Chris@102: 
Chris@102: ///////////////////////////////////////////////////////////////////////////////
Chris@102: //  Copyright 2013 Nikhar Agrawal
Chris@102: //  Copyright 2013 Christopher Kormanyos
Chris@102: //  Copyright 2014 John Maddock
Chris@102: //  Copyright 2013 Paul Bristow
Chris@102: //  Distributed under the Boost
Chris@102: //  Software License, Version 1.0. (See accompanying file
Chris@102: //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
Chris@102: 
Chris@102: #ifndef _BOOST_POLYGAMMA_DETAIL_2013_07_30_HPP_
Chris@102:   #define _BOOST_POLYGAMMA_DETAIL_2013_07_30_HPP_
Chris@102: 
Chris@102:   #include <cmath>
Chris@102:   #include <limits>
Chris@102:   #include <boost/cstdint.hpp>
Chris@102:   #include <boost/math/policies/policy.hpp>
Chris@102:   #include <boost/math/special_functions/bernoulli.hpp>
Chris@102:   #include <boost/math/special_functions/trunc.hpp>
Chris@102:   #include <boost/math/special_functions/zeta.hpp>
Chris@102:   #include <boost/math/special_functions/digamma.hpp>
Chris@102:   #include <boost/math/special_functions/sin_pi.hpp>
Chris@102:   #include <boost/math/special_functions/cos_pi.hpp>
Chris@102:   #include <boost/math/special_functions/pow.hpp>
Chris@102:   #include <boost/mpl/if.hpp>
Chris@102:   #include <boost/mpl/int.hpp>
Chris@102:   #include <boost/static_assert.hpp>
Chris@102:   #include <boost/type_traits/is_convertible.hpp>
Chris@102: 
Chris@102:   namespace boost { namespace math { namespace detail{
Chris@102: 
Chris@102:   template<class T, class Policy>
Chris@102:   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
Chris@102:   {
Chris@102:      // See http://functions.wolfram.com/GammaBetaErf/PolyGamma2/06/02/0001/
Chris@102:      BOOST_MATH_STD_USING
Chris@102:      //
Chris@102:      // sum       == current value of accumulated sum.
Chris@102:      // term      == value of current term to be added to sum.
Chris@102:      // part_term == value of current term excluding the Bernoulli number part
Chris@102:      //
Chris@102:      if(n + x == x)
Chris@102:      {
Chris@102:         // x is crazy large, just concentrate on the first part of the expression and use logs:
Chris@102:         if(n == 1) return 1 / x;
Chris@102:         T nlx = n * log(x);
Chris@102:         if((nlx < tools::log_max_value<T>()) && (n < max_factorial<T>::value))
Chris@102:            return ((n & 1) ? 1 : -1) * boost::math::factorial<T>(n - 1) * pow(x, -n);
Chris@102:         else
Chris@102:          return ((n & 1) ? 1 : -1) * exp(boost::math::lgamma(T(n), pol) - n * log(x));
Chris@102:      }
Chris@102:      T term, sum, part_term;
Chris@102:      T x_squared = x * x;
Chris@102:      //
Chris@102:      // Start by setting part_term to:
Chris@102:      //
Chris@102:      // (n-1)! / x^(n+1)
Chris@102:      //
Chris@102:      // which is common to both the first term of the series (with k = 1)
Chris@102:      // and to the leading part.  
Chris@102:      // We can then get to the leading term by:
Chris@102:      //
Chris@102:      // part_term * (n + 2 * x) / 2
Chris@102:      //
Chris@102:      // and to the first term in the series 
Chris@102:      // (excluding the Bernoulli number) by:
Chris@102:      //
Chris@102:      // part_term n * (n + 1) / (2x)
Chris@102:      //
Chris@102:      // If either the factorial would overflow,
Chris@102:      // or the power term underflows, this just gets set to 0 and then we
Chris@102:      // know that we have to use logs for the initial terms:
Chris@102:      //
Chris@102:      part_term = ((n > boost::math::max_factorial<T>::value) && (T(n) * n > tools::log_max_value<T>())) 
Chris@102:         ? T(0) : static_cast<T>(boost::math::factorial<T>(n - 1, pol) * pow(x, -n - 1));
Chris@102:      if(part_term == 0)
Chris@102:      {
Chris@102:         // Either n is very large, or the power term underflows,
Chris@102:         // set the initial values of part_term, term and sum via logs:
Chris@102:         part_term = boost::math::lgamma(n, pol) - (n + 1) * log(x);
Chris@102:         sum = exp(part_term + log(n + 2 * x) - boost::math::constants::ln_two<T>());
Chris@102:         part_term += log(T(n) * (n + 1)) - boost::math::constants::ln_two<T>() - log(x);
Chris@102:         part_term = exp(part_term);
Chris@102:      }
Chris@102:      else
Chris@102:      {
Chris@102:         sum = part_term * (n + 2 * x) / 2;
Chris@102:         part_term *= (T(n) * (n + 1)) / 2;
Chris@102:         part_term /= x;
Chris@102:      }
Chris@102:      //
Chris@102:      // If the leading term is 0, so is the result:
Chris@102:      //
Chris@102:      if(sum == 0)
Chris@102:         return sum;
Chris@102: 
Chris@102:      for(unsigned k = 1;;)
Chris@102:      {
Chris@102:         term = part_term * boost::math::bernoulli_b2n<T>(k, pol);
Chris@102:         sum += term;
Chris@102:         //
Chris@102:         // Normal termination condition:
Chris@102:         //
Chris@102:         if(fabs(term / sum) < tools::epsilon<T>())
Chris@102:            break;
Chris@102:         //
Chris@102:         // Increment our counter, and move part_term on to the next value:
Chris@102:         //
Chris@102:         ++k;
Chris@102:         part_term *= T(n + 2 * k - 2) * (n - 1 + 2 * k);
Chris@102:         part_term /= (2 * k - 1) * 2 * k;
Chris@102:         part_term /= x_squared;
Chris@102:         //
Chris@102:         // Emergency get out termination condition:
Chris@102:         //
Chris@102:         if(k > policies::get_max_series_iterations<Policy>())
Chris@102:         {
Chris@102:            return policies::raise_evaluation_error(function, "Series did not converge, closest value was %1%", sum, pol);
Chris@102:         }
Chris@102:      }
Chris@102:      
Chris@102:      if((n - 1) & 1)
Chris@102:         sum = -sum;
Chris@102: 
Chris@102:      return sum;
Chris@102:   }
Chris@102: 
Chris@102:   template<class T, class Policy>
Chris@102:   T polygamma_attransitionplus(const int n, const T& x, const Policy& pol, const char* function)
Chris@102:   {
Chris@102:     // See: http://functions.wolfram.com/GammaBetaErf/PolyGamma2/16/01/01/0017/
Chris@102: 
Chris@102:     // Use N = (0.4 * digits) + (4 * n) for target value for x:
Chris@102:     BOOST_MATH_STD_USING
Chris@102:     const int d4d  = static_cast<int>(0.4F * policies::digits_base10<T, Policy>());
Chris@102:     const int N = d4d + (4 * n);
Chris@102:     const int m    = n;
Chris@102:     const int iter = N - itrunc(x);
Chris@102: 
Chris@102:     if(iter > (int)policies::get_max_series_iterations<Policy>())
Chris@102:        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);
Chris@102: 
Chris@102:     const int minus_m_minus_one = -m - 1;
Chris@102: 
Chris@102:     T z(x);
Chris@102:     T sum0(0);
Chris@102:     T z_plus_k_pow_minus_m_minus_one(0);
Chris@102: 
Chris@102:     // Forward recursion to larger x, need to check for overflow first though:
Chris@102:     if(log(z + iter) * minus_m_minus_one > -tools::log_max_value<T>())
Chris@102:     {
Chris@102:        for(int k = 1; k <= iter; ++k)
Chris@102:        {
Chris@102:           z_plus_k_pow_minus_m_minus_one = pow(z, minus_m_minus_one);
Chris@102:           sum0 += z_plus_k_pow_minus_m_minus_one;
Chris@102:           z += 1;
Chris@102:        }
Chris@102:        sum0 *= boost::math::factorial<T>(n);
Chris@102:     }
Chris@102:     else
Chris@102:     {
Chris@102:        for(int k = 1; k <= iter; ++k)
Chris@102:        {
Chris@102:           T log_term = log(z) * minus_m_minus_one + boost::math::lgamma(T(n + 1), pol);
Chris@102:           sum0 += exp(log_term);
Chris@102:           z += 1;
Chris@102:        }
Chris@102:     }
Chris@102:     if((n - 1) & 1)
Chris@102:        sum0 = -sum0;
Chris@102: 
Chris@102:     return sum0 + polygamma_atinfinityplus(n, z, pol, function);
Chris@102:   }
Chris@102: 
Chris@102:   template <class T, class Policy>
Chris@102:   T polygamma_nearzero(int n, T x, const Policy& pol, const char* function)
Chris@102:   {
Chris@102:      BOOST_MATH_STD_USING
Chris@102:      //
Chris@102:      // If we take this expansion for polygamma: http://functions.wolfram.com/06.15.06.0003.02
Chris@102:      // and substitute in this expression for polygamma(n, 1): http://functions.wolfram.com/06.15.03.0009.01
Chris@102:      // we get an alternating series for polygamma when x is small in terms of zeta functions of
Chris@102:      // integer arguments (which are easy to evaluate, at least when the integer is even).
Chris@102:      //
Chris@102:      // In order to avoid spurious overflow, save the n! term for later, and rescale at the end:
Chris@102:      //
Chris@102:      T scale = boost::math::factorial<T>(n, pol);
Chris@102:      //
Chris@102:      // "factorial_part" contains everything except the zeta function
Chris@102:      // evaluations in each term:
Chris@102:      //
Chris@102:      T factorial_part = 1;
Chris@102:      //
Chris@102:      // "prefix" is what we'll be adding the accumulated sum to, it will
Chris@102:      // be n! / z^(n+1), but since we're scaling by n! it's just 
Chris@102:      // 1 / z^(n+1) for now:
Chris@102:      //
Chris@102:      T prefix = pow(x, n + 1);
Chris@102:      if(prefix == 0)
Chris@102:         return boost::math::policies::raise_overflow_error<T>(function, 0, pol);
Chris@102:      prefix = 1 / prefix;
Chris@102:      //
Chris@102:      // First term in the series is necessarily < zeta(2) < 2, so
Chris@102:      // ignore the sum if it will have no effect on the result anyway:
Chris@102:      //
Chris@102:      if(prefix > 2 / policies::get_epsilon<T, Policy>())
Chris@102:         return ((n & 1) ? 1 : -1) * 
Chris@102:          (tools::max_value<T>() / prefix < scale ? policies::raise_overflow_error<T>(function, 0, pol) : prefix * scale);
Chris@102:      //
Chris@102:      // As this is an alternating series we could accelerate it using 
Chris@102:      // "Convergence Acceleration of Alternating Series",
Chris@102:      // Henri Cohen, Fernando Rodriguez Villegas, and Don Zagier, Experimental Mathematics, 1999.
Chris@102:      // In practice however, it appears not to make any difference to the number of terms
Chris@102:      // required except in some edge cases which are filtered out anyway before we get here.
Chris@102:      //
Chris@102:      T sum = prefix;
Chris@102:      for(unsigned k = 0;;)
Chris@102:      {
Chris@102:         // Get the k'th term:
Chris@102:         T term = factorial_part * boost::math::zeta(T(k + n + 1), pol);
Chris@102:         sum += term;
Chris@102:         // Termination condition:
Chris@102:         if(fabs(term) < fabs(sum * boost::math::policies::get_epsilon<T, Policy>()))
Chris@102:            break;
Chris@102:         //
Chris@102:         // Move on k and factorial_part:
Chris@102:         //
Chris@102:         ++k;
Chris@102:         factorial_part *= (-x * (n + k)) / k;
Chris@102:         //
Chris@102:         // Last chance exit:
Chris@102:         //
Chris@102:         if(k > policies::get_max_series_iterations<Policy>())
Chris@102:            return policies::raise_evaluation_error<T>(function, "Series did not converge, best value is %1%", sum, pol);
Chris@102:      }
Chris@102:      //
Chris@102:      // We need to multiply by the scale, at each stage checking for oveflow:
Chris@102:      //
Chris@102:      if(boost::math::tools::max_value<T>() / scale < sum)
Chris@102:         return boost::math::policies::raise_overflow_error<T>(function, 0, pol);
Chris@102:      sum *= scale;
Chris@102:      return n & 1 ? sum : -sum;
Chris@102:   }
Chris@102: 
Chris@102:   //
Chris@102:   // Helper function which figures out which slot our coefficient is in
Chris@102:   // given an angle multiplier for the cosine term of power:
Chris@102:   //
Chris@102:   template <class Table>
Chris@102:   typename Table::value_type::reference dereference_table(Table& table, unsigned row, unsigned power)
Chris@102:   {
Chris@102:      return table[row][power / 2];
Chris@102:   }
Chris@102: 
Chris@102: 
Chris@102: 
Chris@102:   template <class T, class Policy>
Chris@102:   T poly_cot_pi(int n, T x, T xc, const Policy& pol, const char* function)
Chris@102:   {
Chris@102:      BOOST_MATH_STD_USING
Chris@102:      // Return n'th derivative of cot(pi*x) at x, these are simply
Chris@102:      // tabulated for up to n = 9, beyond that it is possible to
Chris@102:      // calculate coefficients as follows:
Chris@102:      //
Chris@102:      // The general form of each derivative is:
Chris@102:      //
Chris@102:      // pi^n * SUM{k=0, n} C[k,n] * cos^k(pi * x) * csc^(n+1)(pi * x)
Chris@102:      //
Chris@102:      // With constant C[0,1] = -1 and all other C[k,n] = 0;
Chris@102:      // Then for each k < n+1:
Chris@102:      // C[k-1, n+1]  -= k * C[k, n];
Chris@102:      // C[k+1, n+1]  += (k-n-1) * C[k, n];
Chris@102:      //
Chris@102:      // Note that there are many different ways of representing this derivative thanks to
Chris@102:      // the many trigomonetric identies available.  In particular, the sum of powers of
Chris@102:      // cosines could be replaced by a sum of cosine multiple angles, and indeed if you
Chris@102:      // plug the derivative into Mathematica this is the form it will give.  The two
Chris@102:      // forms are related via the Chebeshev polynomials of the first kind and
Chris@102:      // T_n(cos(x)) = cos(n x).  The polynomial form has the great advantage that
Chris@102:      // all the cosine terms are zero at half integer arguments - right where this
Chris@102:      // function has it's minumum - thus avoiding cancellation error in this region.
Chris@102:      //
Chris@102:      // And finally, since every other term in the polynomials is zero, we can save
Chris@102:      // space by only storing the non-zero terms.  This greatly complexifies
Chris@102:      // subscripting the tables in the calculation, but halves the storage space
Chris@102:      // (and complexity for that matter).
Chris@102:      //
Chris@102:      T s = fabs(x) < fabs(xc) ? boost::math::sin_pi(x, pol) : boost::math::sin_pi(xc, pol);
Chris@102:      T c = boost::math::cos_pi(x, pol);
Chris@102:      switch(n)
Chris@102:      {
Chris@102:      case 1:
Chris@102:         return -constants::pi<T, Policy>() / (s * s);
Chris@102:      case 2:
Chris@102:      {
Chris@102:         return 2 * constants::pi<T, Policy>() * constants::pi<T, Policy>() * c / boost::math::pow<3>(s, pol);
Chris@102:      }
Chris@102:      case 3:
Chris@102:      {
Chris@102:         int P[] = { -2, -4 };
Chris@102:         return boost::math::pow<3>(constants::pi<T, Policy>(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<4>(s, pol);
Chris@102:      }
Chris@102:      case 4:
Chris@102:      {
Chris@102:         int P[] = { 16, 8 };
Chris@102:         return boost::math::pow<4>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<5>(s, pol);
Chris@102:      }
Chris@102:      case 5:
Chris@102:      {
Chris@102:         int P[] = { -16, -88, -16 };
Chris@102:         return boost::math::pow<5>(constants::pi<T, Policy>(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<6>(s, pol);
Chris@102:      }
Chris@102:      case 6:
Chris@102:      {
Chris@102:         int P[] = { 272, 416, 32 };
Chris@102:         return boost::math::pow<6>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<7>(s, pol);
Chris@102:      }
Chris@102:      case 7:
Chris@102:      {
Chris@102:         int P[] = { -272, -2880, -1824, -64 };
Chris@102:         return boost::math::pow<7>(constants::pi<T, Policy>(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<8>(s, pol);
Chris@102:      }
Chris@102:      case 8:
Chris@102:      {
Chris@102:         int P[] = { 7936, 24576, 7680, 128 };
Chris@102:         return boost::math::pow<8>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<9>(s, pol);
Chris@102:      }
Chris@102:      case 9:
Chris@102:      {
Chris@102:         int P[] = { -7936, -137216, -185856, -31616, -256 };
Chris@102:         return boost::math::pow<9>(constants::pi<T, Policy>(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<10>(s, pol);
Chris@102:      }
Chris@102:      case 10:
Chris@102:      {
Chris@102:         int P[] = { 353792, 1841152, 1304832, 128512, 512 };
Chris@102:         return boost::math::pow<10>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<11>(s, pol);
Chris@102:      }
Chris@102:      case 11:
Chris@102:      {
Chris@102:         int P[] = { -353792, -9061376, -21253376, -8728576, -518656, -1024};
Chris@102:         return boost::math::pow<11>(constants::pi<T, Policy>(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<12>(s, pol);
Chris@102:      }
Chris@102:      case 12:
Chris@102:      {
Chris@102:         int P[] = { 22368256, 175627264, 222398464, 56520704, 2084864, 2048 };
Chris@102:         return boost::math::pow<12>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<13>(s, pol);
Chris@102:      }
Chris@102: #ifndef BOOST_NO_LONG_LONG
Chris@102:      case 13:
Chris@102:      {
Chris@102:         long long P[] = { -22368256LL, -795300864LL, -2868264960LL, -2174832640LL, -357888000LL, -8361984LL, -4096 };
Chris@102:         return boost::math::pow<13>(constants::pi<T, Policy>(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<14>(s, pol);
Chris@102:      }
Chris@102:      case 14:
Chris@102:      {
Chris@102:         long long P[] = { 1903757312LL, 21016670208LL, 41731645440LL, 20261765120LL, 2230947840LL, 33497088LL, 8192 };
Chris@102:         return boost::math::pow<14>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<15>(s, pol);
Chris@102:      }
Chris@102:      case 15:
Chris@102:      {
Chris@102:         long long P[] = { -1903757312LL, -89702612992LL, -460858269696LL, -559148810240LL, -182172651520LL, -13754155008LL, -134094848LL, -16384 };
Chris@102:         return boost::math::pow<15>(constants::pi<T, Policy>(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<16>(s, pol);
Chris@102:      }
Chris@102:      case 16:
Chris@102:      {
Chris@102:         long long P[] = { 209865342976LL, 3099269660672LL, 8885192097792LL, 7048869314560LL, 1594922762240LL, 84134068224LL, 536608768LL, 32768 };
Chris@102:         return boost::math::pow<16>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<17>(s, pol);
Chris@102:      }
Chris@102:      case 17:
Chris@102:      {
Chris@102:         long long P[] = { -209865342976LL, -12655654469632LL, -87815735738368LL, -155964390375424LL, -84842998005760LL, -13684856848384LL, -511780323328LL, -2146926592LL, -65536 };
Chris@102:         return boost::math::pow<17>(constants::pi<T, Policy>(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<18>(s, pol);
Chris@102:      }
Chris@102:      case 18:
Chris@102:      {
Chris@102:         long long P[] = { 29088885112832LL, 553753414467584LL, 2165206642589696LL, 2550316668551168LL, 985278548541440LL, 115620218667008LL, 3100738912256LL, 8588754944LL, 131072 };
Chris@102:         return boost::math::pow<18>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<19>(s, pol);
Chris@102:      }
Chris@102:      case 19:
Chris@102:      {
Chris@102:         long long P[] = { -29088885112832LL, -2184860175433728LL, -19686087844429824LL, -48165109676113920LL, -39471306959486976LL, -11124607890751488LL, -965271355195392LL, -18733264797696LL, -34357248000LL, -262144 };
Chris@102:         return boost::math::pow<19>(constants::pi<T, Policy>(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<20>(s, pol);
Chris@102:      }
Chris@102:      case 20:
Chris@102:      {
Chris@102:         long long P[] = { 4951498053124096LL, 118071834535526400LL, 603968063567560704LL, 990081991141490688LL, 584901762421358592LL, 122829335169859584LL, 7984436548730880LL, 112949304754176LL, 137433710592LL, 524288 };
Chris@102:         return boost::math::pow<20>(constants::pi<T, Policy>(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<21>(s, pol);
Chris@102:      }
Chris@102: #endif
Chris@102:      }
Chris@102: 
Chris@102:      //
Chris@102:      // We'll have to compute the coefficients up to n, 
Chris@102:      // complexity is O(n^2) which we don't worry about for now
Chris@102:      // as the values are computed once and then cached.
Chris@102:      // However, if the final evaluation would have too many
Chris@102:      // terms just bail out right away:
Chris@102:      //
Chris@102:      if((unsigned)n / 2u > policies::get_max_series_iterations<Policy>())
Chris@102:         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);
Chris@102: #ifdef BOOST_HAS_THREADS
Chris@102:      static boost::detail::lightweight_mutex m;
Chris@102:      boost::detail::lightweight_mutex::scoped_lock l(m);
Chris@102: #endif
Chris@102:      static std::vector<std::vector<T> > table(1, std::vector<T>(1, T(-1)));
Chris@102: 
Chris@102:      int index = n - 1;
Chris@102: 
Chris@102:      if(index >= (int)table.size())
Chris@102:      {
Chris@102:         for(int i = (int)table.size() - 1; i < index; ++i)
Chris@102:         {
Chris@102:            int offset = i & 1; // 1 if the first cos power is 0, otherwise 0.
Chris@102:            int sin_order = i + 2;  // order of the sin term
Chris@102:            int max_cos_order = sin_order - 1;  // largest order of the polynomial of cos terms
Chris@102:            int max_columns = (max_cos_order - offset) / 2;  // How many entries there are in the current row.
Chris@102:            int next_offset = offset ? 0 : 1;
Chris@102:            int next_max_columns = (max_cos_order + 1 - next_offset) / 2;  // How many entries there will be in the next row
Chris@102:            table.push_back(std::vector<T>(next_max_columns + 1, T(0)));
Chris@102: 
Chris@102:            for(int column = 0; column <= max_columns; ++column)
Chris@102:            {
Chris@102:               int cos_order = 2 * column + offset;  // order of the cosine term in entry "column"
Chris@102:               BOOST_ASSERT(column < (int)table[i].size());
Chris@102:               BOOST_ASSERT((cos_order + 1) / 2 < (int)table[i + 1].size());
Chris@102:               table[i + 1][(cos_order + 1) / 2] += ((cos_order - sin_order) * table[i][column]) / (sin_order - 1);
Chris@102:               if(cos_order)
Chris@102:                 table[i + 1][(cos_order - 1) / 2] += (-cos_order * table[i][column]) / (sin_order - 1);
Chris@102:            }
Chris@102:         }
Chris@102: 
Chris@102:      }
Chris@102:      T sum = boost::math::tools::evaluate_even_polynomial(&table[index][0], c, table[index].size());
Chris@102:      if(index & 1)
Chris@102:         sum *= c;  // First coeffient is order 1, and really an odd polynomial.
Chris@102:      if(sum == 0)
Chris@102:         return sum;
Chris@102:      //
Chris@102:      // The remaining terms are computed using logs since the powers and factorials
Chris@102:      // get real large real quick:
Chris@102:      //
Chris@102:      T power_terms = n * log(boost::math::constants::pi<T>());
Chris@102:      if(s == 0)
Chris@102:         return sum * boost::math::policies::raise_overflow_error<T>(function, 0, pol);
Chris@102:      power_terms -= log(fabs(s)) * (n + 1);
Chris@102:      power_terms += boost::math::lgamma(T(n));
Chris@102:      power_terms += log(fabs(sum));
Chris@102: 
Chris@102:      if(power_terms > boost::math::tools::log_max_value<T>())
Chris@102:         return sum * boost::math::policies::raise_overflow_error<T>(function, 0, pol);
Chris@102: 
Chris@102:      return exp(power_terms) * ((s < 0) && ((n + 1) & 1) ? -1 : 1) * boost::math::sign(sum);
Chris@102:   }
Chris@102: 
Chris@102:   template <class T, class Policy>
Chris@102:   struct polygamma_initializer
Chris@102:   {
Chris@102:      struct init
Chris@102:      {
Chris@102:         init()
Chris@102:         {
Chris@102:            // Forces initialization of our table of coefficients and mutex:
Chris@102:            boost::math::polygamma(30, T(-2.5f), Policy());
Chris@102:         }
Chris@102:         void force_instantiate()const{}
Chris@102:      };
Chris@102:      static const init initializer;
Chris@102:      static void force_instantiate()
Chris@102:      {
Chris@102:         initializer.force_instantiate();
Chris@102:      }
Chris@102:   };
Chris@102: 
Chris@102:   template <class T, class Policy>
Chris@102:   const typename polygamma_initializer<T, Policy>::init polygamma_initializer<T, Policy>::initializer;
Chris@102:   
Chris@102:   template<class T, class Policy>
Chris@102:   inline T polygamma_imp(const int n, T x, const Policy &pol)
Chris@102:   {
Chris@102:     BOOST_MATH_STD_USING
Chris@102:     static const char* function = "boost::math::polygamma<%1%>(int, %1%)";
Chris@102:     polygamma_initializer<T, Policy>::initializer.force_instantiate();
Chris@102:     if(n < 0)
Chris@102:        return policies::raise_domain_error<T>(function, "Order must be >= 0, but got %1%", static_cast<T>(n), pol);
Chris@102:     if(x < 0)
Chris@102:     {
Chris@102:        if(floor(x) == x)
Chris@102:        {
Chris@102:           //
Chris@102:           // Result is infinity if x is odd, and a pole error if x is even.
Chris@102:           //
Chris@102:           if(lltrunc(x) & 1)
Chris@102:              return policies::raise_overflow_error<T>(function, 0, pol);
Chris@102:           else
Chris@102:              return policies::raise_pole_error<T>(function, "Evaluation at negative integer %1%", x, pol);
Chris@102:        }
Chris@102:        T z = 1 - x;
Chris@102:        T result = polygamma_imp(n, z, pol) + constants::pi<T, Policy>() * poly_cot_pi(n, z, x, pol, function);
Chris@102:        return n & 1 ? T(-result) : result;
Chris@102:     }
Chris@102:     //
Chris@102:     // Limit for use of small-x-series is chosen
Chris@102:     // so that the series doesn't go too divergent
Chris@102:     // in the first few terms.  Ordinarily this
Chris@102:     // would mean setting the limit to ~ 1 / n,
Chris@102:     // but we can tolerate a small amount of divergence:
Chris@102:     //
Chris@102:     T small_x_limit = std::min(T(T(5) / n), T(0.25f));
Chris@102:     if(x < small_x_limit)
Chris@102:     {
Chris@102:       return polygamma_nearzero(n, x, pol, function);
Chris@102:     }
Chris@102:     else if(x > 0.4F * policies::digits_base10<T, Policy>() + 4.0f * n)
Chris@102:     {
Chris@102:       return polygamma_atinfinityplus(n, x, pol, function);
Chris@102:     }
Chris@102:     else if(x == 1)
Chris@102:     {
Chris@102:        return (n & 1 ? 1 : -1) * boost::math::factorial<T>(n, pol) * boost::math::zeta(T(n + 1), pol);
Chris@102:     }
Chris@102:     else if(x == 0.5f)
Chris@102:     {
Chris@102:        T result = (n & 1 ? 1 : -1) * boost::math::factorial<T>(n, pol) * boost::math::zeta(T(n + 1), pol);
Chris@102:        if(fabs(result) >= ldexp(tools::max_value<T>(), -n - 1))
Chris@102:           return boost::math::sign(result) * policies::raise_overflow_error<T>(function, 0, pol);
Chris@102:        result *= ldexp(T(1), n + 1) - 1;
Chris@102:        return result;
Chris@102:     }
Chris@102:     else
Chris@102:     {
Chris@102:       return polygamma_attransitionplus(n, x, pol, function);
Chris@102:     }
Chris@102:   }
Chris@102: 
Chris@102: } } } // namespace boost::math::detail
Chris@102: 
Chris@102: #endif // _BOOST_POLYGAMMA_DETAIL_2013_07_30_HPP_
Chris@102: