beta.hpp

来自「Boost provides free peer-reviewed portab」· HPP 代码 · 共 1,357 行 · 第 1/3 页

   if(normalised)
   {
      T c = a + b;

      // incomplete beta power term, combined with the Lanczos approximation:
      T agh = a + L::g() - T(0.5);
      T bgh = b + L::g() - T(0.5);
      T cgh = c + L::g() - T(0.5);
      result = L::lanczos_sum_expG_scaled(c) / (L::lanczos_sum_expG_scaled(a) * L::lanczos_sum_expG_scaled(b));
      if(a * b < bgh * 10)
         result *= exp((b - 0.5f) * boost::math::log1p(a / bgh, pol));
      else
         result *= pow(cgh / bgh, b - 0.5f);
      result *= pow(x * cgh / agh, a);
      result *= sqrt(agh / boost::math::constants::e<T>());

      if(p_derivative)
      {
         *p_derivative = result * pow(y, b);
         BOOST_ASSERT(*p_derivative >= 0);
      }
   }
   else
   {
      // Non-normalised, just compute the power:
      result = pow(x, a);
   }
   if(result < tools::min_value<T>())
      return s0; // Safeguard: series can't cope with denorms.
   ibeta_series_t<T> s(a, b, x, result);
   boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
   result = boost::math::tools::sum_series(s, boost::math::policies::digits<T, Policy>(), max_iter, s0);
   policies::check_series_iterations("boost::math::ibeta<%1%>(%1%, %1%, %1%) in ibeta_series (with lanczos)", max_iter, pol);
   return result;
}
//
// Incomplete Beta series again, this time without Lanczos support:
//
template <class T, class Policy>
T ibeta_series(T a, T b, T x, T s0, const boost::math::lanczos::undefined_lanczos&, bool normalised, T* p_derivative, T y, const Policy& pol)
{
   BOOST_MATH_STD_USING

   T result;
   BOOST_ASSERT((p_derivative == 0) || normalised);

   if(normalised)
   {
      T prefix = 1;
      T c = a + b;

      // figure out integration limits for the gamma function:
      //T la = (std::max)(T(10), a);
      //T lb = (std::max)(T(10), b);
      //T lc = (std::max)(T(10), a+b);
      T la = a + 5;
      T lb = b + 5;
      T lc = a + b + 5;

      // calculate the gamma parts:
      T sa = detail::lower_gamma_series(a, la, pol) / a;
      sa += detail::upper_gamma_fraction(a, la, ::boost::math::policies::digits<T, Policy>());
      T sb = detail::lower_gamma_series(b, lb, pol) / b;
      sb += detail::upper_gamma_fraction(b, lb, ::boost::math::policies::digits<T, Policy>());
      T sc = detail::lower_gamma_series(c, lc, pol) / c;
      sc += detail::upper_gamma_fraction(c, lc, ::boost::math::policies::digits<T, Policy>());

      // and their combined power-terms:
      T b1 = (x * lc) / la;
      T b2 = lc/lb;
      T e1 = lc - la - lb;
      T lb1 = a * log(b1);
      T lb2 = b * log(b2);

      if((lb1 >= tools::log_max_value<T>())
         || (lb1 <= tools::log_min_value<T>())
         || (lb2 >= tools::log_max_value<T>())
         || (lb2 <= tools::log_min_value<T>())
         || (e1 >= tools::log_max_value<T>())
         || (e1 <= tools::log_min_value<T>()) )
      {
         T p = lb1 + lb2 - e1;
         result = exp(p);
      }
      else
      {
         result = pow(b1, a);
         if(a * b < lb * 10)
            result *= exp(b * boost::math::log1p(a / lb, pol));
         else
            result *= pow(b2, b);
         result /= exp(e1);
      }
      // and combine the results:
      result /= sa * sb / sc;

      if(p_derivative)
      {
         *p_derivative = result * pow(y, b);
         BOOST_ASSERT(*p_derivative >= 0);
      }
   }
   else
   {
      // Non-normalised, just compute the power:
      result = pow(x, a);
   }
   if(result < tools::min_value<T>())
      return s0; // Safeguard: series can't cope with denorms.
   ibeta_series_t<T> s(a, b, x, result);
   boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
   result = boost::math::tools::sum_series(s, boost::math::policies::digits<T, Policy>(), max_iter, s0);
   policies::check_series_iterations("boost::math::ibeta<%1%>(%1%, %1%, %1%) in ibeta_series (without lanczos)", max_iter, pol);
   return result;
}

//
// Continued fraction for the incomplete beta:
//
template <class T>
struct ibeta_fraction2_t
{
   typedef std::pair<T, T> result_type;

   ibeta_fraction2_t(T a_, T b_, T x_) : a(a_), b(b_), x(x_), m(0) {}

   result_type operator()()
   {
      T aN = (a + m - 1) * (a + b + m - 1) * m * (b - m) * x * x;
      T denom = (a + 2 * m - 1);
      aN /= denom * denom;

      T bN = m;
      bN += (m * (b - m) * x) / (a + 2*m - 1);
      bN += ((a + m) * (a - (a + b) * x + 1 + m *(2 - x))) / (a + 2*m + 1);

      ++m;

      return std::make_pair(aN, bN);
   }

private:
   T a, b, x;
   int m;
};
//
// Evaluate the incomplete beta via the continued fraction representation:
//
template <class T, class Policy>
inline T ibeta_fraction2(T a, T b, T x, T y, const Policy& pol, bool normalised, T* p_derivative)
{
   typedef typename lanczos::lanczos<T, Policy>::type lanczos_type;
   BOOST_MATH_STD_USING
   T result = ibeta_power_terms(a, b, x, y, lanczos_type(), normalised, pol);
   if(p_derivative)
   {
      *p_derivative = result;
      BOOST_ASSERT(*p_derivative >= 0);
   }
   if(result == 0)
      return result;

   ibeta_fraction2_t<T> f(a, b, x);
   T fract = boost::math::tools::continued_fraction_b(f, boost::math::policies::digits<T, Policy>());
   return result / fract;
}
//
// Computes the difference between ibeta(a,b,x) and ibeta(a+k,b,x):
//
template <class T, class Policy>
T ibeta_a_step(T a, T b, T x, T y, int k, const Policy& pol, bool normalised, T* p_derivative)
{
   typedef typename lanczos::lanczos<T, Policy>::type lanczos_type;
   T prefix = ibeta_power_terms(a, b, x, y, lanczos_type(), normalised, pol);
   if(p_derivative)
   {
      *p_derivative = prefix;
      BOOST_ASSERT(*p_derivative >= 0);
   }
   prefix /= a;
   if(prefix == 0)
      return prefix;
   T sum = 1;
   T term = 1;
   // series summation from 0 to k-1:
   for(int i = 0; i < k-1; ++i)
   {
      term *= (a+b+i) * x / (a+i+1);
      sum += term;
   }
   prefix *= sum;

   return prefix;
}
//
// This function is only needed for the non-regular incomplete beta,
// it computes the delta in:
// beta(a,b,x) = prefix + delta * beta(a+k,b,x)
// it is currently only called for small k.
//
template <class T>
inline T rising_factorial_ratio(T a, T b, int k)
{
   // calculate:
   // (a)(a+1)(a+2)...(a+k-1)
   // _______________________
   // (b)(b+1)(b+2)...(b+k-1)

   // This is only called with small k, for large k
   // it is grossly inefficient, do not use outside it's
   // intended purpose!!!
   if(k == 0)
      return 1;
   T result = 1;
   for(int i = 0; i < k; ++i)
      result *= (a+i) / (b+i);
   return result;
}
//
// Routine for a > 15, b < 1
//
// Begin by figuring out how large our table of Pn's should be,
// quoted accuracies are "guestimates" based on empiracal observation.
// Note that the table size should never exceed the size of our
// tables of factorials.
//
template <class T>
struct Pn_size
{
   // This is likely to be enough for ~35-50 digit accuracy
   // but it's hard to quantify exactly:
   BOOST_STATIC_CONSTANT(unsigned, value = 50);
   BOOST_STATIC_ASSERT(::boost::math::max_factorial<T>::value >= 100);
};
template <>
struct Pn_size<float>
{
   BOOST_STATIC_CONSTANT(unsigned, value = 15); // ~8-15 digit accuracy
   BOOST_STATIC_ASSERT(::boost::math::max_factorial<float>::value >= 30);
};
template <>
struct Pn_size<double>
{
   BOOST_STATIC_CONSTANT(unsigned, value = 30); // 16-20 digit accuracy
   BOOST_STATIC_ASSERT(::boost::math::max_factorial<double>::value >= 60);
};
template <>
struct Pn_size<long double>
{
   BOOST_STATIC_CONSTANT(unsigned, value = 50); // ~35-50 digit accuracy
   BOOST_STATIC_ASSERT(::boost::math::max_factorial<long double>::value >= 100);
};

template <class T, class Policy>
T beta_small_b_large_a_series(T a, T b, T x, T y, T s0, T mult, const Policy& pol, bool normalised)
{
   typedef typename lanczos::lanczos<T, Policy>::type lanczos_type;
   BOOST_MATH_STD_USING
   //
   // This is DiDonato and Morris's BGRAT routine, see Eq's 9 through 9.6.
   //
   // Some values we'll need later, these are Eq 9.1:
   //
   T bm1 = b - 1;
   T t = a + bm1 / 2;
   T lx, u;
   if(y < 0.35)
      lx = boost::math::log1p(-y, pol);
   else
      lx = log(x);
   u = -t * lx;
   // and from from 9.2:
   T prefix;
   T h = regularised_gamma_prefix(b, u, pol, lanczos_type());
   if(h <= tools::min_value<T>())
      return s0;
   if(normalised)
   {
      prefix = h / boost::math::tgamma_delta_ratio(a, b, pol);
      prefix /= pow(t, b);
   }
   else
   {
      prefix = full_igamma_prefix(b, u, pol) / pow(t, b);
   }
   prefix *= mult;
   //
   // now we need the quantity Pn, unfortunatately this is computed
   // recursively, and requires a full history of all the previous values
   // so no choice but to declare a big table and hope it's big enough...
   //
   T p[ ::boost::math::detail::Pn_size<T>::value ] = { 1 };  // see 9.3.
   //
   // Now an initial value for J, see 9.6:
   //
   T j = boost::math::gamma_q(b, u, pol) / h;
   //
   // Now we can start to pull things together and evaluate the sum in Eq 9:
   //
   T sum = s0 + prefix * j;  // Value at N = 0
   // some variables we'll need:
   unsigned tnp1 = 1; // 2*N+1
   T lx2 = lx / 2;
   lx2 *= lx2;
   T lxp = 1;
   T t4 = 4 * t * t;
   T b2n = b;

   for(unsigned n = 1; n < sizeof(p)/sizeof(p[0]); ++n)
   {
      /*
      // debugging code, enable this if you want to determine whether
      // the table of Pn's is large enough...
      //
      static int max_count = 2;
      if(n > max_count)
      {
         max_count = n;
         std::cerr << "Max iterations in BGRAT was " << n << std::endl;
      }
      */
      //
      // begin by evaluating the next Pn from Eq 9.4:
      //
      tnp1 += 2;
      p[n] = 0;
      T mbn = b - n;
      unsigned tmp1 = 3;
      for(unsigned m = 1; m < n; ++m)
      {
         mbn = m * b - n;
         p[n] += mbn * p[n-m] / boost::math::unchecked_factorial<T>(tmp1);
         tmp1 += 2;
      }
      p[n] /= n;
      p[n] += bm1 / boost::math::unchecked_factorial<T>(tnp1);
      //
      // Now we want Jn from Jn-1 using Eq 9.6:
      //
      j = (b2n * (b2n + 1) * j + (u + b2n + 1) * lxp) / t4;
      lxp *= lx2;
      b2n += 2;
      //
      // pull it together with Eq 9:
      //
      T r = prefix * p[n] * j;
      sum += r;
      if(r > 1)
      {
         if(fabs(r) < fabs(tools::epsilon<T>() * sum))
            break;
      }
      else
      {
         if(fabs(r / tools::epsilon<T>()) < fabs(sum))
            break;
      }
   }
   return sum;
} // template <class T, class L>T beta_small_b_large_a_series(T a, T b, T x, T y, T s0, T mult, const L& l, bool normalised)

//
// For integer arguments we can relate the incomplete beta to the
// complement of the binomial distribution cdf and use this finite sum.
//
template <class T>
inline T binomial_ccdf(T n, T k, T x, T y)
{
   BOOST_MATH_STD_USING // ADL of std names
   T result = pow(x, n);
   T term = result;
   for(unsigned i = itrunc(n - 1); i > k; --i)
   {
      term *= ((i + 1) * y) / ((n - i) * x) ;
      result += term;
   }

   return result;
}


//
// The incomplete beta function implementation:
// This is just a big bunch of spagetti code to divide up the
// input range and select the right implementation method for
// each domain:
//
template <class T, class Policy>
T ibeta_imp(T a, T b, T x, const Policy& pol, bool inv, bool normalised, T* p_derivative)
{
   static const char* function = "boost::math::ibeta<%1%>(%1%, %1%, %1%)";
   typedef typename lanczos::lanczos<T, Policy>::type lanczos_type;
   BOOST_MATH_STD_USING // for ADL of std math functions.

   bool invert = inv;
   T fract;
   T y = 1 - x;

   BOOST_ASSERT((p_derivative == 0) || normalised);

   if(p_derivative)
      *p_derivative = -1; // value not set.

   if(normalised)
   {
      // extend to a few very special cases:
      if((a == 0) && (b != 0))
         return inv ? 0 : 1;
      else if(b == 0)
         return inv ? 1 : 0;
   }

   if(a <= 0)
      policies::raise_domain_error<T>(function, "The argument a to the incomplete beta function must be greater than zero (got a=%1%).", a, pol);
   if(b <= 0)
      policies::raise_domain_error<T>(function, "The argument b to the incomplete beta function must be greater than zero (got b=%1%).", b, pol);
   if((x < 0) || (x > 1))
      policies::raise_domain_error<T>(function, "Parameter x outside the range [0,1] in the incomplete beta function (got x=%1%).", x, pol);

   if(x == 0)
   {
      if(p_derivative)
      {
         *p_derivative = (a == 1) ? 1 : (a < 1) ? tools::max_value<T>() / 2 : tools::min_value<T>() * 2;
      }
      return (invert ? (normalised ? 1 : boost::math::beta(a, b, pol)) : 0);
   }
   if(x == 1)
   {
      if(p_derivative)
      {
         *p_derivative = (b == 1) ? 1 : (b < 1) ? tools::max_value<T>() / 2 : tools::min_value<T>() * 2;
      }
      return (invert == 0 ? (normalised ? 1 : boost::math::beta(a, b, pol)) : 0);
   }

   if((std::min)(a, b) <= 1)
   {
      if(x > 0.5)
      {
         std::swap(a, b);
         std::swap(x, y);
         invert = !invert;
      }
      if((std::max)(a, b) <= 1)
      {
         // Both a,b < 1:
         if((a >= (std::min)(T(0.2), b)) || (pow(x, a) <= 0.9))
         {
            if(!invert)
               fract = ibeta_series(a, b, x, T(0), lanczos_type(), normalised, p_derivative, y, pol);
            else
            {