| |
| /////////////////////////////////////////////////////////////////////////////// |
| // 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 <boost/cstdint.hpp> |
| #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/mpl/if.hpp> |
| #include <boost/mpl/int.hpp> |
| #include <boost/static_assert.hpp> |
| #include <boost/type_traits/is_convertible.hpp> |
| |
| 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 < max_factorial<T>::value)) |
| return ((n & 1) ? 1 : -1) * boost::math::factorial<T>(n - 1) * 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 > 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 = 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 = " + boost::lexical_cast<std::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); |
| } |
| 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, 0, 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, 0, 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 oveflow: |
| // |
| if(boost::math::tools::max_value<T>() / scale < sum) |
| return boost::math::policies::raise_overflow_error<T>(function, 0, pol); |
| sum *= scale; |
| return n & 1 ? sum : -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 trigomonetric 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 minumum - 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: |
| { |
| 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: |
| { |
| 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: |
| { |
| 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: |
| { |
| 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: |
| { |
| 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: |
| { |
| 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: |
| { |
| 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: |
| { |
| 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: |
| { |
| 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: |
| { |
| 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: |
| { |
| 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: |
| { |
| 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: |
| { |
| 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: |
| { |
| 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: |
| { |
| 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: |
| { |
| 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: |
| { |
| 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: |
| { |
| 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 boost::detail::lightweight_mutex m; |
| boost::detail::lightweight_mutex::scoped_lock l(m); |
| #endif |
| static std::vector<std::vector<T> > table(1, std::vector<T>(1, T(-1))); |
| |
| 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_ASSERT(column < (int)table[i].size()); |
| BOOST_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 coeffient 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, 0, pol); |
| power_terms -= log(fabs(s)) * (n + 1); |
| power_terms += boost::math::lgamma(T(n)); |
| 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, 0, 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, 0, 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, 0, pol); |
| result *= ldexp(T(1), n + 1) - 1; |
| return result; |
| } |
| else |
| { |
| return polygamma_attransitionplus(n, x, pol, function); |
| } |
| } |
| |
| } } } // namespace boost::math::detail |
| |
| #endif // _BOOST_POLYGAMMA_DETAIL_2013_07_30_HPP_ |
| |
