| // Copyright John Maddock 2007. |
| // 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_ZETA_HPP |
| #define BOOST_MATH_ZETA_HPP |
| |
| #ifdef _MSC_VER |
| #pragma once |
| #endif |
| |
| #include <boost/math/tools/precision.hpp> |
| #include <boost/math/tools/series.hpp> |
| #include <boost/math/policies/error_handling.hpp> |
| #include <boost/math/special_functions/gamma.hpp> |
| #include <boost/math/special_functions/sin_pi.hpp> |
| |
| namespace boost{ namespace math{ namespace detail{ |
| |
| template <class T, class Policy> |
| struct zeta_series_cache_size |
| { |
| // |
| // Work how large to make our cache size when evaluating the series |
| // evaluation: normally this is just large enough for the series |
| // to have converged, but for arbitrary precision types we need a |
| // really large cache to achieve reasonable precision in a reasonable |
| // time. This is important when constructing rational approximations |
| // to zeta for example. |
| // |
| typedef typename boost::math::policies::precision<T,Policy>::type precision_type; |
| typedef typename mpl::if_< |
| mpl::less_equal<precision_type, mpl::int_<0> >, |
| mpl::int_<5000>, |
| typename mpl::if_< |
| mpl::less_equal<precision_type, mpl::int_<64> >, |
| mpl::int_<70>, |
| typename mpl::if_< |
| mpl::less_equal<precision_type, mpl::int_<113> >, |
| mpl::int_<100>, |
| mpl::int_<5000> |
| >::type |
| >::type |
| >::type type; |
| }; |
| |
| template <class T, class Policy> |
| T zeta_series_imp(T s, T sc, const Policy&) |
| { |
| // |
| // Series evaluation from: |
| // Havil, J. Gamma: Exploring Euler's Constant. |
| // Princeton, NJ: Princeton University Press, 2003. |
| // |
| // See also http://mathworld.wolfram.com/RiemannZetaFunction.html |
| // |
| BOOST_MATH_STD_USING |
| T sum = 0; |
| T mult = 0.5; |
| T change; |
| typedef typename zeta_series_cache_size<T,Policy>::type cache_size; |
| T powers[cache_size::value] = { 0, }; |
| unsigned n = 0; |
| do{ |
| T binom = -static_cast<T>(n); |
| T nested_sum = 1; |
| if(n < sizeof(powers) / sizeof(powers[0])) |
| powers[n] = pow(static_cast<T>(n + 1), -s); |
| for(unsigned k = 1; k <= n; ++k) |
| { |
| T p; |
| if(k < sizeof(powers) / sizeof(powers[0])) |
| { |
| p = powers[k]; |
| //p = pow(k + 1, -s); |
| } |
| else |
| p = pow(static_cast<T>(k + 1), -s); |
| nested_sum += binom * p; |
| binom *= (k - static_cast<T>(n)) / (k + 1); |
| } |
| change = mult * nested_sum; |
| sum += change; |
| mult /= 2; |
| ++n; |
| }while(fabs(change / sum) > tools::epsilon<T>()); |
| |
| return sum * 1 / -boost::math::powm1(T(2), sc); |
| } |
| // |
| // Classical p-series: |
| // |
| template <class T> |
| struct zeta_series2 |
| { |
| typedef T result_type; |
| zeta_series2(T _s) : s(-_s), k(1){} |
| T operator()() |
| { |
| BOOST_MATH_STD_USING |
| return pow(static_cast<T>(k++), s); |
| } |
| private: |
| T s; |
| unsigned k; |
| }; |
| |
| template <class T, class Policy> |
| inline T zeta_series2_imp(T s, const Policy& pol) |
| { |
| boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();; |
| zeta_series2<T> f(s); |
| T result = tools::sum_series( |
| f, |
| policies::get_epsilon<T, Policy>(), |
| max_iter); |
| policies::check_series_iterations("boost::math::zeta_series2<%1%>(%1%)", max_iter, pol); |
| return result; |
| } |
| |
| template <class T, class Policy> |
| T zeta_imp_prec(T s, T sc, const Policy& pol, const mpl::int_<0>&) |
| { |
| BOOST_MATH_STD_USING |
| T result; |
| // |
| // Only use power series if it will converge in 100 |
| // iterations or less: the more iterations it consumes |
| // the slower convergence becomes so we have to be very |
| // careful in it's usage. |
| // |
| if (s > -log(tools::epsilon<T>()) / 4.5) |
| result = detail::zeta_series2_imp(s, pol); |
| else |
| result = detail::zeta_series_imp(s, sc, pol); |
| return result; |
| } |
| |
| template <class T, class Policy> |
| inline T zeta_imp_prec(T s, T sc, const Policy&, const mpl::int_<53>&) |
| { |
| BOOST_MATH_STD_USING |
| T result; |
| if(s < 1) |
| { |
| // Rational Approximation |
| // Maximum Deviation Found: 2.020e-18 |
| // Expected Error Term: -2.020e-18 |
| // Max error found at double precision: 3.994987e-17 |
| static const T P[6] = { |
| 0.24339294433593750202L, |
| -0.49092470516353571651L, |
| 0.0557616214776046784287L, |
| -0.00320912498879085894856L, |
| 0.000451534528645796438704L, |
| -0.933241270357061460782e-5L, |
| }; |
| static const T Q[6] = { |
| 1L, |
| -0.279960334310344432495L, |
| 0.0419676223309986037706L, |
| -0.00413421406552171059003L, |
| 0.00024978985622317935355L, |
| -0.101855788418564031874e-4L, |
| }; |
| result = tools::evaluate_polynomial(P, sc) / tools::evaluate_polynomial(Q, sc); |
| result -= 1.2433929443359375F; |
| result += (sc); |
| result /= (sc); |
| } |
| else if(s <= 2) |
| { |
| // Maximum Deviation Found: 9.007e-20 |
| // Expected Error Term: 9.007e-20 |
| static const T P[6] = { |
| 0.577215664901532860516, |
| 0.243210646940107164097, |
| 0.0417364673988216497593, |
| 0.00390252087072843288378, |
| 0.000249606367151877175456, |
| 0.110108440976732897969e-4, |
| }; |
| static const T Q[6] = { |
| 1, |
| 0.295201277126631761737, |
| 0.043460910607305495864, |
| 0.00434930582085826330659, |
| 0.000255784226140488490982, |
| 0.10991819782396112081e-4, |
| }; |
| result = tools::evaluate_polynomial(P, -sc) / tools::evaluate_polynomial(Q, -sc); |
| result += 1 / (-sc); |
| } |
| else if(s <= 4) |
| { |
| // Maximum Deviation Found: 5.946e-22 |
| // Expected Error Term: -5.946e-22 |
| static const float Y = 0.6986598968505859375; |
| static const T P[6] = { |
| -0.0537258300023595030676, |
| 0.0445163473292365591906, |
| 0.0128677673534519952905, |
| 0.00097541770457391752726, |
| 0.769875101573654070925e-4, |
| 0.328032510000383084155e-5, |
| }; |
| static const T Q[7] = { |
| 1, |
| 0.33383194553034051422, |
| 0.0487798431291407621462, |
| 0.00479039708573558490716, |
| 0.000270776703956336357707, |
| 0.106951867532057341359e-4, |
| 0.236276623974978646399e-7, |
| }; |
| result = tools::evaluate_polynomial(P, s - 2) / tools::evaluate_polynomial(Q, s - 2); |
| result += Y + 1 / (-sc); |
| } |
| else if(s <= 7) |
| { |
| // Maximum Deviation Found: 2.955e-17 |
| // Expected Error Term: 2.955e-17 |
| // Max error found at double precision: 2.009135e-16 |
| |
| static const T P[6] = { |
| -2.49710190602259410021, |
| -2.60013301809475665334, |
| -0.939260435377109939261, |
| -0.138448617995741530935, |
| -0.00701721240549802377623, |
| -0.229257310594893932383e-4, |
| }; |
| static const T Q[9] = { |
| 1, |
| 0.706039025937745133628, |
| 0.15739599649558626358, |
| 0.0106117950976845084417, |
| -0.36910273311764618902e-4, |
| 0.493409563927590008943e-5, |
| -0.234055487025287216506e-6, |
| 0.718833729365459760664e-8, |
| -0.1129200113474947419e-9, |
| }; |
| result = tools::evaluate_polynomial(P, s - 4) / tools::evaluate_polynomial(Q, s - 4); |
| result = 1 + exp(result); |
| } |
| else if(s < 15) |
| { |
| // Maximum Deviation Found: 7.117e-16 |
| // Expected Error Term: 7.117e-16 |
| // Max error found at double precision: 9.387771e-16 |
| static const T P[7] = { |
| -4.78558028495135619286, |
| -1.89197364881972536382, |
| -0.211407134874412820099, |
| -0.000189204758260076688518, |
| 0.00115140923889178742086, |
| 0.639949204213164496988e-4, |
| 0.139348932445324888343e-5, |
| }; |
| static const T Q[9] = { |
| 1, |
| 0.244345337378188557777, |
| 0.00873370754492288653669, |
| -0.00117592765334434471562, |
| -0.743743682899933180415e-4, |
| -0.21750464515767984778e-5, |
| 0.471001264003076486547e-8, |
| -0.833378440625385520576e-10, |
| 0.699841545204845636531e-12, |
| }; |
| result = tools::evaluate_polynomial(P, s - 7) / tools::evaluate_polynomial(Q, s - 7); |
| result = 1 + exp(result); |
| } |
| else if(s < 36) |
| { |
| // Max error in interpolated form: 1.668e-17 |
| // Max error found at long double precision: 1.669714e-17 |
| static const T P[8] = { |
| -10.3948950573308896825, |
| -2.85827219671106697179, |
| -0.347728266539245787271, |
| -0.0251156064655346341766, |
| -0.00119459173416968685689, |
| -0.382529323507967522614e-4, |
| -0.785523633796723466968e-6, |
| -0.821465709095465524192e-8, |
| }; |
| static const T Q[10] = { |
| 1, |
| 0.208196333572671890965, |
| 0.0195687657317205033485, |
| 0.00111079638102485921877, |
| 0.408507746266039256231e-4, |
| 0.955561123065693483991e-6, |
| 0.118507153474022900583e-7, |
| 0.222609483627352615142e-14, |
| }; |
| result = tools::evaluate_polynomial(P, s - 15) / tools::evaluate_polynomial(Q, s - 15); |
| result = 1 + exp(result); |
| } |
| else if(s < 56) |
| { |
| result = 1 + pow(T(2), -s); |
| } |
| else |
| { |
| result = 1; |
| } |
| return result; |
| } |
| |
| template <class T, class Policy> |
| T zeta_imp_prec(T s, T sc, const Policy&, const mpl::int_<64>&) |
| { |
| BOOST_MATH_STD_USING |
| T result; |
| if(s < 1) |
| { |
| // Rational Approximation |
| // Maximum Deviation Found: 3.099e-20 |
| // Expected Error Term: 3.099e-20 |
| // Max error found at long double precision: 5.890498e-20 |
| static const T P[6] = { |
| 0.243392944335937499969L, |
| -0.496837806864865688082L, |
| 0.0680008039723709987107L, |
| -0.00511620413006619942112L, |
| 0.000455369899250053003335L, |
| -0.279496685273033761927e-4L, |
| }; |
| static const T Q[7] = { |
| 1L, |
| -0.30425480068225790522L, |
| 0.050052748580371598736L, |
| -0.00519355671064700627862L, |
| 0.000360623385771198350257L, |
| -0.159600883054550987633e-4L, |
| 0.339770279812410586032e-6L, |
| }; |
| result = tools::evaluate_polynomial(P, sc) / tools::evaluate_polynomial(Q, sc); |
| result -= 1.2433929443359375F; |
| result += (sc); |
| result /= (sc); |
| } |
| else if(s <= 2) |
| { |
| // Maximum Deviation Found: 1.059e-21 |
| // Expected Error Term: 1.059e-21 |
| // Max error found at long double precision: 1.626303e-19 |
| |
| static const T P[6] = { |
| 0.577215664901532860605L, |
| 0.222537368917162139445L, |
| 0.0356286324033215682729L, |
| 0.00304465292366350081446L, |
| 0.000178102511649069421904L, |
| 0.700867470265983665042e-5L, |
| }; |
| static const T Q[7] = { |
| 1L, |
| 0.259385759149531030085L, |
| 0.0373974962106091316854L, |
| 0.00332735159183332820617L, |
| 0.000188690420706998606469L, |
| 0.635994377921861930071e-5L, |
| 0.226583954978371199405e-7L, |
| }; |
| result = tools::evaluate_polynomial(P, -sc) / tools::evaluate_polynomial(Q, -sc); |
| result += 1 / (-sc); |
| } |
| else if(s <= 4) |
| { |
| // Maximum Deviation Found: 5.946e-22 |
| // Expected Error Term: -5.946e-22 |
| static const float Y = 0.6986598968505859375; |
| static const T P[7] = { |
| -0.053725830002359501027L, |
| 0.0470551187571475844778L, |
| 0.0101339410415759517471L, |
| 0.00100240326666092854528L, |
| 0.685027119098122814867e-4L, |
| 0.390972820219765942117e-5L, |
| 0.540319769113543934483e-7L, |
| }; |
| static const T Q[8] = { |
| 1, |
| 0.286577739726542730421L, |
| 0.0447355811517733225843L, |
| 0.00430125107610252363302L, |
| 0.000284956969089786662045L, |
| 0.116188101609848411329e-4L, |
| 0.278090318191657278204e-6L, |
| -0.19683620233222028478e-8L, |
| }; |
| result = tools::evaluate_polynomial(P, s - 2) / tools::evaluate_polynomial(Q, s - 2); |
| result += Y + 1 / (-sc); |
| } |
| else if(s <= 7) |
| { |
| // Max error found at long double precision: 8.132216e-19 |
| static const T P[8] = { |
| -2.49710190602259407065L, |
| -3.36664913245960625334L, |
| -1.77180020623777595452L, |
| -0.464717885249654313933L, |
| -0.0643694921293579472583L, |
| -0.00464265386202805715487L, |
| -0.000165556579779704340166L, |
| -0.252884970740994069582e-5L, |
| }; |
| static const T Q[9] = { |
| 1, |
| 1.01300131390690459085L, |
| 0.387898115758643503827L, |
| 0.0695071490045701135188L, |
| 0.00586908595251442839291L, |
| 0.000217752974064612188616L, |
| 0.397626583349419011731e-5L, |
| -0.927884739284359700764e-8L, |
| 0.119810501805618894381e-9L, |
| }; |
| result = tools::evaluate_polynomial(P, s - 4) / tools::evaluate_polynomial(Q, s - 4); |
| result = 1 + exp(result); |
| } |
| else if(s < 15) |
| { |
| // Max error in interpolated form: 1.133e-18 |
| // Max error found at long double precision: 2.183198e-18 |
| static const T P[9] = { |
| -4.78558028495135548083L, |
| -3.23873322238609358947L, |
| -0.892338582881021799922L, |
| -0.131326296217965913809L, |
| -0.0115651591773783712996L, |
| -0.000657728968362695775205L, |
| -0.252051328129449973047e-4L, |
| -0.626503445372641798925e-6L, |
| -0.815696314790853893484e-8L, |
| }; |
| static const T Q[9] = { |
| 1, |
| 0.525765665400123515036L, |
| 0.10852641753657122787L, |
| 0.0115669945375362045249L, |
| 0.000732896513858274091966L, |
| 0.30683952282420248448e-4L, |
| 0.819649214609633126119e-6L, |
| 0.117957556472335968146e-7L, |
| -0.193432300973017671137e-12L, |
| }; |
| result = tools::evaluate_polynomial(P, s - 7) / tools::evaluate_polynomial(Q, s - 7); |
| result = 1 + exp(result); |
| } |
| else if(s < 42) |
| { |
| // Max error in interpolated form: 1.668e-17 |
| // Max error found at long double precision: 1.669714e-17 |
| static const T P[9] = { |
| -10.3948950573308861781L, |
| -2.82646012777913950108L, |
| -0.342144362739570333665L, |
| -0.0249285145498722647472L, |
| -0.00122493108848097114118L, |
| -0.423055371192592850196e-4L, |
| -0.1025215577185967488e-5L, |
| -0.165096762663509467061e-7L, |
| -0.145392555873022044329e-9L, |
| }; |
| static const T Q[10] = { |
| 1, |
| 0.205135978585281988052L, |
| 0.0192359357875879453602L, |
| 0.00111496452029715514119L, |
| 0.434928449016693986857e-4L, |
| 0.116911068726610725891e-5L, |
| 0.206704342290235237475e-7L, |
| 0.209772836100827647474e-9L, |
| -0.939798249922234703384e-16L, |
| 0.264584017421245080294e-18L, |
| }; |
| result = tools::evaluate_polynomial(P, s - 15) / tools::evaluate_polynomial(Q, s - 15); |
| result = 1 + exp(result); |
| } |
| else if(s < 63) |
| { |
| result = 1 + pow(T(2), -s); |
| } |
| else |
| { |
| result = 1; |
| } |
| return result; |
| } |
| |
| template <class T, class Policy> |
| T zeta_imp_prec(T s, T sc, const Policy& pol, const mpl::int_<113>&) |
| { |
| BOOST_MATH_STD_USING |
| T result; |
| if(s < 1) |
| { |
| // Rational Approximation |
| // Maximum Deviation Found: 9.493e-37 |
| // Expected Error Term: 9.492e-37 |
| // Max error found at long double precision: 7.281332e-31 |
| |
| static const T P[10] = { |
| -1L, |
| -0.0353008629988648122808504280990313668L, |
| 0.0107795651204927743049369868548706909L, |
| 0.000523961870530500751114866884685172975L, |
| -0.661805838304910731947595897966487515e-4L, |
| -0.658932670403818558510656304189164638e-5L, |
| -0.103437265642266106533814021041010453e-6L, |
| 0.116818787212666457105375746642927737e-7L, |
| 0.660690993901506912123512551294239036e-9L, |
| 0.113103113698388531428914333768142527e-10L, |
| }; |
| static const T Q[11] = { |
| 1L, |
| -0.387483472099602327112637481818565459L, |
| 0.0802265315091063135271497708694776875L, |
| -0.0110727276164171919280036408995078164L, |
| 0.00112552716946286252000434849173787243L, |
| -0.874554160748626916455655180296834352e-4L, |
| 0.530097847491828379568636739662278322e-5L, |
| -0.248461553590496154705565904497247452e-6L, |
| 0.881834921354014787309644951507523899e-8L, |
| -0.217062446168217797598596496310953025e-9L, |
| 0.315823200002384492377987848307151168e-11L, |
| }; |
| result = tools::evaluate_polynomial(P, sc) / tools::evaluate_polynomial(Q, sc); |
| result += (sc); |
| result /= (sc); |
| } |
| else if(s <= 2) |
| { |
| // Maximum Deviation Found: 1.616e-37 |
| // Expected Error Term: -1.615e-37 |
| |
| static const T P[10] = { |
| 0.577215664901532860606512090082402431L, |
| 0.255597968739771510415479842335906308L, |
| 0.0494056503552807274142218876983542205L, |
| 0.00551372778611700965268920983472292325L, |
| 0.00043667616723970574871427830895192731L, |
| 0.268562259154821957743669387915239528e-4L, |
| 0.109249633923016310141743084480436612e-5L, |
| 0.273895554345300227466534378753023924e-7L, |
| 0.583103205551702720149237384027795038e-9L, |
| -0.835774625259919268768735944711219256e-11L, |
| }; |
| static const T Q[11] = { |
| 1L, |
| 0.316661751179735502065583176348292881L, |
| 0.0540401806533507064453851182728635272L, |
| 0.00598621274107420237785899476374043797L, |
| 0.000474907812321704156213038740142079615L, |
| 0.272125421722314389581695715835862418e-4L, |
| 0.112649552156479800925522445229212933e-5L, |
| 0.301838975502992622733000078063330461e-7L, |
| 0.422960728687211282539769943184270106e-9L, |
| -0.377105263588822468076813329270698909e-11L, |
| -0.581926559304525152432462127383600681e-13L, |
| }; |
| result = tools::evaluate_polynomial(P, -sc) / tools::evaluate_polynomial(Q, -sc); |
| result += 1 / (-sc); |
| } |
| else if(s <= 4) |
| { |
| // Maximum Deviation Found: 1.891e-36 |
| // Expected Error Term: -1.891e-36 |
| // Max error found: 2.171527e-35 |
| |
| static const float Y = 0.6986598968505859375; |
| static const T P[11] = { |
| -0.0537258300023595010275848333539748089L, |
| 0.0429086930802630159457448174466342553L, |
| 0.0136148228754303412510213395034056857L, |
| 0.00190231601036042925183751238033763915L, |
| 0.000186880390916311438818302549192456581L, |
| 0.145347370745893262394287982691323657e-4L, |
| 0.805843276446813106414036600485884885e-6L, |
| 0.340818159286739137503297172091882574e-7L, |
| 0.115762357488748996526167305116837246e-8L, |
| 0.231904754577648077579913403645767214e-10L, |
| 0.340169592866058506675897646629036044e-12L, |
| }; |
| static const T Q[12] = { |
| 1L, |
| 0.363755247765087100018556983050520554L, |
| 0.0696581979014242539385695131258321598L, |
| 0.00882208914484611029571547753782014817L, |
| 0.000815405623261946661762236085660996718L, |
| 0.571366167062457197282642344940445452e-4L, |
| 0.309278269271853502353954062051797838e-5L, |
| 0.12822982083479010834070516053794262e-6L, |
| 0.397876357325018976733953479182110033e-8L, |
| 0.8484432107648683277598472295289279e-10L, |
| 0.105677416606909614301995218444080615e-11L, |
| 0.547223964564003701979951154093005354e-15L, |
| }; |
| result = tools::evaluate_polynomial(P, s - 2) / tools::evaluate_polynomial(Q, s - 2); |
| result += Y + 1 / (-sc); |
| } |
| else if(s <= 6) |
| { |
| // Max error in interpolated form: 1.510e-37 |
| // Max error found at long double precision: 2.769266e-34 |
| |
| static const T Y = 3.28348541259765625F; |
| |
| static const T P[13] = { |
| 0.786383506575062179339611614117697622L, |
| 0.495766593395271370974685959652073976L, |
| -0.409116737851754766422360889037532228L, |
| -0.57340744006238263817895456842655987L, |
| -0.280479899797421910694892949057963111L, |
| -0.0753148409447590257157585696212649869L, |
| -0.0122934003684672788499099362823748632L, |
| -0.00126148398446193639247961370266962927L, |
| -0.828465038179772939844657040917364896e-4L, |
| -0.361008916706050977143208468690645684e-5L, |
| -0.109879825497910544424797771195928112e-6L, |
| -0.214539416789686920918063075528797059e-8L, |
| -0.15090220092460596872172844424267351e-10L, |
| }; |
| static const T Q[14] = { |
| 1L, |
| 1.69490865837142338462982225731926485L, |
| 1.22697696630994080733321401255942464L, |
| 0.495409420862526540074366618006341533L, |
| 0.122368084916843823462872905024259633L, |
| 0.0191412993625268971656513890888208623L, |
| 0.00191401538628980617753082598351559642L, |
| 0.000123318142456272424148930280876444459L, |
| 0.531945488232526067889835342277595709e-5L, |
| 0.161843184071894368337068779669116236e-6L, |
| 0.305796079600152506743828859577462778e-8L, |
| 0.233582592298450202680170811044408894e-10L, |
| -0.275363878344548055574209713637734269e-13L, |
| 0.221564186807357535475441900517843892e-15L, |
| }; |
| result = tools::evaluate_polynomial(P, s - 4) / tools::evaluate_polynomial(Q, s - 4); |
| result -= Y; |
| result = 1 + exp(result); |
| } |
| else if(s < 10) |
| { |
| // Max error in interpolated form: 1.999e-34 |
| // Max error found at long double precision: 2.156186e-33 |
| |
| static const T P[13] = { |
| -4.0545627381873738086704293881227365L, |
| -4.70088348734699134347906176097717782L, |
| -2.36921550900925512951976617607678789L, |
| -0.684322583796369508367726293719322866L, |
| -0.126026534540165129870721937592996324L, |
| -0.015636903921778316147260572008619549L, |
| -0.00135442294754728549644376325814460807L, |
| -0.842793965853572134365031384646117061e-4L, |
| -0.385602133791111663372015460784978351e-5L, |
| -0.130458500394692067189883214401478539e-6L, |
| -0.315861074947230418778143153383660035e-8L, |
| -0.500334720512030826996373077844707164e-10L, |
| -0.420204769185233365849253969097184005e-12L, |
| }; |
| static const T Q[14] = { |
| 1L, |
| 0.97663511666410096104783358493318814L, |
| 0.40878780231201806504987368939673249L, |
| 0.0963890666609396058945084107597727252L, |
| 0.0142207619090854604824116070866614505L, |
| 0.00139010220902667918476773423995750877L, |
| 0.940669540194694997889636696089994734e-4L, |
| 0.458220848507517004399292480807026602e-5L, |
| 0.16345521617741789012782420625435495e-6L, |
| 0.414007452533083304371566316901024114e-8L, |
| 0.68701473543366328016953742622661377e-10L, |
| 0.603461891080716585087883971886075863e-12L, |
| 0.294670713571839023181857795866134957e-16L, |
| -0.147003914536437243143096875069813451e-18L, |
| }; |
| result = tools::evaluate_polynomial(P, s - 6) / tools::evaluate_polynomial(Q, s - 6); |
| result = 1 + exp(result); |
| } |
| else if(s < 17) |
| { |
| // Max error in interpolated form: 1.641e-32 |
| // Max error found at long double precision: 1.696121e-32 |
| static const T P[13] = { |
| -6.91319491921722925920883787894829678L, |
| -3.65491257639481960248690596951049048L, |
| -0.813557553449954526442644544105257881L, |
| -0.0994317301685870959473658713841138083L, |
| -0.00726896610245676520248617014211734906L, |
| -0.000317253318715075854811266230916762929L, |
| -0.66851422826636750855184211580127133e-5L, |
| 0.879464154730985406003332577806849971e-7L, |
| 0.113838903158254250631678791998294628e-7L, |
| 0.379184410304927316385211327537817583e-9L, |
| 0.612992858643904887150527613446403867e-11L, |
| 0.347873737198164757035457841688594788e-13L, |
| -0.289187187441625868404494665572279364e-15L, |
| }; |
| static const T Q[14] = { |
| 1L, |
| 0.427310044448071818775721584949868806L, |
| 0.074602514873055756201435421385243062L, |
| 0.00688651562174480772901425121653945942L, |
| 0.000360174847635115036351323894321880445L, |
| 0.973556847713307543918865405758248777e-5L, |
| -0.853455848314516117964634714780874197e-8L, |
| -0.118203513654855112421673192194622826e-7L, |
| -0.462521662511754117095006543363328159e-9L, |
| -0.834212591919475633107355719369463143e-11L, |
| -0.5354594751002702935740220218582929e-13L, |
| 0.406451690742991192964889603000756203e-15L, |
| 0.887948682401000153828241615760146728e-19L, |
| -0.34980761098820347103967203948619072e-21L, |
| }; |
| result = tools::evaluate_polynomial(P, s - 10) / tools::evaluate_polynomial(Q, s - 10); |
| result = 1 + exp(result); |
| } |
| else if(s < 30) |
| { |
| // Max error in interpolated form: 1.563e-31 |
| // Max error found at long double precision: 1.562725e-31 |
| |
| static const T P[13] = { |
| -11.7824798233959252791987402769438322L, |
| -4.36131215284987731928174218354118102L, |
| -0.732260980060982349410898496846972204L, |
| -0.0744985185694913074484248803015717388L, |
| -0.00517228281320594683022294996292250527L, |
| -0.000260897206152101522569969046299309939L, |
| -0.989553462123121764865178453128769948e-5L, |
| -0.286916799741891410827712096608826167e-6L, |
| -0.637262477796046963617949532211619729e-8L, |
| -0.106796831465628373325491288787760494e-9L, |
| -0.129343095511091870860498356205376823e-11L, |
| -0.102397936697965977221267881716672084e-13L, |
| -0.402663128248642002351627980255756363e-16L, |
| }; |
| static const T Q[14] = { |
| 1L, |
| 0.311288325355705609096155335186466508L, |
| 0.0438318468940415543546769437752132748L, |
| 0.00374396349183199548610264222242269536L, |
| 0.000218707451200585197339671707189281302L, |
| 0.927578767487930747532953583797351219e-5L, |
| 0.294145760625753561951137473484889639e-6L, |
| 0.704618586690874460082739479535985395e-8L, |
| 0.126333332872897336219649130062221257e-9L, |
| 0.16317315713773503718315435769352765e-11L, |
| 0.137846712823719515148344938160275695e-13L, |
| 0.580975420554224366450994232723910583e-16L, |
| -0.291354445847552426900293580511392459e-22L, |
| 0.73614324724785855925025452085443636e-25L, |
| }; |
| result = tools::evaluate_polynomial(P, s - 17) / tools::evaluate_polynomial(Q, s - 17); |
| result = 1 + exp(result); |
| } |
| else if(s < 74) |
| { |
| // Max error in interpolated form: 2.311e-27 |
| // Max error found at long double precision: 2.297544e-27 |
| static const T P[14] = { |
| -20.7944102007844314586649688802236072L, |
| -4.95759941987499442499908748130192187L, |
| -0.563290752832461751889194629200298688L, |
| -0.0406197001137935911912457120706122877L, |
| -0.0020846534789473022216888863613422293L, |
| -0.808095978462109173749395599401375667e-4L, |
| -0.244706022206249301640890603610060959e-5L, |
| -0.589477682919645930544382616501666572e-7L, |
| -0.113699573675553496343617442433027672e-8L, |
| -0.174767860183598149649901223128011828e-10L, |
| -0.210051620306761367764549971980026474e-12L, |
| -0.189187969537370950337212675466400599e-14L, |
| -0.116313253429564048145641663778121898e-16L, |
| -0.376708747782400769427057630528578187e-19L, |
| }; |
| static const T Q[16] = { |
| 1L, |
| 0.205076752981410805177554569784219717L, |
| 0.0202526722696670378999575738524540269L, |
| 0.001278305290005994980069466658219057L, |
| 0.576404779858501791742255670403304787e-4L, |
| 0.196477049872253010859712483984252067e-5L, |
| 0.521863830500876189501054079974475762e-7L, |
| 0.109524209196868135198775445228552059e-8L, |
| 0.181698713448644481083966260949267825e-10L, |
| 0.234793316975091282090312036524695562e-12L, |
| 0.227490441461460571047545264251399048e-14L, |
| 0.151500292036937400913870642638520668e-16L, |
| 0.543475775154780935815530649335936121e-19L, |
| 0.241647013434111434636554455083309352e-28L, |
| -0.557103423021951053707162364713587374e-31L, |
| 0.618708773442584843384712258199645166e-34L, |
| }; |
| result = tools::evaluate_polynomial(P, s - 30) / tools::evaluate_polynomial(Q, s - 30); |
| result = 1 + exp(result); |
| } |
| else if(s < 117) |
| { |
| result = 1 + pow(T(2), -s); |
| } |
| else |
| { |
| result = 1; |
| } |
| return result; |
| } |
| |
| template <class T, class Policy, class Tag> |
| T zeta_imp(T s, T sc, const Policy& pol, const Tag& tag) |
| { |
| BOOST_MATH_STD_USING |
| if(s == 1) |
| return policies::raise_pole_error<T>( |
| "boost::math::zeta<%1%>", |
| "Evaluation of zeta function at pole %1%", |
| s, pol); |
| T result; |
| if(s == 0) |
| { |
| result = -0.5; |
| } |
| else if(s < 0) |
| { |
| std::swap(s, sc); |
| if(floor(sc/2) == sc/2) |
| result = 0; |
| else |
| { |
| result = boost::math::sin_pi(0.5f * sc, pol) |
| * 2 * pow(2 * constants::pi<T>(), -s) |
| * boost::math::tgamma(s, pol) |
| * zeta_imp(s, sc, pol, tag); |
| } |
| } |
| else |
| { |
| result = zeta_imp_prec(s, sc, pol, tag); |
| } |
| return result; |
| } |
| |
| } // detail |
| |
| template <class T, class Policy> |
| inline typename tools::promote_args<T>::type zeta(T s, const Policy&) |
| { |
| typedef typename tools::promote_args<T>::type result_type; |
| typedef typename policies::evaluation<result_type, Policy>::type value_type; |
| typedef typename policies::precision<result_type, Policy>::type precision_type; |
| typedef typename policies::normalise< |
| Policy, |
| policies::promote_float<false>, |
| policies::promote_double<false>, |
| policies::discrete_quantile<>, |
| policies::assert_undefined<> >::type forwarding_policy; |
| typedef typename mpl::if_< |
| mpl::less_equal<precision_type, mpl::int_<0> >, |
| mpl::int_<0>, |
| typename mpl::if_< |
| mpl::less_equal<precision_type, mpl::int_<53> >, |
| mpl::int_<53>, // double |
| typename mpl::if_< |
| mpl::less_equal<precision_type, mpl::int_<64> >, |
| mpl::int_<64>, // 80-bit long double |
| typename mpl::if_< |
| mpl::less_equal<precision_type, mpl::int_<113> >, |
| mpl::int_<113>, // 128-bit long double |
| mpl::int_<0> // too many bits, use generic version. |
| >::type |
| >::type |
| >::type |
| >::type tag_type; |
| //typedef mpl::int_<0> tag_type; |
| |
| return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::zeta_imp( |
| static_cast<value_type>(s), |
| static_cast<value_type>(1 - static_cast<value_type>(s)), |
| forwarding_policy(), |
| tag_type()), "boost::math::zeta<%1%>(%1%)"); |
| } |
| |
| template <class T> |
| inline typename tools::promote_args<T>::type zeta(T s) |
| { |
| return zeta(s, policies::policy<>()); |
| } |
| |
| }} // namespaces |
| |
| #endif // BOOST_MATH_ZETA_HPP |
| |
| |
| |