ibeta_inverse.hpp

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T temme_method_3_ibeta_inverse(T a, T b, T p, T q, const Policy& pol)
{
   BOOST_MATH_STD_USING // ADL of std names

   //
   // Begin by getting an initial approximation for the quantity
   // eta from the dominant part of the incomplete beta:
   //
   T eta0;
   if(p < q)
      eta0 = boost::math::gamma_q_inv(b, p, pol);
   else
      eta0 = boost::math::gamma_p_inv(b, q, pol);
   eta0 /= a;
   //
   // Define the variables and powers we'll need later on:
   //
   T mu = b / a;
   T w = sqrt(1 + mu);
   T w_2 = w * w;
   T w_3 = w_2 * w;
   T w_4 = w_2 * w_2;
   T w_5 = w_3 * w_2;
   T w_6 = w_3 * w_3;
   T w_7 = w_4 * w_3;
   T w_8 = w_4 * w_4;
   T w_9 = w_5 * w_4;
   T w_10 = w_5 * w_5;
   T d = eta0 - mu;
   T d_2 = d * d;
   T d_3 = d_2 * d;
   T d_4 = d_2 * d_2;
   T w1 = w + 1;
   T w1_2 = w1 * w1;
   T w1_3 = w1 * w1_2;
   T w1_4 = w1_2 * w1_2;
   //
   // Now we need to compute the purturbation error terms that
   // convert eta0 to eta, these are all polynomials of polynomials.
   // Probably these should be re-written to use tabulated data
   // (see examples above), but it's less of a win in this case as we
   // need to calculate the individual powers for the denominator terms
   // anyway, so we might as well use them for the numerator-polynomials
   // as well....
   //
   // Refer to p154-p155 for the details of these expansions:
   //
   T e1 = (w + 2) * (w - 1) / (3 * w);
   e1 += (w_3 + 9 * w_2 + 21 * w + 5) * d / (36 * w_2 * w1);
   e1 -= (w_4 - 13 * w_3 + 69 * w_2 + 167 * w + 46) * d_2 / (1620 * w1_2 * w_3);
   e1 -= (7 * w_5 + 21 * w_4 + 70 * w_3 + 26 * w_2 - 93 * w - 31) * d_3 / (6480 * w1_3 * w_4);
   e1 -= (75 * w_6 + 202 * w_5 + 188 * w_4 - 888 * w_3 - 1345 * w_2 + 118 * w + 138) * d_4 / (272160 * w1_4 * w_5);

   T e2 = (28 * w_4 + 131 * w_3 + 402 * w_2 + 581 * w + 208) * (w - 1) / (1620 * w1 * w_3);
   e2 -= (35 * w_6 - 154 * w_5 - 623 * w_4 - 1636 * w_3 - 3983 * w_2 - 3514 * w - 925) * d / (12960 * w1_2 * w_4);
   e2 -= (2132 * w_7 + 7915 * w_6 + 16821 * w_5 + 35066 * w_4 + 87490 * w_3 + 141183 * w_2 + 95993 * w + 21640) * d_2  / (816480 * w_5 * w1_3);
   e2 -= (11053 * w_8 + 53308 * w_7 + 117010 * w_6 + 163924 * w_5 + 116188 * w_4 - 258428 * w_3 - 677042 * w_2 - 481940 * w - 105497) * d_3 / (14696640 * w1_4 * w_6);

   T e3 = -((3592 * w_7 + 8375 * w_6 - 1323 * w_5 - 29198 * w_4 - 89578 * w_3 - 154413 * w_2 - 116063 * w - 29632) * (w - 1)) / (816480 * w_5 * w1_2);
   e3 -= (442043 * w_9 + 2054169 * w_8 + 3803094 * w_7 + 3470754 * w_6 + 2141568 * w_5 - 2393568 * w_4 - 19904934 * w_3 - 34714674 * w_2 - 23128299 * w - 5253353) * d / (146966400 * w_6 * w1_3);
   e3 -= (116932 * w_10 + 819281 * w_9 + 2378172 * w_8 + 4341330 * w_7 + 6806004 * w_6 + 10622748 * w_5 + 18739500 * w_4 + 30651894 * w_3 + 30869976 * w_2 + 15431867 * w + 2919016) * d_2 / (146966400 * w1_4 * w_7);
   //
   // Combine eta0 and the error terms to compute eta (Second eqaution p155):
   //
   T eta = eta0 + e1 / a + e2 / (a * a) + e3 / (a * a * a);
   //
   // Now we need to solve Eq 4.2 to obtain x.  For any given value of
   // eta there are two solutions to this equation, and since the distribtion
   // may be very skewed, these are not related by x ~ 1-x we used when
   // implementing section 3 above.  However we know that:
   //
   //  cross < x <= 1       ; iff eta < mu
   //          x == cross   ; iff eta == mu
   //     0 <= x < cross    ; iff eta > mu
   //
   // Where cross == 1 / (1 + mu)
   // Many thanks to Prof Temme for clarifying this point.
   //
   // Therefore we'll just jump straight into Newton iterations
   // to solve Eq 4.2 using these bounds, and simple bisection
   // as the first guess, in practice this converges pretty quickly
   // and we only need a few digits correct anyway:
   //
   if(eta <= 0)
      eta = tools::min_value<T>();
   T u = eta - mu * log(eta) + (1 + mu) * log(1 + mu) - mu;
   T cross = 1 / (1 + mu);
   T lower = eta < mu ? cross : 0;
   T upper = eta < mu ? 1 : cross;
   T x = (lower + upper) / 2;
   x = tools::newton_raphson_iterate(
      temme_root_finder<T>(u, mu), x, lower, upper, policies::digits<T, Policy>() / 2);
#ifdef BOOST_INSTRUMENT
   std::cout << "Estimating x with Temme method 3: " << x << std::endl;
#endif
   return x;
}

template <class T, class Policy>
struct ibeta_roots
{
   ibeta_roots(T _a, T _b, T t, bool inv = false)
      : a(_a), b(_b), target(t), invert(inv) {}

   std::tr1::tuple<T, T, T> operator()(T x)
   {
      BOOST_MATH_STD_USING // ADL of std names

      BOOST_FPU_EXCEPTION_GUARD
      
      T f1;
      T y = 1 - x;
      T f = ibeta_imp(a, b, x, Policy(), invert, true, &f1) - target;
      if(invert)
         f1 = -f1;
      if(y == 0)
         y = tools::min_value<T>() * 64;
      if(x == 0)
         x = tools::min_value<T>() * 64;

      T f2 = f1 * (-y * a + (b - 2) * x + 1);
      if(fabs(f2) < y * x * tools::max_value<T>())
         f2 /= (y * x);
      if(invert)
         f2 = -f2;

      // make sure we don't have a zero derivative:
      if(f1 == 0)
         f1 = (invert ? -1 : 1) * tools::min_value<T>() * 64;

      return std::tr1::make_tuple(f, f1, f2);
   }
private:
   T a, b, target;
   bool invert;
};

template <class T, class Policy>
T ibeta_inv_imp(T a, T b, T p, T q, const Policy& pol, T* py)
{
   BOOST_MATH_STD_USING  // For ADL of math functions.

   //
   // The flag invert is set to true if we swap a for b and p for q,
   // in which case the result has to be subtracted from 1:
   //
   bool invert = false;
   //
   // Depending upon which approximation method we use, we may end up
   // calculating either x or y initially (where y = 1-x):
   //
   T x = 0; // Set to a safe zero to avoid a
   // MSVC 2005 warning C4701: potentially uninitialized local variable 'x' used
   // But code inspection appears to ensure that x IS assigned whatever the code path.
   T y; 

   // For some of the methods we can put tighter bounds
   // on the result than simply [0,1]:
   //
   T lower = 0;
   T upper = 1;
   //
   // Student's T with b = 0.5 gets handled as a special case, swap
   // around if the arguments are in the "wrong" order:
   //
   if(a == 0.5f)
   {
      std::swap(a, b);
      std::swap(p, q);
      invert = !invert;
   }
   //
   // Handle trivial cases first:
   //
   if(q == 0)
   {
      if(py) *py = 0;
      return 1;
   }
   else if(p == 0)
   {
      if(py) *py = 1;
      return 0;
   }
   else if((a == 1) && (b == 1))
   {
      if(py) *py = 1 - p;
      return p;
   }
   else if((b == 0.5f) && (a >= 0.5f))
   {
      //
      // We have a Student's T distribution:
      x = find_ibeta_inv_from_t_dist(a, p, q, &y, pol);
   }
   else if(a + b > 5)
   {
      //
      // When a+b is large then we can use one of Prof Temme's
      // asymptotic expansions, begin by swapping things around
      // so that p < 0.5, we do this to avoid cancellations errors
      // when p is large.
      //
      if(p > 0.5)
      {
         std::swap(a, b);
         std::swap(p, q);
         invert = !invert;
      }
      T minv = (std::min)(a, b);
      T maxv = (std::max)(a, b);
      if((sqrt(minv) > (maxv - minv)) && (minv > 5))
      {
         //
         // When a and b differ by a small amount
         // the curve is quite symmetrical and we can use an error
         // function to approximate the inverse. This is the cheapest
         // of the three Temme expantions, and the calculated value
         // for x will never be much larger than p, so we don't have
         // to worry about cancellation as long as p is small.
         //
         x = temme_method_1_ibeta_inverse(a, b, p, pol);
         y = 1 - x;
      }
      else
      {
         T r = a + b;
         T theta = asin(sqrt(a / r));
         T lambda = minv / r;
         if((lambda >= 0.2) && (lambda <= 0.8) && (lambda >= 10))
         {
            //
            // The second error function case is the next cheapest
            // to use, it brakes down when the result is likely to be
            // very small, if a+b is also small, but we can use a
            // cheaper expansion there in any case.  As before x won't
            // be much larger than p, so as long as p is small we should
            // be free of cancellation error.
            //
            T ppa = pow(p, 1/a);
            if((ppa < 0.0025) && (a + b < 200))
            {
               x = ppa * pow(a * boost::math::beta(a, b, pol), 1/a);
            }
            else
               x = temme_method_2_ibeta_inverse(a, b, p, r, theta, pol);
            y = 1 - x;
         }
         else
         {
            //
            // If we get here then a and b are very different in magnitude
            // and we need to use the third of Temme's methods which
            // involves inverting the incomplete gamma.  This is much more
            // expensive than the other methods.  We also can only use this
            // method when a > b, which can lead to cancellation errors
            // if we really want y (as we will when x is close to 1), so
            // a different expansion is used in that case.
            //
            if(a < b)
            {
               std::swap(a, b);
               std::swap(p, q);
               invert = !invert;
            }
            //
            // Try and compute the easy way first:
            //
            T bet = 0;
            if(b < 2)
               bet = boost::math::beta(a, b, pol);
            if(bet != 0)
            {
               y = pow(b * q * bet, 1/b);
               x = 1 - y;
            }
            else 
               y = 1;
            if(y > 1e-5)
            {
               x = temme_method_3_ibeta_inverse(a, b, p, q, pol);
               y = 1 - x;
            }
         }
      }
   }
   else if((a < 1) && (b < 1))
   {
      //
      // Both a and b less than 1,
      // there is a point of inflection at xs:
      //
      T xs = (1 - a) / (2 - a - b);
      //
      // Now we need to ensure that we start our iteration from the
      // right side of the inflection point:
      //
      T fs = boost::math::ibeta(a, b, xs, pol) - p;
      if(fabs(fs) / p < tools::epsilon<T>() * 3)
      {
         // The result is at the point of inflection, best just return it:
         *py = invert ? xs : 1 - xs;
         return invert ? 1-xs : xs;
      }
      if(fs < 0)
      {
         std::swap(a, b);
         std::swap(p, q);
         invert = true;
         xs = 1 - xs;
      }
      T xg = pow(a * p * boost::math::beta(a, b, pol), 1/a);
      x = xg / (1 + xg);
      y = 1 / (1 + xg);