gamma.hpp

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   }
   BOOST_MATH_INSTRUMENT_CODE(prefix);
   if((floor(z) == z) && (z < max_factorial<T>::value))
   {
      prefix *= unchecked_factorial<T>(itrunc(z, pol) - 1);
   }
   else
   {
      prefix = prefix * pow(z / boost::math::constants::e<T>(), z);
      BOOST_MATH_INSTRUMENT_CODE(prefix);
      T sum = detail::lower_gamma_series(z, z, pol) / z;
      BOOST_MATH_INSTRUMENT_CODE(sum);
      sum += detail::upper_gamma_fraction(z, z, ::boost::math::policies::digits<T, Policy>());
      BOOST_MATH_INSTRUMENT_CODE(sum);
      if(fabs(tools::max_value<T>() / prefix) < fabs(sum))
         return policies::raise_overflow_error<T>(function, "Result of tgamma is too large to represent.", pol);
      BOOST_MATH_INSTRUMENT_CODE((sum * prefix));
      return sum * prefix;
   }
   return prefix;
}

template <class T, class Policy>
T lgamma_imp(T z, const Policy& pol, const lanczos::undefined_lanczos& l, int*sign)
{
   BOOST_MATH_STD_USING

   static const char* function = "boost::math::lgamma<%1%>(%1%)";
   T result = 0;
   int sresult = 1;
   if(z <= 0)
   {
      if(floor(z) == z)
         return policies::raise_pole_error<T>(function, "Evaluation of tgamma at a negative integer %1%.", z, pol);
      T t = detail::sinpx(z);
      z = -z;
      if(t < 0)
      {
         t = -t;
      }
      else
      {
         sresult = -sresult;
      }
      result = log(boost::math::constants::pi<T>()) - lgamma_imp(z, pol, l, 0) - log(t);
   }
   else if((z != 1) && (z != 2))
   {
      T limit = (std::max)(z+1, T(10));
      T prefix = z * log(limit) - limit;
      T sum = detail::lower_gamma_series(z, limit, pol) / z;
      sum += detail::upper_gamma_fraction(z, limit, ::boost::math::policies::digits<T, Policy>());
      result = log(sum) + prefix;
   }
   if(sign)
      *sign = sresult;
   return result;
}
//
// This helper calculates tgamma(dz+1)-1 without cancellation errors,
// used by the upper incomplete gamma with z < 1:
//
template <class T, class Policy, class L>
T tgammap1m1_imp(T dz, Policy const& pol, const L& l)
{
   BOOST_MATH_STD_USING

   typedef typename policies::precision<T,Policy>::type precision_type;

   typedef typename mpl::if_<
      mpl::or_<
         mpl::less_equal<precision_type, mpl::int_<0> >,
         mpl::greater<precision_type, mpl::int_<113> >
      >,
      typename mpl::if_<
         is_same<L, lanczos::lanczos24m113>,
         mpl::int_<113>,
         mpl::int_<0>
      >::type,
      typename mpl::if_<
         mpl::less_equal<precision_type, mpl::int_<64> >,
         mpl::int_<64>, mpl::int_<113> >::type
       >::type tag_type;

   T result;
   if(dz < 0)
   {
      if(dz < -0.5)
      {
         // Best method is simply to subtract 1 from tgamma:
         result = boost::math::tgamma(1+dz, pol) - 1;
         BOOST_MATH_INSTRUMENT_CODE(result);
      }
      else
      {
         // Use expm1 on lgamma:
         result = boost::math::expm1(-boost::math::log1p(dz, pol) 
            + lgamma_small_imp(dz+2, dz + 1, dz, tag_type(), pol, l));
         BOOST_MATH_INSTRUMENT_CODE(result);
      }
   }
   else
   {
      if(dz < 2)
      {
         // Use expm1 on lgamma:
         result = boost::math::expm1(lgamma_small_imp(dz+1, dz, dz-1, tag_type(), pol, l), pol);
         BOOST_MATH_INSTRUMENT_CODE(result);
      }
      else
      {
         // Best method is simply to subtract 1 from tgamma:
         result = boost::math::tgamma(1+dz, pol) - 1;
         BOOST_MATH_INSTRUMENT_CODE(result);
      }
   }

   return result;
}

template <class T, class Policy>
inline T tgammap1m1_imp(T dz, Policy const& pol,
                 const ::boost::math::lanczos::undefined_lanczos& l)
{
   BOOST_MATH_STD_USING // ADL of std names
   //
   // There should be a better solution than this, but the
   // algebra isn't easy for the general case....
   // Start by subracting 1 from tgamma:
   //
   T result = gamma_imp(1 + dz, pol, l) - 1;
   BOOST_MATH_INSTRUMENT_CODE(result);
   //
   // Test the level of cancellation error observed: we loose one bit
   // for each power of 2 the result is less than 1.  If we would get
   // more bits from our most precise lgamma rational approximation, 
   // then use that instead:
   //
   BOOST_MATH_INSTRUMENT_CODE((dz > -0.5));
   BOOST_MATH_INSTRUMENT_CODE((dz < 2));
   BOOST_MATH_INSTRUMENT_CODE((ldexp(1.0, boost::math::policies::digits<T, Policy>()) * fabs(result) < 1e34));
   if((dz > -0.5) && (dz < 2) && (ldexp(1.0, boost::math::policies::digits<T, Policy>()) * fabs(result) < 1e34))
   {
      result = tgammap1m1_imp(dz, pol, boost::math::lanczos::lanczos24m113());
      BOOST_MATH_INSTRUMENT_CODE(result);
   }
   return result;
}

//
// Series representation for upper fraction when z is small:
//
template <class T>
struct small_gamma2_series
{
   typedef T result_type;

   small_gamma2_series(T a_, T x_) : result(-x_), x(-x_), apn(a_+1), n(1){}

   T operator()()
   {
      T r = result / (apn);
      result *= x;
      result /= ++n;
      apn += 1;
      return r;
   }

private:
   T result, x, apn;
   int n;
};
//
// calculate power term prefix (z^a)(e^-z) used in the non-normalised
// incomplete gammas:
//
template <class T, class Policy>
T full_igamma_prefix(T a, T z, const Policy& pol)
{
   BOOST_MATH_STD_USING

   T prefix;
   T alz = a * log(z);

   if(z >= 1)
   {
      if((alz < tools::log_max_value<T>()) && (-z > tools::log_min_value<T>()))
      {
         prefix = pow(z, a) * exp(-z);
      }
      else if(a >= 1)
      {
         prefix = pow(z / exp(z/a), a);
      }
      else
      {
         prefix = exp(alz - z);
      }
   }
   else
   {
      if(alz > tools::log_min_value<T>())
      {
         prefix = pow(z, a) * exp(-z);
      }
      else if(z/a < tools::log_max_value<T>())
      {
         prefix = pow(z / exp(z/a), a);
      }
      else
      {
         prefix = exp(alz - z);
      }
   }
   //
   // This error handling isn't very good: it happens after the fact
   // rather than before it...
   //
   if((boost::math::fpclassify)(prefix) == (int)FP_INFINITE)
      policies::raise_overflow_error<T>("boost::math::detail::full_igamma_prefix<%1%>(%1%, %1%)", "Result of incomplete gamma function is too large to represent.", pol);

   return prefix;
}
//
// Compute (z^a)(e^-z)/tgamma(a)
// most if the error occurs in this function:
//
template <class T, class Policy, class L>
T regularised_gamma_prefix(T a, T z, const Policy& pol, const L& l)
{
   BOOST_MATH_STD_USING
   T agh = a + static_cast<T>(L::g()) - T(0.5);
   T prefix;
   T d = ((z - a) - static_cast<T>(L::g()) + T(0.5)) / agh;

   if(a < 1)
   {
      //
      // We have to treat a < 1 as a special case because our Lanczos
      // approximations are optimised against the factorials with a > 1,
      // and for high precision types especially (128-bit reals for example)
      // very small values of a can give rather eroneous results for gamma
      // unless we do this:
      //
      // TODO: is this still required?  Lanczos approx should be better now?
      //
      if(z <= tools::log_min_value<T>())
      {
         // Oh dear, have to use logs, should be free of cancellation errors though:
         return exp(a * log(z) - z - lgamma_imp(a, pol, l));
      }
      else
      {
         // direct calculation, no danger of overflow as gamma(a) < 1/a
         // for small a.
         return pow(z, a) * exp(-z) / gamma_imp(a, pol, l);
      }
   }
   else if((fabs(d*d*a) <= 100) && (a > 150))
   {
      // special case for large a and a ~ z.
      prefix = a * boost::math::log1pmx(d, pol) + z * static_cast<T>(0.5 - L::g()) / agh;
      prefix = exp(prefix);
   }
   else
   {
      //
      // general case.
      // direct computation is most accurate, but use various fallbacks
      // for different parts of the problem domain:
      //
      T alz = a * log(z / agh);
      T amz = a - z;
      if(((std::min)(alz, amz) <= tools::log_min_value<T>()) || ((std::max)(alz, amz) >= tools::log_max_value<T>()))
      {
         T amza = amz / a;
         if(((std::min)(alz, amz)/2 > tools::log_min_value<T>()) && ((std::max)(alz, amz)/2 < tools::log_max_value<T>()))
         {
            // compute square root of the result and then square it:
            T sq = pow(z / agh, a / 2) * exp(amz / 2);
            prefix = sq * sq;
         }
         else if(((std::min)(alz, amz)/4 > tools::log_min_value<T>()) && ((std::max)(alz, amz)/4 < tools::log_max_value<T>()) && (z > a))
         {
            // compute the 4th root of the result then square it twice:
            T sq = pow(z / agh, a / 4) * exp(amz / 4);
            prefix = sq * sq;
            prefix *= prefix;
         }
         else if((amza > tools::log_min_value<T>()) && (amza < tools::log_max_value<T>()))
         {
            prefix = pow((z * exp(amza)) / agh, a);
         }
         else
         {
            prefix = exp(alz + amz);
         }
      }
      else
      {
         prefix = pow(z / agh, a) * exp(amz);
      }
   }
   prefix *= sqrt(agh / boost::math::constants::e<T>()) / L::lanczos_sum_expG_scaled(a);
   return prefix;
}
//
// And again, without Lanczos support:
//
template <class T, class Policy>
T regularised_gamma_prefix(T a, T z, const Policy& pol, const lanczos::undefined_lanczos&)
{
   BOOST_MATH_STD_USING

   T limit = (std::max)(T(10), a);
   T sum = detail::lower_gamma_series(a, limit, pol) / a;
   sum += detail::upper_gamma_fraction(a, limit, ::boost::math::policies::digits<T, Policy>());

   if(a < 10)
   {
      // special case for small a:
      T prefix = pow(z / 10, a);
      prefix *= exp(10-z);
      if(0 == prefix)
      {
         prefix = pow((z * exp((10-z)/a)) / 10, a);
      }
      prefix /= sum;
      return prefix;
   }

   T zoa = z / a;
   T amz = a - z;
   T alzoa = a * log(zoa);
   T prefix;
   if(((std::min)(alzoa, amz) <= tools::log_min_value<T>()) || ((std::max)(alzoa, amz) >= tools::log_max_value<T>()))
   {
      T amza = amz / a;
      if((amza <= tools::log_min_value<T>()) || (amza >= tools::log_max_value<T>()))
      {
         prefix = exp(alzoa + amz);
      }
      else
      {
         prefix = pow(zoa * exp(amza), a);
      }
   }
   else
   {
      prefix = pow(zoa, a) * exp(amz);
   }
   prefix /= sum;
   return prefix;
}
//
// Upper gamma fraction for very small a:
//
template <class T, class Policy>
inline T tgamma_small_upper_part(T a, T x, const Policy& pol)
{
   BOOST_MATH_STD_USING  // ADL of std functions.
   //
   // Compute the full upper fraction (Q) when a is very small:
   //
   T result;
   result = boost::math::tgamma1pm1(a, pol) - boost::math::powm1(x, a, pol);
   result /= a;
   detail::small_gamma2_series<T> s(a, x);
   boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
#if BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582))