64_t bc(Int i, Int j) { - assert(i < Int(fac.size())); - assert(j < Int(fac.size())); + 6190283353629375LL, + 42849873690624000LL, + 191898783962510625LL, + 1371195958099968000LL, + 6332659870762850625LL}}; + +/// @brief fac_ld[k] = static_cast(k!), `0<=k<=170` +/// @note This is an inexact representation for most values, so should only be +/// used when `fac` does not +/// suffice, i.e. for `k>=21`. +/// @note This should be sufficient for computing with `double` reals +static constexpr std::array df_Kminus1_ld = { + {1.l, + 1.l, + 1.l, + 2.l, + 3.l, + 8.l, + 15.l, + 48.l, + 105.l, + 384.l, + 945.l, + 3840.l, + 10395.l, + 46080.l, + 135135.l, + 645120.l, + 2.027025e6l, + 1.032192e7l, + 3.4459425e7l, + 1.8579456e8l, + 6.54729075e8l, + 3.7158912e9l, + 1.3749310575e10l, + 8.17496064e10l, + 3.16234143225e11l, + 1.9619905536e12l, + 7.905853580625e12l, + 5.10117543936e13l, + 2.13458046676875e14l, + 1.4283291230208e15l, + 6.190283353629375e15l, + 4.2849873690624e16l, + 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3.1874758976922247332371523359727913e273l, + 4.205968339406864392707700592495767e274l, + 8.73368395967669576906979740056544817e275l, + 1.15664129333688770799461766293633592e277l, + 2.41049677287076803226326408255606369e278l, + 3.20389638254317895114509092633365051e279l, + 6.70118102858073512969187414950585707e280l, + 8.93887090729546927369480368447088492e281l, + 1.87633068800260583631372476186163998e283l, + 2.51182272495002686590823983533631866e284l, + 5.29125254016734845840470382844982474e285l, + 7.10845831160857603052031873400178182e286l, + 1.50271572140752696218693588727975023e288l, + 2.02591061880844416869829083919050782e289l, + 4.29776696322552711185463663762008565e290l, + 5.81436347598023476416409470847675744e291l, + 1.23775688540895180821413535163458467e293l, + 1.6803510445582878468434233707497829e294l, + 3.58949496768596024382099251974029553e295l, + 4.88982153966461763431436200888186824e296l, + 1.0481325305643003911957298157641663e298l, + 1.43271771112173296685410806860238739e299l, + 3.08150963985904315011544565834664891e300l, + 4.22651724780911225221961880237704281e301l, + 9.12126853398276772434171914870608078e302l, + 1.25527562259930633890922678430598171e304l, + 2.71813802312686478185383230631441207e305l, + 3.75327411157192595333858808507488533e306l, + 8.15441406938059434556149691894323621e307l}}; + +/// @brief computes double factorial as an integer + +/// @param k non-negative integer +/// @pre `k::value>::type> +constexpr int64_t df_int(Int k) { +#if LIBINT2_CPLUSPLUS_STD >= 2014 + assert(!std::is_signed::value || k >= 0); + assert(k < df_Kminus1.size() - 1); +#endif + return df_Kminus1[k + 1]; +} + +/// @brief computes double factorial as a real number + +/// @param k non-negative integer +/// @pre `k::value>::type> +constexpr Real df_real(Int k) { +#if LIBINT2_CPLUSPLUS_STD >= 2014 + assert(!std::is_signed::value || k >= 0); + assert(k < df_Kminus1.size() - 1 || k < df_Kminus1_ld.size() - 1); + // for extended-precision types should do better ... + assert(k < df_Kminus1.size() - 1 || + (std::is_same_v || std::is_same_v)) +#endif + // use exact representation when available + return (k < df_Kminus1.size() - 1) + ? static_cast(df_Kminus1[k + 1]) + : static_cast(df_Kminus1_ld[k + 1]); +} + +/// @brief computes binomial coefficient as an integer + +/// @param i non-negative integer +/// @param j non-negative integer +/// @pre `i>=j` +/// return `i! / (j! * (i-j)!)` +template ::value>::type> +constexpr int64_t bc_int(Int i, Int j) { +#if LIBINT2_CPLUSPLUS_STD >= 2014 assert(i >= j); - return fac[i] / (fac[j] * fac[i - j]); +#endif + return fac_int(i) / (fac_int(j) * fac_int(i - j)); } + +/// @brief computes binomial coefficient as a real number + +/// @param i non-negative integer +/// @param j non-negative integer +/// @pre `i>=j` +/// @return binomial coefficient, (\p i , \p j), uses exact representation when +/// possible +template < + typename Real, typename Int, + typename = typename std::enable_if::value>::type> +constexpr Real bc_real(Int i, Int j) { +#if LIBINT2_CPLUSPLUS_STD >= 2014 + assert(!std::is_signed::value || (i >= 0 && j >= 0)); + assert(i >= j); +#endif + return fac_real(i) / (fac_real(j) * fac_real(i - j)); +} + } // namespace math /// generally-contracted Solid-Harmonic/Cartesion Gaussian Shell diff --git a/include/libint2/solidharmonics.h b/include/libint2/solidharmonics.h index f97dc7632..5fdf47dc6 100644 --- a/include/libint2/solidharmonics.h +++ b/include/libint2/solidharmonics.h @@ -83,8 +83,7 @@ class SolidHarmonicsCoefficients { static const SolidHarmonicsCoefficients& instance(unsigned int l) { static std::vector shg_coefs( SolidHarmonicsCoefficients::CtorHelperIter(0), - SolidHarmonicsCoefficients::CtorHelperIter(11)); - assert(l <= 10); // see coeff() for explanation of the upper limit on l + SolidHarmonicsCoefficients::CtorHelperIter(13)); return shg_coefs[l]; } @@ -108,9 +107,9 @@ class SolidHarmonicsCoefficients { harmonic Ylm is requested. ---------------------------------------------------------------------------------------------*/ static Real coeff(int l, int m, int lx, int ly, int lz) { - using libint2::math::bc; - using libint2::math::df_Kminus1; - using libint2::math::fac; + using libint2::math::bc_real; + using libint2::math::df_real; + using libint2::math::fac_real; auto abs_m = std::abs(m); if ((lx + ly - abs_m) % 2) return 0.0; @@ -127,13 +126,17 @@ class SolidHarmonicsCoefficients { auto i = abs_m - lx; if (comp != parity(abs(i))) return 0.0; - assert(l <= 10); // libint2::math::fac[] is only defined up to 20 - Real pfac = - sqrt(((Real(fac[2 * lx]) * Real(fac[2 * ly]) * Real(fac[2 * lz])) / - fac[2 * l]) * - ((Real(fac[l - abs_m])) / (fac[l])) * (Real(1) / fac[l + abs_m]) * - (Real(1) / (fac[lx] * fac[ly] * fac[lz]))); - /* pfac = sqrt(fac[l-abs_m]/(fac[l]*fac[l]*fac[l+abs_m]));*/ + Real pfac; + pfac = sqrt(((fac_real(2 * lx) * fac_real(2 * ly) * + fac_real(2 * lz)) / + fac_real(2 * l)) * + ((fac_real(l - abs_m)) / (fac_real(l))) * + (Real(1) / fac_real(l + abs_m)) * + (Real(1) / (fac_real(lx) * fac_real(ly) * + fac_real(lz)))); + + /* pfac = + * sqrt(fac_real(l-abs_m)/(fac_real(l)*fac_real(l)*fac[l+abs_m]));*/ pfac /= (1L << l); if (m < 0) pfac *= parity((i - 1) / 2); @@ -144,19 +147,22 @@ class SolidHarmonicsCoefficients { auto i_max = (l - abs_m) / 2; Real sum = 0; for (auto i = i_min; i <= i_max; i++) { - Real pfac1 = bc(l, i) * bc(i, j); - pfac1 *= (Real(parity(i) * fac[2 * (l - i)]) / fac[l - abs_m - 2 * i]); + Real pfac1 = bc_real(l, i) * bc_real(i, j); + pfac1 *= (Real(parity(i) * fac_real(2 * (l - i))) / + fac_real(l - abs_m - 2 * i)); Real sum1 = 0.0; const int k_min = std::max((lx - abs_m) / 2, 0); const int k_max = std::min(j, lx / 2); for (int k = k_min; k <= k_max; k++) { if (lx - 2 * k <= abs_m) - sum1 += bc(j, k) * bc(abs_m, lx - 2 * k) * parity(k); + sum1 += bc_real(j, k) * bc_real(abs_m, lx - 2 * k) * + parity(k); } sum += pfac1 * sum1; } - sum *= sqrt(Real(df_Kminus1[2 * l]) / - (df_Kminus1[2 * lx] * df_Kminus1[2 * ly] * df_Kminus1[2 * lz])); + sum *= sqrt(df_real(2 * l - 1) / + (df_real(2 * lx - 1) * df_real(2 * ly - 1) * + df_real(2 * lz - 1))); Real result = (m == 0) ? pfac * sum : M_SQRT2 * pfac * sum; return result; diff --git a/tests/unit/test-core.cc b/tests/unit/test-core.cc index dd7e0d6f0..4370962ba 100644 --- a/tests/unit/test-core.cc +++ b/tests/unit/test-core.cc @@ -18,9 +18,21 @@ * */ +#include + #include "catch.hpp" #include "fixture.h" +TEST_CASE("SolidHarmonicsCoefficients ctor", "[shell]") { + using libint2::solidharmonics::SolidHarmonicsCoefficients; + REQUIRE_NOTHROW(SolidHarmonicsCoefficients::instance(12)); + auto sh12 = SolidHarmonicsCoefficients::instance(12); + CHECK_NOTHROW(sh12.coeff(12, 0, 0, 0, 12)); + CHECK(sh12.coeff(12, 0, 0, 0, 12) == 1.); + CHECK_NOTHROW(SolidHarmonicsCoefficients::instance(25).coeff( + 25, 0, 0, 0, 25)); +} + TEST_CASE("Shell ctor", "[shell]") { REQUIRE_NOTHROW(Shell{}); auto s0 = Shell{};