boost/math/special_functions/detail/polygamma.hpp
/////////////////////////////////////////////////////////////////////////////// // Copyright 2013 Nikhar Agrawal // Copyright 2013 Christopher Kormanyos // Copyright 2014 John Maddock // Copyright 2013 Paul Bristow // Distributed under 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_POLYGAMMA_DETAIL_2013_07_30_HPP_ #define _BOOST_POLYGAMMA_DETAIL_2013_07_30_HPP_ #include <cmath> #include <limits> #include <string> #include <boost/math/policies/policy.hpp> #include <boost/math/special_functions/bernoulli.hpp> #include <boost/math/special_functions/trunc.hpp> #include <boost/math/special_functions/zeta.hpp> #include <boost/math/special_functions/digamma.hpp> #include <boost/math/special_functions/sin_pi.hpp> #include <boost/math/special_functions/cos_pi.hpp> #include <boost/math/special_functions/pow.hpp> #include <boost/math/tools/assert.hpp> #include <boost/math/tools/config.hpp> #ifdef BOOST_HAS_THREADS #include <mutex> #endif #ifdef _MSC_VER #pragma once #pragma warning(push) #pragma warning(disable:4702) // Unreachable code (release mode only warning) #endif namespace boost { namespace math { namespace detail{ template<class T, class Policy> T polygamma_atinfinityplus(const int n, const T& x, const Policy& pol, const char* function) // for large values of x such as for x> 400 { // See http://functions.wolfram.com/GammaBetaErf/PolyGamma2/06/02/0001/ BOOST_MATH_STD_USING // // sum == current value of accumulated sum. // term == value of current term to be added to sum. // part_term == value of current term excluding the Bernoulli number part // if(n + x == x) { // x is crazy large, just concentrate on the first part of the expression and use logs: if(n == 1) return 1 / x; T nlx = n * log(x); if((nlx < tools::log_max_value<T>()) && (n < (int)max_factorial<T>::value)) return ((n & 1) ? 1 : -1) * boost::math::factorial<T>(n - 1, pol) * pow(x, -n); else return ((n & 1) ? 1 : -1) * exp(boost::math::lgamma(T(n), pol) - n * log(x)); } T term, sum, part_term; T x_squared = x * x; // // Start by setting part_term to: // // (n-1)! / x^(n+1) // // which is common to both the first term of the series (with k = 1) // and to the leading part. // We can then get to the leading term by: // // part_term * (n + 2 * x) / 2 // // and to the first term in the series // (excluding the Bernoulli number) by: // // part_term n * (n + 1) / (2x) // // If either the factorial would overflow, // or the power term underflows, this just gets set to 0 and then we // know that we have to use logs for the initial terms: // part_term = ((n > (int)boost::math::max_factorial<T>::value) && (T(n) * n > tools::log_max_value<T>())) ? T(0) : static_cast<T>(boost::math::factorial<T>(n - 1, pol) * pow(x, -n - 1)); if(part_term == 0) { // Either n is very large, or the power term underflows, // set the initial values of part_term, term and sum via logs: part_term = static_cast<T>(boost::math::lgamma(n, pol) - (n + 1) * log(x)); sum = exp(part_term + log(n + 2 * x) - boost::math::constants::ln_two<T>()); part_term += log(T(n) * (n + 1)) - boost::math::constants::ln_two<T>() - log(x); part_term = exp(part_term); } else { sum = part_term * (n + 2 * x) / 2; part_term *= (T(n) * (n + 1)) / 2; part_term /= x; } // // If the leading term is 0, so is the result: // if(sum == 0) return sum; for(unsigned k = 1;;) { term = part_term * boost::math::bernoulli_b2n<T>(k, pol); sum += term; // // Normal termination condition: // if(fabs(term / sum) < tools::epsilon<T>()) break; // // Increment our counter, and move part_term on to the next value: // ++k; part_term *= T(n + 2 * k - 2) * (n - 1 + 2 * k); part_term /= (2 * k - 1) * 2 * k; part_term /= x_squared; // // Emergency get out termination condition: // if(k > policies::get_max_series_iterations<Policy>()) { return policies::raise_evaluation_error(function, "Series did not converge, closest value was %1%", sum, pol); } } if((n - 1) & 1) sum = -sum; return sum; } template<class T, class Policy> T polygamma_attransitionplus(const int n, const T& x, const Policy& pol, const char* function) { // See: http://functions.wolfram.com/GammaBetaErf/PolyGamma2/16/01/01/0017/ // Use N = (0.4 * digits) + (4 * n) for target value for x: BOOST_MATH_STD_USING const int d4d = static_cast<int>(0.4F * policies::digits_base10<T, Policy>()); const int N = d4d + (4 * n); const int m = n; const int iter = N - itrunc(x); if(iter > (int)policies::get_max_series_iterations<Policy>()) return policies::raise_evaluation_error<T>(function, ("Exceeded maximum series evaluations evaluating at n = " + std::to_string(n) + " and x = %1%").c_str(), x, pol); const int minus_m_minus_one = -m - 1; T z(x); T sum0(0); T z_plus_k_pow_minus_m_minus_one(0); // Forward recursion to larger x, need to check for overflow first though: if(log(z + iter) * minus_m_minus_one > -tools::log_max_value<T>()) { for(int k = 1; k <= iter; ++k) { z_plus_k_pow_minus_m_minus_one = pow(z, minus_m_minus_one); sum0 += z_plus_k_pow_minus_m_minus_one; z += 1; } sum0 *= boost::math::factorial<T>(n, pol); } else { for(int k = 1; k <= iter; ++k) { T log_term = log(z) * minus_m_minus_one + boost::math::lgamma(T(n + 1), pol); sum0 += exp(log_term); z += 1; } } if((n - 1) & 1) sum0 = -sum0; return sum0 + polygamma_atinfinityplus(n, z, pol, function); } template <class T, class Policy> T polygamma_nearzero(int n, T x, const Policy& pol, const char* function) { BOOST_MATH_STD_USING // // If we take this expansion for polygamma: http://functions.wolfram.com/06.15.06.0003.02 // and substitute in this expression for polygamma(n, 1): http://functions.wolfram.com/06.15.03.0009.01 // we get an alternating series for polygamma when x is small in terms of zeta functions of // integer arguments (which are easy to evaluate, at least when the integer is even). // // In order to avoid spurious overflow, save the n! term for later, and rescale at the end: // T scale = boost::math::factorial<T>(n, pol); // // "factorial_part" contains everything except the zeta function // evaluations in each term: // T factorial_part = 1; // // "prefix" is what we'll be adding the accumulated sum to, it will // be n! / z^(n+1), but since we're scaling by n! it's just // 1 / z^(n+1) for now: // T prefix = pow(x, n + 1); if(prefix == 0) return boost::math::policies::raise_overflow_error<T>(function, nullptr, pol); prefix = 1 / prefix; // // First term in the series is necessarily < zeta(2) < 2, so // ignore the sum if it will have no effect on the result anyway: // if(prefix > 2 / policies::get_epsilon<T, Policy>()) return ((n & 1) ? 1 : -1) * (tools::max_value<T>() / prefix < scale ? policies::raise_overflow_error<T>(function, nullptr, pol) : prefix * scale); // // As this is an alternating series we could accelerate it using // "Convergence Acceleration of Alternating Series", // Henri Cohen, Fernando Rodriguez Villegas, and Don Zagier, Experimental Mathematics, 1999. // In practice however, it appears not to make any difference to the number of terms // required except in some edge cases which are filtered out anyway before we get here. // T sum = prefix; for(unsigned k = 0;;) { // Get the k'th term: T term = factorial_part * boost::math::zeta(T(k + n + 1), pol); sum += term; // Termination condition: if(fabs(term) < fabs(sum * boost::math::policies::get_epsilon<T, Policy>())) break; // // Move on k and factorial_part: // ++k; factorial_part *= (-x * (n + k)) / k; // // Last chance exit: // if(k > policies::get_max_series_iterations<Policy>()) return policies::raise_evaluation_error<T>(function, "Series did not converge, best value is %1%", sum, pol); } // // We need to multiply by the scale, at each stage checking for overflow: // if(boost::math::tools::max_value<T>() / scale < sum) return boost::math::policies::raise_overflow_error<T>(function, nullptr, pol); sum *= scale; return n & 1 ? sum : T(-sum); } // // Helper function which figures out which slot our coefficient is in // given an angle multiplier for the cosine term of power: // template <class Table> typename Table::value_type::reference dereference_table(Table& table, unsigned row, unsigned power) { return table[row][power / 2]; } template <class T, class Policy> T poly_cot_pi(int n, T x, T xc, const Policy& pol, const char* function) { BOOST_MATH_STD_USING // Return n'th derivative of cot(pi*x) at x, these are simply // tabulated for up to n = 9, beyond that it is possible to // calculate coefficients as follows: // // The general form of each derivative is: // // pi^n * SUM{k=0, n} C[k,n] * cos^k(pi * x) * csc^(n+1)(pi * x) // // With constant C[0,1] = -1 and all other C[k,n] = 0; // Then for each k < n+1: // C[k-1, n+1] -= k * C[k, n]; // C[k+1, n+1] += (k-n-1) * C[k, n]; // // Note that there are many different ways of representing this derivative thanks to // the many trigonometric identies available. In particular, the sum of powers of // cosines could be replaced by a sum of cosine multiple angles, and indeed if you // plug the derivative into Mathematica this is the form it will give. The two // forms are related via the Chebeshev polynomials of the first kind and // T_n(cos(x)) = cos(n x). The polynomial form has the great advantage that // all the cosine terms are zero at half integer arguments - right where this // function has it's minimum - thus avoiding cancellation error in this region. // // And finally, since every other term in the polynomials is zero, we can save // space by only storing the non-zero terms. This greatly complexifies // subscripting the tables in the calculation, but halves the storage space // (and complexity for that matter). // T s = fabs(x) < fabs(xc) ? boost::math::sin_pi(x, pol) : boost::math::sin_pi(xc, pol); T c = boost::math::cos_pi(x, pol); switch(n) { case 1: return -constants::pi<T, Policy>() / (s * s); case 2: { return 2 * constants::pi<T, Policy>() * constants::pi<T, Policy>() * c / boost::math::pow<3>(s, pol); } case 3: { constexpr int P[] = { -2, -4 }; return boost::math::pow<3>(constants::pi<T, Policy>(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<4>(s, pol); } case 4: { constexpr int P[] = { 16, 8 }; return boost::math::pow<4>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<5>(s, pol); } case 5: { constexpr int P[] = { -16, -88, -16 }; return boost::math::pow<5>(constants::pi<T, Policy>(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<6>(s, pol); } case 6: { constexpr int P[] = { 272, 416, 32 }; return boost::math::pow<6>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<7>(s, pol); } case 7: { constexpr int P[] = { -272, -2880, -1824, -64 }; return boost::math::pow<7>(constants::pi<T, Policy>(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<8>(s, pol); } case 8: { constexpr int P[] = { 7936, 24576, 7680, 128 }; return boost::math::pow<8>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<9>(s, pol); } case 9: { constexpr int P[] = { -7936, -137216, -185856, -31616, -256 }; return boost::math::pow<9>(constants::pi<T, Policy>(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<10>(s, pol); } case 10: { constexpr int P[] = { 353792, 1841152, 1304832, 128512, 512 }; return boost::math::pow<10>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<11>(s, pol); } case 11: { constexpr int P[] = { -353792, -9061376, -21253376, -8728576, -518656, -1024}; return boost::math::pow<11>(constants::pi<T, Policy>(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<12>(s, pol); } case 12: { constexpr int P[] = { 22368256, 175627264, 222398464, 56520704, 2084864, 2048 }; return boost::math::pow<12>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<13>(s, pol); } #ifndef BOOST_NO_LONG_LONG case 13: { constexpr long long P[] = { -22368256LL, -795300864LL, -2868264960LL, -2174832640LL, -357888000LL, -8361984LL, -4096 }; return boost::math::pow<13>(constants::pi<T, Policy>(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<14>(s, pol); } case 14: { constexpr long long P[] = { 1903757312LL, 21016670208LL, 41731645440LL, 20261765120LL, 2230947840LL, 33497088LL, 8192 }; return boost::math::pow<14>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<15>(s, pol); } case 15: { constexpr long long P[] = { -1903757312LL, -89702612992LL, -460858269696LL, -559148810240LL, -182172651520LL, -13754155008LL, -134094848LL, -16384 }; return boost::math::pow<15>(constants::pi<T, Policy>(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<16>(s, pol); } case 16: { constexpr long long P[] = { 209865342976LL, 3099269660672LL, 8885192097792LL, 7048869314560LL, 1594922762240LL, 84134068224LL, 536608768LL, 32768 }; return boost::math::pow<16>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<17>(s, pol); } case 17: { constexpr long long P[] = { -209865342976LL, -12655654469632LL, -87815735738368LL, -155964390375424LL, -84842998005760LL, -13684856848384LL, -511780323328LL, -2146926592LL, -65536 }; return boost::math::pow<17>(constants::pi<T, Policy>(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<18>(s, pol); } case 18: { constexpr long long P[] = { 29088885112832LL, 553753414467584LL, 2165206642589696LL, 2550316668551168LL, 985278548541440LL, 115620218667008LL, 3100738912256LL, 8588754944LL, 131072 }; return boost::math::pow<18>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<19>(s, pol); } case 19: { constexpr long long P[] = { -29088885112832LL, -2184860175433728LL, -19686087844429824LL, -48165109676113920LL, -39471306959486976LL, -11124607890751488LL, -965271355195392LL, -18733264797696LL, -34357248000LL, -262144 }; return boost::math::pow<19>(constants::pi<T, Policy>(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<20>(s, pol); } case 20: { constexpr long long P[] = { 4951498053124096LL, 118071834535526400LL, 603968063567560704LL, 990081991141490688LL, 584901762421358592LL, 122829335169859584LL, 7984436548730880LL, 112949304754176LL, 137433710592LL, 524288 }; return boost::math::pow<20>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<21>(s, pol); } #endif } // // We'll have to compute the coefficients up to n, // complexity is O(n^2) which we don't worry about for now // as the values are computed once and then cached. // However, if the final evaluation would have too many // terms just bail out right away: // if((unsigned)n / 2u > policies::get_max_series_iterations<Policy>()) return policies::raise_evaluation_error<T>(function, "The value of n is so large that we're unable to compute the result in reasonable time, best guess is %1%", 0, pol); #ifdef BOOST_HAS_THREADS static std::mutex m; std::lock_guard<std::mutex> l(m); #endif static int digits = tools::digits<T>(); static std::vector<std::vector<T> > table(1, std::vector<T>(1, T(-1))); int current_digits = tools::digits<T>(); if(digits != current_digits) { // Oh my... our precision has changed! table = std::vector<std::vector<T> >(1, std::vector<T>(1, T(-1))); digits = current_digits; } int index = n - 1; if(index >= (int)table.size()) { for(int i = (int)table.size() - 1; i < index; ++i) { int offset = i & 1; // 1 if the first cos power is 0, otherwise 0. int sin_order = i + 2; // order of the sin term int max_cos_order = sin_order - 1; // largest order of the polynomial of cos terms int max_columns = (max_cos_order - offset) / 2; // How many entries there are in the current row. int next_offset = offset ? 0 : 1; int next_max_columns = (max_cos_order + 1 - next_offset) / 2; // How many entries there will be in the next row table.push_back(std::vector<T>(next_max_columns + 1, T(0))); for(int column = 0; column <= max_columns; ++column) { int cos_order = 2 * column + offset; // order of the cosine term in entry "column" BOOST_MATH_ASSERT(column < (int)table[i].size()); BOOST_MATH_ASSERT((cos_order + 1) / 2 < (int)table[i + 1].size()); table[i + 1][(cos_order + 1) / 2] += ((cos_order - sin_order) * table[i][column]) / (sin_order - 1); if(cos_order) table[i + 1][(cos_order - 1) / 2] += (-cos_order * table[i][column]) / (sin_order - 1); } } } T sum = boost::math::tools::evaluate_even_polynomial(&table[index][0], c, table[index].size()); if(index & 1) sum *= c; // First coefficient is order 1, and really an odd polynomial. if(sum == 0) return sum; // // The remaining terms are computed using logs since the powers and factorials // get real large real quick: // T power_terms = n * log(boost::math::constants::pi<T>()); if(s == 0) return sum * boost::math::policies::raise_overflow_error<T>(function, nullptr, pol); power_terms -= log(fabs(s)) * (n + 1); power_terms += boost::math::lgamma(T(n), pol); power_terms += log(fabs(sum)); if(power_terms > boost::math::tools::log_max_value<T>()) return sum * boost::math::policies::raise_overflow_error<T>(function, nullptr, pol); return exp(power_terms) * ((s < 0) && ((n + 1) & 1) ? -1 : 1) * boost::math::sign(sum); } template <class T, class Policy> struct polygamma_initializer { struct init { init() { // Forces initialization of our table of coefficients and mutex: boost::math::polygamma(30, T(-2.5f), Policy()); } void force_instantiate()const{} }; static const init initializer; static void force_instantiate() { initializer.force_instantiate(); } }; template <class T, class Policy> const typename polygamma_initializer<T, Policy>::init polygamma_initializer<T, Policy>::initializer; template<class T, class Policy> inline T polygamma_imp(const int n, T x, const Policy &pol) { BOOST_MATH_STD_USING static const char* function = "boost::math::polygamma<%1%>(int, %1%)"; polygamma_initializer<T, Policy>::initializer.force_instantiate(); if(n < 0) return policies::raise_domain_error<T>(function, "Order must be >= 0, but got %1%", static_cast<T>(n), pol); if(x < 0) { if(floor(x) == x) { // // Result is infinity if x is odd, and a pole error if x is even. // if(lltrunc(x) & 1) return policies::raise_overflow_error<T>(function, nullptr, pol); else return policies::raise_pole_error<T>(function, "Evaluation at negative integer %1%", x, pol); } T z = 1 - x; T result = polygamma_imp(n, z, pol) + constants::pi<T, Policy>() * poly_cot_pi(n, z, x, pol, function); return n & 1 ? T(-result) : result; } // // Limit for use of small-x-series is chosen // so that the series doesn't go too divergent // in the first few terms. Ordinarily this // would mean setting the limit to ~ 1 / n, // but we can tolerate a small amount of divergence: // T small_x_limit = (std::min)(T(T(5) / n), T(0.25f)); if(x < small_x_limit) { return polygamma_nearzero(n, x, pol, function); } else if(x > 0.4F * policies::digits_base10<T, Policy>() + 4.0f * n) { return polygamma_atinfinityplus(n, x, pol, function); } else if(x == 1) { return (n & 1 ? 1 : -1) * boost::math::factorial<T>(n, pol) * boost::math::zeta(T(n + 1), pol); } else if(x == 0.5f) { T result = (n & 1 ? 1 : -1) * boost::math::factorial<T>(n, pol) * boost::math::zeta(T(n + 1), pol); if(fabs(result) >= ldexp(tools::max_value<T>(), -n - 1)) return boost::math::sign(result) * policies::raise_overflow_error<T>(function, nullptr, pol); result *= ldexp(T(1), n + 1) - 1; return result; } else { return polygamma_attransitionplus(n, x, pol, function); } } } } } // namespace boost::math::detail #ifdef _MSC_VER #pragma warning(pop) #endif #endif // _BOOST_POLYGAMMA_DETAIL_2013_07_30_HPP_