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Diffstat (limited to 'boost/math/constants/calculate_constants.hpp')
| -rw-r--r-- | boost/math/constants/calculate_constants.hpp | 959 |
1 files changed, 959 insertions, 0 deletions
diff --git a/boost/math/constants/calculate_constants.hpp b/boost/math/constants/calculate_constants.hpp new file mode 100644 index 0000000000..0b78929e71 --- /dev/null +++ b/boost/math/constants/calculate_constants.hpp @@ -0,0 +1,959 @@ +// Copyright John Maddock 2010, 2012. +// Copyright Paul A. Bristow 2011, 2012. + +// 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_CALCULATE_CONSTANTS_CONSTANTS_INCLUDED +#define BOOST_MATH_CALCULATE_CONSTANTS_CONSTANTS_INCLUDED + +#include <boost/math/special_functions/trunc.hpp> + +namespace boost{ namespace math{ namespace constants{ namespace detail{ + +template <class T> +template<int N> +inline T constant_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + + return ldexp(acos(T(0)), 1); + + /* + // Although this code works well, it's usually more accurate to just call acos + // and access the number types own representation of PI which is usually calculated + // at slightly higher precision... + + T result; + T a = 1; + T b; + T A(a); + T B = 0.5f; + T D = 0.25f; + + T lim; + lim = boost::math::tools::epsilon<T>(); + + unsigned k = 1; + + do + { + result = A + B; + result = ldexp(result, -2); + b = sqrt(B); + a += b; + a = ldexp(a, -1); + A = a * a; + B = A - result; + B = ldexp(B, 1); + result = A - B; + bool neg = boost::math::sign(result) < 0; + if(neg) + result = -result; + if(result <= lim) + break; + if(neg) + result = -result; + result = ldexp(result, k - 1); + D -= result; + ++k; + lim = ldexp(lim, 1); + } + while(true); + + result = B / D; + return result; + */ +} + +template <class T> +template<int N> +inline T constant_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + return 2 * pi<T, policies::policy<policies::digits2<N> > >(); +} + +template <class T> // 2 / pi +template<int N> +inline T constant_two_div_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + return 2 / pi<T, policies::policy<policies::digits2<N> > >(); +} + +template <class T> // sqrt(2/pi) +template <int N> +inline T constant_root_two_div_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + return sqrt((2 / pi<T, policies::policy<policies::digits2<N> > >())); +} + +template <class T> +template<int N> +inline T constant_one_div_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + return 1 / two_pi<T, policies::policy<policies::digits2<N> > >(); +} + +template <class T> +template<int N> +inline T constant_root_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return sqrt(pi<T, policies::policy<policies::digits2<N> > >()); +} + +template <class T> +template<int N> +inline T constant_root_half_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return sqrt(pi<T, policies::policy<policies::digits2<N> > >() / 2); +} + +template <class T> +template<int N> +inline T constant_root_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return sqrt(two_pi<T, policies::policy<policies::digits2<N> > >()); +} + +template <class T> +template<int N> +inline T constant_root_ln_four<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return sqrt(log(static_cast<T>(4))); +} + +template <class T> +template<int N> +inline T constant_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + // + // Although we can clearly calculate this from first principles, this hooks into + // T's own notion of e, which hopefully will more accurate than one calculated to + // a few epsilon: + // + BOOST_MATH_STD_USING + return exp(static_cast<T>(1)); +} + +template <class T> +template<int N> +inline T constant_half<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + return static_cast<T>(1) / static_cast<T>(2); +} + +template <class T> +template<int M> +inline T constant_euler<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<M>)) +{ + BOOST_MATH_STD_USING + // + // This is the method described in: + // "Some New Algorithms for High-Precision Computation of Euler's Constant" + // Richard P Brent and Edwin M McMillan. + // Mathematics of Comnputation, Volume 34, Number 149, Jan 1980, pages 305-312. + // See equation 17 with p = 2. + // + T n = 3 + (M ? (std::min)(M, tools::digits<T>()) : tools::digits<T>()) / 4; + T lim = M ? ldexp(T(1), (std::min)(M, tools::digits<T>())) : tools::epsilon<T>(); + T lnn = log(n); + T term = 1; + T N = -lnn; + T D = 1; + T Hk = 0; + T one = 1; + + for(unsigned k = 1;; ++k) + { + term *= n * n; + term /= k * k; + Hk += one / k; + N += term * (Hk - lnn); + D += term; + + if(term < D * lim) + break; + } + return N / D; +} + +template <class T> +template<int N> +inline T constant_euler_sqr<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return euler<T, policies::policy<policies::digits2<N> > >() + * euler<T, policies::policy<policies::digits2<N> > >(); +} + +template <class T> +template<int N> +inline T constant_one_div_euler<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return static_cast<T>(1) + / euler<T, policies::policy<policies::digits2<N> > >(); +} + + +template <class T> +template<int N> +inline T constant_root_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return sqrt(static_cast<T>(2)); +} + + +template <class T> +template<int N> +inline T constant_root_three<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return sqrt(static_cast<T>(3)); +} + +template <class T> +template<int N> +inline T constant_half_root_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return sqrt(static_cast<T>(2)) / 2; +} + +template <class T> +template<int N> +inline T constant_ln_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + // + // Although there are good ways to calculate this from scratch, this hooks into + // T's own notion of log(2) which will hopefully be accurate to the full precision + // of T: + // + BOOST_MATH_STD_USING + return log(static_cast<T>(2)); +} + +template <class T> +template<int N> +inline T constant_ln_ten<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return log(static_cast<T>(10)); +} + +template <class T> +template<int N> +inline T constant_ln_ln_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return log(log(static_cast<T>(2))); +} + +template <class T> +template<int N> +inline T constant_third<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return static_cast<T>(1) / static_cast<T>(3); +} + +template <class T> +template<int N> +inline T constant_twothirds<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return static_cast<T>(2) / static_cast<T>(3); +} + +template <class T> +template<int N> +inline T constant_two_thirds<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return static_cast<T>(2) / static_cast<T>(3); +} + +template <class T> +template<int N> +inline T constant_three_quarters<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return static_cast<T>(3) / static_cast<T>(4); +} + +template <class T> +template<int N> +inline T constant_pi_minus_three<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + return pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(3); +} + +template <class T> +template<int N> +inline T constant_four_minus_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + return static_cast<T>(4) - pi<T, policies::policy<policies::digits2<N> > >(); +} + +template <class T> +template<int N> +inline T constant_pow23_four_minus_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return pow(four_minus_pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(1.5)); +} + +template <class T> +template<int N> +inline T constant_exp_minus_half<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return exp(static_cast<T>(-0.5)); +} + +// Pi +template <class T> +template<int N> +inline T constant_one_div_root_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + return static_cast<T>(1) / root_two<T, policies::policy<policies::digits2<N> > >(); +} + +template <class T> +template<int N> +inline T constant_one_div_root_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + return static_cast<T>(1) / root_pi<T, policies::policy<policies::digits2<N> > >(); +} + +template <class T> +template<int N> +inline T constant_one_div_root_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + return static_cast<T>(1) / root_two_pi<T, policies::policy<policies::digits2<N> > >(); +} + +template <class T> +template<int N> +inline T constant_root_one_div_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return sqrt(static_cast<T>(1) / pi<T, policies::policy<policies::digits2<N> > >()); +} + + +template <class T> +template<int N> +inline T constant_four_thirds_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return pi<T, policies::policy<policies::digits2<N> > >() * static_cast<T>(4) / static_cast<T>(3); +} + +template <class T> +template<int N> +inline T constant_half_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return pi<T, policies::policy<policies::digits2<N> > >() / static_cast<T>(2); +} + + +template <class T> +template<int N> +inline T constant_third_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return pi<T, policies::policy<policies::digits2<N> > >() / static_cast<T>(3); +} + +template <class T> +template<int N> +inline T constant_sixth_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return pi<T, policies::policy<policies::digits2<N> > >() / static_cast<T>(6); +} + +template <class T> +template<int N> +inline T constant_two_thirds_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return pi<T, policies::policy<policies::digits2<N> > >() * static_cast<T>(2) / static_cast<T>(3); +} + +template <class T> +template<int N> +inline T constant_three_quarters_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return pi<T, policies::policy<policies::digits2<N> > >() * static_cast<T>(3) / static_cast<T>(4); +} + +template <class T> +template<int N> +inline T constant_pi_pow_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return pow(pi<T, policies::policy<policies::digits2<N> > >(), e<T, policies::policy<policies::digits2<N> > >()); // +} + +template <class T> +template<int N> +inline T constant_pi_sqr<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return pi<T, policies::policy<policies::digits2<N> > >() + * pi<T, policies::policy<policies::digits2<N> > >() ; // +} + +template <class T> +template<int N> +inline T constant_pi_sqr_div_six<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return pi<T, policies::policy<policies::digits2<N> > >() + * pi<T, policies::policy<policies::digits2<N> > >() + / static_cast<T>(6); // +} + + +template <class T> +template<int N> +inline T constant_pi_cubed<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return pi<T, policies::policy<policies::digits2<N> > >() + * pi<T, policies::policy<policies::digits2<N> > >() + * pi<T, policies::policy<policies::digits2<N> > >() + ; // +} + +template <class T> +template<int N> +inline T constant_cbrt_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return pow(pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(1)/ static_cast<T>(3)); +} + +template <class T> +template<int N> +inline T constant_one_div_cbrt_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return static_cast<T>(1) + / pow(pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(1)/ static_cast<T>(3)); +} + +// Euler's e + +template <class T> +template<int N> +inline T constant_e_pow_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return pow(e<T, policies::policy<policies::digits2<N> > >(), pi<T, policies::policy<policies::digits2<N> > >()); // +} + +template <class T> +template<int N> +inline T constant_root_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return sqrt(e<T, policies::policy<policies::digits2<N> > >()); +} + +template <class T> +template<int N> +inline T constant_log10_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return log10(e<T, policies::policy<policies::digits2<N> > >()); +} + +template <class T> +template<int N> +inline T constant_one_div_log10_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return static_cast<T>(1) / + log10(e<T, policies::policy<policies::digits2<N> > >()); +} + +// Trigonometric + +template <class T> +template<int N> +inline T constant_degree<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return pi<T, policies::policy<policies::digits2<N> > >() + / static_cast<T>(180) + ; // +} + +template <class T> +template<int N> +inline T constant_radian<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return static_cast<T>(180) + / pi<T, policies::policy<policies::digits2<N> > >() + ; // +} + +template <class T> +template<int N> +inline T constant_sin_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return sin(static_cast<T>(1)) ; // +} + +template <class T> +template<int N> +inline T constant_cos_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return cos(static_cast<T>(1)) ; // +} + +template <class T> +template<int N> +inline T constant_sinh_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return sinh(static_cast<T>(1)) ; // +} + +template <class T> +template<int N> +inline T constant_cosh_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return cosh(static_cast<T>(1)) ; // +} + +template <class T> +template<int N> +inline T constant_phi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return (static_cast<T>(1) + sqrt(static_cast<T>(5)) )/static_cast<T>(2) ; // +} + +template <class T> +template<int N> +inline T constant_ln_phi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + //return log(phi<T, policies::policy<policies::digits2<N> > >()); // ??? + return log((static_cast<T>(1) + sqrt(static_cast<T>(5)) )/static_cast<T>(2) ); +} +template <class T> +template<int N> +inline T constant_one_div_ln_phi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + return static_cast<T>(1) / + log((static_cast<T>(1) + sqrt(static_cast<T>(5)) )/static_cast<T>(2) ); +} + +// Zeta + +template <class T> +template<int N> +inline T constant_zeta_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + BOOST_MATH_STD_USING + + return pi<T, policies::policy<policies::digits2<N> > >() + * pi<T, policies::policy<policies::digits2<N> > >() + /static_cast<T>(6); +} + +template <class T> +template<int N> +inline T constant_zeta_three<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + // http://mathworld.wolfram.com/AperysConstant.html + // http://en.wikipedia.org/wiki/Mathematical_constant + + // http://oeis.org/A002117/constant + //T zeta3("1.20205690315959428539973816151144999076" + // "4986292340498881792271555341838205786313" + // "09018645587360933525814619915"); + + //"1.202056903159594285399738161511449990, 76498629234049888179227155534183820578631309018645587360933525814619915" A002117 + // 1.202056903159594285399738161511449990, 76498629234049888179227155534183820578631309018645587360933525814619915780, +00); + //"1.2020569031595942 double + // http://www.spaennare.se/SSPROG/ssnum.pdf // section 11, Algorithmfor Apery’s constant zeta(3). + // Programs to Calculate some Mathematical Constants to Large Precision, Document Version 1.50 + + // by Stefan Spannare September 19, 2007 + // zeta(3) = 1/64 * sum + BOOST_MATH_STD_USING + T n_fact=static_cast<T>(1); // build n! for n = 0. + T sum = static_cast<double>(77); // Start with n = 0 case. + // for n = 0, (77/1) /64 = 1.203125 + //double lim = std::numeric_limits<double>::epsilon(); + T lim = N ? ldexp(T(1), 1 - (std::min)(N, tools::digits<T>())) : tools::epsilon<T>(); + for(unsigned int n = 1; n < 40; ++n) + { // three to five decimal digits per term, so 40 should be plenty for 100 decimal digits. + //cout << "n = " << n << endl; + n_fact *= n; // n! + T n_fact_p10 = n_fact * n_fact * n_fact * n_fact * n_fact * n_fact * n_fact * n_fact * n_fact * n_fact; // (n!)^10 + T num = ((205 * n * n) + (250 * n) + 77) * n_fact_p10; // 205n^2 + 250n + 77 + // int nn = (2 * n + 1); + // T d = factorial(nn); // inline factorial. + T d = 1; + for(unsigned int i = 1; i <= (n+n + 1); ++i) // (2n + 1) + { + d *= i; + } + T den = d * d * d * d * d; // [(2n+1)!]^5 + //cout << "den = " << den << endl; + T term = num/den; + if (n % 2 != 0) + { //term *= -1; + sum -= term; + } + else + { + sum += term; + } + //cout << "term = " << term << endl; + //cout << "sum/64 = " << sum/64 << endl; + if(abs(term) < lim) + { + break; + } + } + return sum / 64; +} + +template <class T> +template<int N> +inline T constant_catalan<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ // http://oeis.org/A006752/constant + //T c("0.915965594177219015054603514932384110774" + //"149374281672134266498119621763019776254769479356512926115106248574"); + + // 9.159655941772190150546035149323841107, 74149374281672134266498119621763019776254769479356512926115106248574422619, -01); + + // This is equation (entry) 31 from + // http://www-2.cs.cmu.edu/~adamchik/articles/catalan/catalan.htm + // See also http://www.mpfr.org/algorithms.pdf + BOOST_MATH_STD_USING + T k_fact = 1; + T tk_fact = 1; + T sum = 1; + T term; + T lim = N ? ldexp(T(1), 1 - (std::min)(N, tools::digits<T>())) : tools::epsilon<T>(); + + for(unsigned k = 1;; ++k) + { + k_fact *= k; + tk_fact *= (2 * k) * (2 * k - 1); + term = k_fact * k_fact / (tk_fact * (2 * k + 1) * (2 * k + 1)); + sum += term; + if(term < lim) + { + break; + } + } + return boost::math::constants::pi<T, boost::math::policies::policy<> >() + * log(2 + boost::math::constants::root_three<T, boost::math::policies::policy<> >()) + / 8 + + 3 * sum / 8; +} + +namespace khinchin_detail{ + +template <class T> +T zeta_polynomial_series(T s, T sc, int digits) +{ + BOOST_MATH_STD_USING + // + // This is algorithm 3 from: + // + // "An Efficient Algorithm for the Riemann Zeta Function", P. Borwein, + // Canadian Mathematical Society, Conference Proceedings, 2000. + // See: http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf + // + BOOST_MATH_STD_USING + int n = (digits * 19) / 53; + 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 * (1 - pow(T(2), sc))); +} + +template <class T> +T khinchin(int digits) +{ + BOOST_MATH_STD_USING + T sum = 0; + T term; + T lim = ldexp(T(1), 1-digits); + T factor = 0; + unsigned last_k = 1; + T num = 1; + for(unsigned n = 1;; ++n) + { + for(unsigned k = last_k; k <= 2 * n - 1; ++k) + { + factor += num / k; + num = -num; + } + last_k = 2 * n; + term = (zeta_polynomial_series(T(2 * n), T(1 - T(2 * n)), digits) - 1) * factor / n; + sum += term; + if(term < lim) + break; + } + return exp(sum / boost::math::constants::ln_two<T, boost::math::policies::policy<> >()); +} + +} + +template <class T> +template<int N> +inline T constant_khinchin<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + int n = N ? (std::min)(N, tools::digits<T>()) : tools::digits<T>(); + return khinchin_detail::khinchin<T>(n); +} + +template <class T> +template<int N> +inline T constant_extreme_value_skewness<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ // from e_float constants.cpp + // Mathematica: N[12 Sqrt[6] Zeta[3]/Pi^3, 1101] + BOOST_MATH_STD_USING + T ev(12 * sqrt(static_cast<T>(6)) * zeta_three<T, policies::policy<policies::digits2<N> > >() + / pi_cubed<T, policies::policy<policies::digits2<N> > >() ); + +//T ev( +//"1.1395470994046486574927930193898461120875997958365518247216557100852480077060706857071875468869385150" +//"1894272048688553376986765366075828644841024041679714157616857834895702411080704529137366329462558680" +//"2015498788776135705587959418756809080074611906006528647805347822929577145038743873949415294942796280" +//"0895597703063466053535550338267721294164578901640163603544404938283861127819804918174973533694090594" +//"3094963822672055237678432023017824416203652657301470473548274848068762500300316769691474974950757965" +//"8640779777748741897542093874605477776538884083378029488863880220988107155275203245233994097178778984" +//"3488995668362387892097897322246698071290011857605809901090220903955815127463328974447572119951192970" +//"3684453635456559086126406960279692862247058250100678008419431185138019869693206366891639436908462809" +//"9756051372711251054914491837034685476095423926553367264355374652153595857163724698198860485357368964" +//"3807049634423621246870868566707915720704996296083373077647528285782964567312903914752617978405994377" +//"9064157147206717895272199736902453130842229559980076472936976287378945035706933650987259357729800315"); + + return ev; +} + +namespace detail{ +// +// Calculation of the Glaisher constant depends upon calculating the +// derivative of the zeta function at 2, we can then use the relation: +// zeta'(2) = 1/6 pi^2 [euler + ln(2pi)-12ln(A)] +// To get the constant A. +// See equation 45 at http://mathworld.wolfram.com/RiemannZetaFunction.html. +// +// The derivative of the zeta function is computed by direct differentiation +// of the relation: +// (1-2^(1-s))zeta(s) = SUM(n=0, INF){ (-n)^n / (n+1)^s } +// Which gives us 2 slowly converging but alternating sums to compute, +// for this we use Algorithm 1 from "Convergent Acceleration of Alternating Series", +// Henri Cohen, Fernando Rodriguez Villegas and Don Zagier, Experimental Mathematics 9:1 (1999). +// See http://www.math.utexas.edu/users/villegas/publications/conv-accel.pdf +// +template <class T> +T zeta_series_derivative_2(unsigned digits) +{ + // Derivative of the series part, evaluated at 2: + BOOST_MATH_STD_USING + int n = digits * 301 * 13 / 10000; + boost::math::itrunc((std::numeric_limits<T>::digits10 + 1) * 1.3); + T d = pow(3 + sqrt(T(8)), n); + d = (d + 1 / d) / 2; + T b = -1; + T c = -d; + T s = 0; + for(int k = 0; k < n; ++k) + { + T a = -log(T(k+1)) / ((k+1) * (k+1)); + c = b - c; + s = s + c * a; + b = (k + n) * (k - n) * b / ((k + T(0.5f)) * (k + 1)); + } + return s / d; +} + +template <class T> +T zeta_series_2(unsigned digits) +{ + // Series part of zeta at 2: + BOOST_MATH_STD_USING + int n = digits * 301 * 13 / 10000; + T d = pow(3 + sqrt(T(8)), n); + d = (d + 1 / d) / 2; + T b = -1; + T c = -d; + T s = 0; + for(int k = 0; k < n; ++k) + { + T a = T(1) / ((k + 1) * (k + 1)); + c = b - c; + s = s + c * a; + b = (k + n) * (k - n) * b / ((k + T(0.5f)) * (k + 1)); + } + return s / d; +} + +template <class T> +inline T zeta_series_lead_2() +{ + // lead part at 2: + return 2; +} + +template <class T> +inline T zeta_series_derivative_lead_2() +{ + // derivative of lead part at 2: + return -2 * boost::math::constants::ln_two<T>(); +} + +template <class T> +inline T zeta_derivative_2(unsigned n) +{ + // zeta derivative at 2: + return zeta_series_derivative_2<T>(n) * zeta_series_lead_2<T>() + + zeta_series_derivative_lead_2<T>() * zeta_series_2<T>(n); +} + +} // namespace detail + +template <class T> +template<int N> +inline T constant_glaisher<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ + + BOOST_MATH_STD_USING + typedef policies::policy<policies::digits2<N> > forwarding_policy; + int n = N ? (std::min)(N, tools::digits<T>()) : tools::digits<T>(); + T v = detail::zeta_derivative_2<T>(n); + v *= 6; + v /= boost::math::constants::pi<T, forwarding_policy>() * boost::math::constants::pi<T, forwarding_policy>(); + v -= boost::math::constants::euler<T, forwarding_policy>(); + v -= log(2 * boost::math::constants::pi<T, forwarding_policy>()); + v /= -12; + return exp(v); + + /* + // from http://mpmath.googlecode.com/svn/data/glaisher.txt + // 20,000 digits of the Glaisher-Kinkelin constant A = exp(1/2 - zeta'(-1)) + // Computed using A = exp((6 (-zeta'(2))/pi^2 + log 2 pi + gamma)/12) + // with Euler-Maclaurin summation for zeta'(2). + T g( + "1.282427129100622636875342568869791727767688927325001192063740021740406308858826" + "46112973649195820237439420646120399000748933157791362775280404159072573861727522" + "14334327143439787335067915257366856907876561146686449997784962754518174312394652" + "76128213808180219264516851546143919901083573730703504903888123418813674978133050" + "93770833682222494115874837348064399978830070125567001286994157705432053927585405" + "81731588155481762970384743250467775147374600031616023046613296342991558095879293" + "36343887288701988953460725233184702489001091776941712153569193674967261270398013" + "52652668868978218897401729375840750167472114895288815996668743164513890306962645" + "59870469543740253099606800842447417554061490189444139386196089129682173528798629" + "88434220366989900606980888785849587494085307347117090132667567503310523405221054" + "14176776156308191919997185237047761312315374135304725819814797451761027540834943" + "14384965234139453373065832325673954957601692256427736926358821692159870775858274" + "69575162841550648585890834128227556209547002918593263079373376942077522290940187"); + + return g; + */ +} + +template <class T> +template<int N> +inline T constant_rayleigh_skewness<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ // From e_float + // 1100 digits of the Rayleigh distribution skewness + // Mathematica: N[2 Sqrt[Pi] (Pi - 3)/((4 - Pi)^(3/2)), 1100] + + BOOST_MATH_STD_USING + T rs(2 * root_pi<T, policies::policy<policies::digits2<N> > >() + * pi_minus_three<T, policies::policy<policies::digits2<N> > >() + / pow(four_minus_pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(3./2)) + ); + // 6.31110657818937138191899351544227779844042203134719497658094585692926819617473725459905027032537306794400047264, + + //"0.6311106578189371381918993515442277798440422031347194976580945856929268196174737254599050270325373067" + //"9440004726436754739597525250317640394102954301685809920213808351450851396781817932734836994829371322" + //"5797376021347531983451654130317032832308462278373358624120822253764532674177325950686466133508511968" + //"2389168716630349407238090652663422922072397393006683401992961569208109477307776249225072042971818671" + //"4058887072693437217879039875871765635655476241624825389439481561152126886932506682176611183750503553" + //"1218982627032068396407180216351425758181396562859085306247387212297187006230007438534686340210168288" + //"8956816965453815849613622117088096547521391672977226658826566757207615552041767516828171274858145957" + //"6137539156656005855905288420585194082284972984285863898582313048515484073396332610565441264220790791" + //"0194897267890422924599776483890102027823328602965235306539844007677157873140562950510028206251529523" + //"7428049693650605954398446899724157486062545281504433364675815915402937209673727753199567661561209251" + //"4695589950526053470201635372590001578503476490223746511106018091907936826431407434894024396366284848"); ; + return rs; +} + +template <class T> +template<int N> +inline T constant_rayleigh_kurtosis_excess<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ // - (6 Pi^2 - 24 Pi + 16)/((Pi - 4)^2) + // Might provide provide and calculate this using pi_minus_four. + BOOST_MATH_STD_USING + return - (((static_cast<T>(6) * pi<T, policies::policy<policies::digits2<N> > >() + * pi<T, policies::policy<policies::digits2<N> > >()) + - (static_cast<T>(24) * pi<T, policies::policy<policies::digits2<N> > >()) + static_cast<T>(16) ) + / + ((pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(4)) + * (pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(4))) + ); +} + +template <class T> +template<int N> +inline T constant_rayleigh_kurtosis<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>)) +{ // 3 - (6 Pi^2 - 24 Pi + 16)/((Pi - 4)^2) + // Might provide provide and calculate this using pi_minus_four. + BOOST_MATH_STD_USING + return static_cast<T>(3) - (((static_cast<T>(6) * pi<T, policies::policy<policies::digits2<N> > >() + * pi<T, policies::policy<policies::digits2<N> > >()) + - (static_cast<T>(24) * pi<T, policies::policy<policies::digits2<N> > >()) + static_cast<T>(16) ) + / + ((pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(4)) + * (pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(4))) + ); +} + +}}}} // namespaces + +#endif // BOOST_MATH_CALCULATE_CONSTANTS_CONSTANTS_INCLUDED |