// 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/tools/big_constant.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{
#if 0
//
// This code is commented out because we have a better more rapidly converging series
// now. Retained for future reference and in case the new code causes any issues down the line....
//
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<T>("boost::math::zeta_series2<%1%>(%1%)", max_iter, pol);
return result;
}
#endif
template <class T, class Policy>
T zeta_polynomial_series(T s, T sc, Policy const &)
{
//
// This is algorithm 3 from:
//
// "An Efficient Algorithm for the Riemann Zeta Function", P. Borwein,
// Canadian Mathematical Society, Conference Proceedings.
// See: http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf
//
BOOST_MATH_STD_USING
int n = itrunc(T(log(boost::math::tools::epsilon<T>()) / -2));
T sum = 0;
T two_n = ldexp(T(1), n);
int ej_sign = 1;
for(int j = 0; j < n; ++j)
{
sum += ej_sign * -two_n / pow(T(j + 1), s);
ej_sign = -ej_sign;
}
T ej_sum = 1;
T ej_term = 1;
for(int j = n; j <= 2 * n - 1; ++j)
{
sum += ej_sign * (ej_sum - two_n) / pow(T(j + 1), s);
ej_sign = -ej_sign;
ej_term *= 2 * n - j;
ej_term /= j - n + 1;
ej_sum += ej_term;
}
return -sum / (two_n * (-powm1(T(2), sc)));
}
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;
result = zeta_polynomial_series(s, sc, pol);
#if 0
// Old code archived for future reference:
//
// 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);
#endif
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, T(-sc)) / tools::evaluate_polynomial(Q, T(-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, T(s - 2)) / tools::evaluate_polynomial(Q, T(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, T(s - 4)) / tools::evaluate_polynomial(Q, T(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, T(s - 7)) / tools::evaluate_polynomial(Q, T(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, T(s - 15)) / tools::evaluate_polynomial(Q, T(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] = {
BOOST_MATH_BIG_CONSTANT(T, 64, 0.243392944335937499969),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.496837806864865688082),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.0680008039723709987107),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.00511620413006619942112),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.000455369899250053003335),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.279496685273033761927e-4),
};
static const T Q[7] = {
BOOST_MATH_BIG_CONSTANT(T, 64, 1),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.30425480068225790522),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.050052748580371598736),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.00519355671064700627862),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.000360623385771198350257),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.159600883054550987633e-4),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.339770279812410586032e-6),
};
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] = {
BOOST_MATH_BIG_CONSTANT(T, 64, 0.577215664901532860605),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.222537368917162139445),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.0356286324033215682729),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.00304465292366350081446),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.000178102511649069421904),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.700867470265983665042e-5),
};
static const T Q[7] = {
BOOST_MATH_BIG_CONSTANT(T, 64, 1),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.259385759149531030085),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.0373974962106091316854),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.00332735159183332820617),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.000188690420706998606469),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.635994377921861930071e-5),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.226583954978371199405e-7),
};
result = tools::evaluate_polynomial(P, T(-sc)) / tools::evaluate_polynomial(Q, T(-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] = {
BOOST_MATH_BIG_CONSTANT(T, 64, -0.053725830002359501027),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.0470551187571475844778),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.0101339410415759517471),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.00100240326666092854528),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.685027119098122814867e-4),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.390972820219765942117e-5),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.540319769113543934483e-7),
};
static const T Q[8] = {
1,
BOOST_MATH_BIG_CONSTANT(T, 64, 0.286577739726542730421),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.0447355811517733225843),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.00430125107610252363302),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.000284956969089786662045),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.116188101609848411329e-4),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.278090318191657278204e-6),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.19683620233222028478e-8),
};
result = tools::evaluate_polynomial(P, T(s - 2)) / tools::evaluate_polynomial(Q, T(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] = {
BOOST_MATH_BIG_CONSTANT(T, 64, -2.49710190602259407065),
BOOST_MATH_BIG_CONSTANT(T, 64, -3.36664913245960625334),
BOOST_MATH_BIG_CONSTANT(T, 64, -1.77180020623777595452),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.464717885249654313933),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.0643694921293579472583),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.00464265386202805715487),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.000165556579779704340166),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.252884970740994069582e-5),
};
static const T Q[9] = {
1,
BOOST_MATH_BIG_CONSTANT(T, 64, 1.01300131390690459085),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.387898115758643503827),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.0695071490045701135188),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.00586908595251442839291),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.000217752974064612188616),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.397626583349419011731e-5),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.927884739284359700764e-8),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.119810501805618894381e-9),
};
result = tools::evaluate_polynomial(P, T(s - 4)) / tools::evaluate_polynomial(Q, T(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] = {
BOOST_MATH_BIG_CONSTANT(T, 64, -4.78558028495135548083),
BOOST_MATH_BIG_CONSTANT(T, 64, -3.23873322238609358947),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.892338582881021799922),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.131326296217965913809),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.0115651591773783712996),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.000657728968362695775205),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.252051328129449973047e-4),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.626503445372641798925e-6),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.815696314790853893484e-8),
};
static const T Q[9] = {
1,
BOOST_MATH_BIG_CONSTANT(T, 64, 0.525765665400123515036),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.10852641753657122787),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.0115669945375362045249),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.000732896513858274091966),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.30683952282420248448e-4),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.819649214609633126119e-6),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.117957556472335968146e-7),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.193432300973017671137e-12),
};
result = tools::evaluate_polynomial(P, T(s - 7)) / tools::evaluate_polynomial(Q, T(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] = {
BOOST_MATH_BIG_CONSTANT(T, 64, -10.3948950573308861781),
BOOST_MATH_BIG_CONSTANT(T, 64, -2.82646012777913950108),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.342144362739570333665),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.0249285145498722647472),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.00122493108848097114118),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.423055371192592850196e-4),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.1025215577185967488e-5),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.165096762663509467061e-7),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.145392555873022044329e-9),
};
static const T Q[10] = {
1,
BOOST_MATH_BIG_CONSTANT(T, 64, 0.205135978585281988052),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.0192359357875879453602),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.00111496452029715514119),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.434928449016693986857e-4),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.116911068726610725891e-5),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.206704342290235237475e-7),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.209772836100827647474e-9),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.939798249922234703384e-16),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.264584017421245080294e-18),
};
result = tools::evaluate_polynomial(P, T(s - 15)) / tools::evaluate_polynomial(Q, T(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] = {
BOOST_MATH_BIG_CONSTANT(T, 113, -1),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.0353008629988648122808504280990313668),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0107795651204927743049369868548706909),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.000523961870530500751114866884685172975),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.661805838304910731947595897966487515e-4),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.658932670403818558510656304189164638e-5),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.103437265642266106533814021041010453e-6),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.116818787212666457105375746642927737e-7),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.660690993901506912123512551294239036e-9),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.113103113698388531428914333768142527e-10),
};
static const T Q[11] = {
BOOST_MATH_BIG_CONSTANT(T, 113, 1),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.387483472099602327112637481818565459),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0802265315091063135271497708694776875),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.0110727276164171919280036408995078164),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00112552716946286252000434849173787243),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.874554160748626916455655180296834352e-4),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.530097847491828379568636739662278322e-5),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.248461553590496154705565904497247452e-6),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.881834921354014787309644951507523899e-8),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.217062446168217797598596496310953025e-9),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.315823200002384492377987848307151168e-11),
};
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] = {
BOOST_MATH_BIG_CONSTANT(T, 113, 0.577215664901532860606512090082402431),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.255597968739771510415479842335906308),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0494056503552807274142218876983542205),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00551372778611700965268920983472292325),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00043667616723970574871427830895192731),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.268562259154821957743669387915239528e-4),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.109249633923016310141743084480436612e-5),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.273895554345300227466534378753023924e-7),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.583103205551702720149237384027795038e-9),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.835774625259919268768735944711219256e-11),
};
static const T Q[11] = {
BOOST_MATH_BIG_CONSTANT(T, 113, 1),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.316661751179735502065583176348292881),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0540401806533507064453851182728635272),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00598621274107420237785899476374043797),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.000474907812321704156213038740142079615),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.272125421722314389581695715835862418e-4),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.112649552156479800925522445229212933e-5),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.301838975502992622733000078063330461e-7),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.422960728687211282539769943184270106e-9),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.377105263588822468076813329270698909e-11),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.581926559304525152432462127383600681e-13),
};
result = tools::evaluate_polynomial(P, T(-sc)) / tools::evaluate_polynomial(Q, T(-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] = {
BOOST_MATH_BIG_CONSTANT(T, 113, -0.0537258300023595010275848333539748089),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0429086930802630159457448174466342553),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0136148228754303412510213395034056857),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00190231601036042925183751238033763915),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.000186880390916311438818302549192456581),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.145347370745893262394287982691323657e-4),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.805843276446813106414036600485884885e-6),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.340818159286739137503297172091882574e-7),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.115762357488748996526167305116837246e-8),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.231904754577648077579913403645767214e-10),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.340169592866058506675897646629036044e-12),
};
static const T Q[12] = {
BOOST_MATH_BIG_CONSTANT(T, 113, 1),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.363755247765087100018556983050520554),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0696581979014242539385695131258321598),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00882208914484611029571547753782014817),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.000815405623261946661762236085660996718),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.571366167062457197282642344940445452e-4),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.309278269271853502353954062051797838e-5),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.12822982083479010834070516053794262e-6),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.397876357325018976733953479182110033e-8),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.8484432107648683277598472295289279e-10),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.105677416606909614301995218444080615e-11),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.547223964564003701979951154093005354e-15),
};
result = tools::evaluate_polynomial(P, T(s - 2)) / tools::evaluate_polynomial(Q, T(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] = {
BOOST_MATH_BIG_CONSTANT(T, 113, 0.786383506575062179339611614117697622),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.495766593395271370974685959652073976),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.409116737851754766422360889037532228),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.57340744006238263817895456842655987),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.280479899797421910694892949057963111),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.0753148409447590257157585696212649869),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.0122934003684672788499099362823748632),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.00126148398446193639247961370266962927),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.828465038179772939844657040917364896e-4),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.361008916706050977143208468690645684e-5),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.109879825497910544424797771195928112e-6),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.214539416789686920918063075528797059e-8),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.15090220092460596872172844424267351e-10),
};
static const T Q[14] = {
BOOST_MATH_BIG_CONSTANT(T, 113, 1),
BOOST_MATH_BIG_CONSTANT(T, 113, 1.69490865837142338462982225731926485),
BOOST_MATH_BIG_CONSTANT(T, 113, 1.22697696630994080733321401255942464),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.495409420862526540074366618006341533),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.122368084916843823462872905024259633),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0191412993625268971656513890888208623),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00191401538628980617753082598351559642),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.000123318142456272424148930280876444459),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.531945488232526067889835342277595709e-5),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.161843184071894368337068779669116236e-6),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.305796079600152506743828859577462778e-8),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.233582592298450202680170811044408894e-10),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.275363878344548055574209713637734269e-13),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.221564186807357535475441900517843892e-15),
};
result = tools::evaluate_polynomial(P, T(s - 4)) / tools::evaluate_polynomial(Q, T(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] = {
BOOST_MATH_BIG_CONSTANT(T, 113, -4.0545627381873738086704293881227365),
BOOST_MATH_BIG_CONSTANT(T, 113, -4.70088348734699134347906176097717782),
BOOST_MATH_BIG_CONSTANT(T, 113, -2.36921550900925512951976617607678789),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.684322583796369508367726293719322866),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.126026534540165129870721937592996324),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.015636903921778316147260572008619549),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.00135442294754728549644376325814460807),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.842793965853572134365031384646117061e-4),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.385602133791111663372015460784978351e-5),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.130458500394692067189883214401478539e-6),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.315861074947230418778143153383660035e-8),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.500334720512030826996373077844707164e-10),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.420204769185233365849253969097184005e-12),
};
static const T Q[14] = {
BOOST_MATH_BIG_CONSTANT(T, 113, 1),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.97663511666410096104783358493318814),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.40878780231201806504987368939673249),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0963890666609396058945084107597727252),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0142207619090854604824116070866614505),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00139010220902667918476773423995750877),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.940669540194694997889636696089994734e-4),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.458220848507517004399292480807026602e-5),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.16345521617741789012782420625435495e-6),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.414007452533083304371566316901024114e-8),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.68701473543366328016953742622661377e-10),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.603461891080716585087883971886075863e-12),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.294670713571839023181857795866134957e-16),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.147003914536437243143096875069813451e-18),
};
result = tools::evaluate_polynomial(P, T(s - 6)) / tools::evaluate_polynomial(Q, T(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] = {
BOOST_MATH_BIG_CONSTANT(T, 113, -6.91319491921722925920883787894829678),
BOOST_MATH_BIG_CONSTANT(T, 113, -3.65491257639481960248690596951049048),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.813557553449954526442644544105257881),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.0994317301685870959473658713841138083),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.00726896610245676520248617014211734906),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.000317253318715075854811266230916762929),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.66851422826636750855184211580127133e-5),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.879464154730985406003332577806849971e-7),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.113838903158254250631678791998294628e-7),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.379184410304927316385211327537817583e-9),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.612992858643904887150527613446403867e-11),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.347873737198164757035457841688594788e-13),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.289187187441625868404494665572279364e-15),
};
static const T Q[14] = {
BOOST_MATH_BIG_CONSTANT(T, 113, 1),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.427310044448071818775721584949868806),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.074602514873055756201435421385243062),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00688651562174480772901425121653945942),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.000360174847635115036351323894321880445),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.973556847713307543918865405758248777e-5),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.853455848314516117964634714780874197e-8),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.118203513654855112421673192194622826e-7),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.462521662511754117095006543363328159e-9),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.834212591919475633107355719369463143e-11),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.5354594751002702935740220218582929e-13),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.406451690742991192964889603000756203e-15),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.887948682401000153828241615760146728e-19),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.34980761098820347103967203948619072e-21),
};
result = tools::evaluate_polynomial(P, T(s - 10)) / tools::evaluate_polynomial(Q, T(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] = {
BOOST_MATH_BIG_CONSTANT(T, 113, -11.7824798233959252791987402769438322),
BOOST_MATH_BIG_CONSTANT(T, 113, -4.36131215284987731928174218354118102),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.732260980060982349410898496846972204),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.0744985185694913074484248803015717388),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.00517228281320594683022294996292250527),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.000260897206152101522569969046299309939),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.989553462123121764865178453128769948e-5),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.286916799741891410827712096608826167e-6),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.637262477796046963617949532211619729e-8),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.106796831465628373325491288787760494e-9),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.129343095511091870860498356205376823e-11),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.102397936697965977221267881716672084e-13),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.402663128248642002351627980255756363e-16),
};
static const T Q[14] = {
BOOST_MATH_BIG_CONSTANT(T, 113, 1),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.311288325355705609096155335186466508),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0438318468940415543546769437752132748),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00374396349183199548610264222242269536),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.000218707451200585197339671707189281302),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.927578767487930747532953583797351219e-5),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.294145760625753561951137473484889639e-6),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.704618586690874460082739479535985395e-8),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.126333332872897336219649130062221257e-9),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.16317315713773503718315435769352765e-11),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.137846712823719515148344938160275695e-13),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.580975420554224366450994232723910583e-16),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.291354445847552426900293580511392459e-22),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.73614324724785855925025452085443636e-25),
};
result = tools::evaluate_polynomial(P, T(s - 17)) / tools::evaluate_polynomial(Q, T(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] = {
BOOST_MATH_BIG_CONSTANT(T, 113, -20.7944102007844314586649688802236072),
BOOST_MATH_BIG_CONSTANT(T, 113, -4.95759941987499442499908748130192187),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.563290752832461751889194629200298688),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.0406197001137935911912457120706122877),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.0020846534789473022216888863613422293),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.808095978462109173749395599401375667e-4),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.244706022206249301640890603610060959e-5),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.589477682919645930544382616501666572e-7),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.113699573675553496343617442433027672e-8),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.174767860183598149649901223128011828e-10),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.210051620306761367764549971980026474e-12),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.189187969537370950337212675466400599e-14),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.116313253429564048145641663778121898e-16),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.376708747782400769427057630528578187e-19),
};
static const T Q[16] = {
BOOST_MATH_BIG_CONSTANT(T, 113, 1),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.205076752981410805177554569784219717),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0202526722696670378999575738524540269),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.001278305290005994980069466658219057),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.576404779858501791742255670403304787e-4),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.196477049872253010859712483984252067e-5),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.521863830500876189501054079974475762e-7),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.109524209196868135198775445228552059e-8),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.181698713448644481083966260949267825e-10),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.234793316975091282090312036524695562e-12),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.227490441461460571047545264251399048e-14),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.151500292036937400913870642638520668e-16),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.543475775154780935815530649335936121e-19),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.241647013434111434636554455083309352e-28),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.557103423021951053707162364713587374e-31),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.618708773442584843384712258199645166e-34),
};
result = tools::evaluate_polynomial(P, T(s - 30)) / tools::evaluate_polynomial(Q, T(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