boost/math/special_functions/detail/hypergeometric_rational.hpp
///////////////////////////////////////////////////////////////////////////////
// Copyright 2014 Anton Bikineev
// Copyright 2014 Christopher Kormanyos
// Copyright 2014 John Maddock
// Copyright 2014 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_MATH_HYPERGEOMETRIC_RATIONAL_HPP
#define BOOST_MATH_HYPERGEOMETRIC_RATIONAL_HPP
#include <array>
namespace boost{ namespace math{ namespace detail{
// Luke: C ------- SUBROUTINE R1F1P(AP, CP, Z, A, B, N) ---------
// Luke: C --- RATIONAL APPROXIMATION OF 1F1( AP ; CP ; -Z ) ----
template <class T, class Policy>
inline T hypergeometric_1F1_rational(const T& ap, const T& cp, const T& zp, const Policy& )
{
BOOST_MATH_STD_USING
static const T zero = T(0), one = T(1), two = T(2), three = T(3);
// Luke: C ------------- INITIALIZATION -------------
const T z = -zp;
const T z2 = z / two;
T ct1 = ap * (z / cp);
T ct2 = z2 / (one + cp);
T xn3 = zero;
T xn2 = one;
T xn1 = two;
T xn0 = three;
T b1 = one;
T a1 = one;
T b2 = one + ((one + ap) * (z2 / cp));
T a2 = b2 - ct1;
T b3 = one + ((two + b2) * (((two + ap) / three) * ct2));
T a3 = b3 - ((one + ct2) * ct1);
ct1 = three;
const unsigned max_iterations = boost::math::policies::get_max_series_iterations<Policy>();
T a4 = T(0), b4 = T(0);
T result = T(0), prev_result = a3 / b3;
for (unsigned k = 2; k < max_iterations; ++k)
{
// Luke: C ----- CALCULATION OF THE MULTIPLIERS -----
// Luke: C ----------- FOR THE RECURSION ------------
ct2 = (z2 / ct1) / (cp + xn1);
const T g1 = one + (ct2 * (xn2 - ap));
ct2 *= ((ap + xn1) / (cp + xn2));
const T g2 = ct2 * ((cp - xn1) + (((ap + xn0) / (ct1 + two)) * z2));
const T g3 = ((ct2 * z2) * (((z2 / ct1) / (ct1 - two)) * ((ap + xn2)) / (cp + xn3))) * (ap - xn2);
// Luke: C ------- THE RECURRENCE RELATIONS ---------
// Luke: C ------------ ARE AS FOLLOWS --------------
b4 = (g1 * b3) + (g2 * b2) + (g3 * b1);
a4 = (g1 * a3) + (g2 * a2) + (g3 * a1);
prev_result = result;
result = a4 / b4;
// condition for interruption
if ((fabs(result) * boost::math::tools::epsilon<T>()) > fabs(result - prev_result) / fabs(result))
break;
b1 = b2; b2 = b3; b3 = b4;
a1 = a2; a2 = a3; a3 = a4;
xn3 = xn2;
xn2 = xn1;
xn1 = xn0;
xn0 += 1;
ct1 += two;
}
return result;
}
// Luke: C ----- SUBROUTINE R2F1P(AB, BP, CP, Z, A, B, N) -------
// Luke: C -- RATIONAL APPROXIMATION OF 2F1( AB , BP; CP ; -Z ) -
template <class T, class Policy>
inline T hypergeometric_2F1_rational(const T& ap, const T& bp, const T& cp, const T& zp, const unsigned n, const Policy& )
{
BOOST_MATH_STD_USING
static const T one = T(1), two = T(2), three = T(3), four = T(4),
six = T(6), half_7 = T(3.5), half_3 = T(1.5), forth_3 = T(0.75);
// Luke: C ------------- INITIALIZATION -------------
const T z = -zp;
const T z2 = z / two;
T sabz = (ap + bp) * z;
const T ab = ap * bp;
const T abz = ab * z;
const T abz1 = z + (abz + sabz);
const T abz2 = abz1 + (sabz + (three * z));
const T cp1 = cp + one;
const T ct1 = cp1 + cp1;
T b1 = one;
T a1 = one;
T b2 = one + (abz1 / (cp + cp));
T a2 = b2 - (abz / cp);
T b3 = one + ((abz2 / ct1) * (one + (abz1 / ((-six) + (three * ct1)))));
T a3 = b3 - ((abz / cp) * (one + ((abz2 - abz1) / ct1)));
sabz /= four;
const T abz1_div_4 = abz1 / four;
const T cp1_inc = cp1 + one;
const T cp1_mul_cp1_inc = cp1 * cp1_inc;
std::array<T, 9u> d = {{
((half_7 - ab) * z2) - sabz,
abz1_div_4,
abz1_div_4 - (two * sabz),
cp1_inc,
cp1_mul_cp1_inc,
cp * cp1_mul_cp1_inc,
half_3,
forth_3,
forth_3 * z
}};
T xi = three;
T a4 = T(0), b4 = T(0);
for (unsigned k = 2; k < n; ++k)
{
// Luke: C ----- CALCULATION OF THE MULTIPLIERS -----
// Luke: C ----------- FOR THE RECURSION ------------
T g3 = (d[2] / d[7]) * (d[1] / d[5]);
d[1] += d[8] + sabz;
d[2] += d[8] - sabz;
g3 *= d[1] / d[6];
T g1 = one + (((d[1] + d[0]) / d[6]) / d[3]);
T g2 = (d[1] / d[4]) / d[6];
d[7] += two * d[6];
++d[6];
g2 *= cp1 - (xi + ((d[2] + d[0]) / d[6]));
// Luke: C ------- THE RECURRENCE RELATIONS ---------
// Luke: C ------------ ARE AS FOLLOWS --------------
b4 = (g1 * b3) + (g2 * b2) + (g3 * b1);
a4 = (g1 * a3) + (g2 * a2) + (g3 * a1);
b1 = b2; b2 = b3; b3 = b4;
a1 = a2; a2 = a3; a3 = a4;
d[8] += z2;
d[0] += two * d[8];
d[5] += three * d[4];
d[4] += two * d[3];
++d[3];
++xi;
}
return a4 / b4;
}
} } } // namespaces
#endif // BOOST_MATH_HYPERGEOMETRIC_RATIONAL_HPP