///////////////////////////////////////////////////////////////////////////////
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// Copyright 2014 Anton Bikineev
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// Copyright 2014 Christopher Kormanyos
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// Copyright 2014 John Maddock
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// Copyright 2014 Paul Bristow
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// Distributed under the Boost
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// Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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//
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#ifndef BOOST_MATH_HYPERGEOMETRIC_RATIONAL_HPP
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#define BOOST_MATH_HYPERGEOMETRIC_RATIONAL_HPP
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#include <boost/array.hpp>
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namespace boost{ namespace math{ namespace detail{
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// Luke: C ------- SUBROUTINE R1F1P(AP, CP, Z, A, B, N) ---------
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// Luke: C --- RATIONAL APPROXIMATION OF 1F1( AP ; CP ; -Z ) ----
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template <class T, class Policy>
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inline T hypergeometric_1F1_rational(const T& ap, const T& cp, const T& zp, const Policy& )
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{
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BOOST_MATH_STD_USING
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static const T zero = T(0), one = T(1), two = T(2), three = T(3);
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// Luke: C ------------- INITIALIZATION -------------
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const T z = -zp;
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const T z2 = z / two;
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T ct1 = ap * (z / cp);
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T ct2 = z2 / (one + cp);
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T xn3 = zero;
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T xn2 = one;
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T xn1 = two;
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T xn0 = three;
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T b1 = one;
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T a1 = one;
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T b2 = one + ((one + ap) * (z2 / cp));
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T a2 = b2 - ct1;
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T b3 = one + ((two + b2) * (((two + ap) / three) * ct2));
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T a3 = b3 - ((one + ct2) * ct1);
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ct1 = three;
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const unsigned max_iterations = boost::math::policies::get_max_series_iterations<Policy>();
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T a4 = T(0), b4 = T(0);
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T result = T(0), prev_result = a3 / b3;
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for (unsigned k = 2; k < max_iterations; ++k)
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{
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// Luke: C ----- CALCULATION OF THE MULTIPLIERS -----
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// Luke: C ----------- FOR THE RECURSION ------------
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ct2 = (z2 / ct1) / (cp + xn1);
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const T g1 = one + (ct2 * (xn2 - ap));
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ct2 *= ((ap + xn1) / (cp + xn2));
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const T g2 = ct2 * ((cp - xn1) + (((ap + xn0) / (ct1 + two)) * z2));
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const T g3 = ((ct2 * z2) * (((z2 / ct1) / (ct1 - two)) * ((ap + xn2)) / (cp + xn3))) * (ap - xn2);
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// Luke: C ------- THE RECURRENCE RELATIONS ---------
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// Luke: C ------------ ARE AS FOLLOWS --------------
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b4 = (g1 * b3) + (g2 * b2) + (g3 * b1);
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a4 = (g1 * a3) + (g2 * a2) + (g3 * a1);
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prev_result = result;
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result = a4 / b4;
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// condition for interruption
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if ((fabs(result) * boost::math::tools::epsilon<T>()) > fabs(result - prev_result) / fabs(result))
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break;
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b1 = b2; b2 = b3; b3 = b4;
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a1 = a2; a2 = a3; a3 = a4;
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xn3 = xn2;
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xn2 = xn1;
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xn1 = xn0;
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xn0 += 1;
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ct1 += two;
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}
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return result;
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}
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// Luke: C ----- SUBROUTINE R2F1P(AB, BP, CP, Z, A, B, N) -------
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// Luke: C -- RATIONAL APPROXIMATION OF 2F1( AB , BP; CP ; -Z ) -
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template <class T, class Policy>
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inline T hypergeometric_2F1_rational(const T& ap, const T& bp, const T& cp, const T& zp, const unsigned n, const Policy& )
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{
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BOOST_MATH_STD_USING
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static const T one = T(1), two = T(2), three = T(3), four = T(4),
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six = T(6), half_7 = T(3.5), half_3 = T(1.5), forth_3 = T(0.75);
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// Luke: C ------------- INITIALIZATION -------------
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const T z = -zp;
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const T z2 = z / two;
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T sabz = (ap + bp) * z;
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const T ab = ap * bp;
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const T abz = ab * z;
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const T abz1 = z + (abz + sabz);
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const T abz2 = abz1 + (sabz + (three * z));
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const T cp1 = cp + one;
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const T ct1 = cp1 + cp1;
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T b1 = one;
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T a1 = one;
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T b2 = one + (abz1 / (cp + cp));
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T a2 = b2 - (abz / cp);
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T b3 = one + ((abz2 / ct1) * (one + (abz1 / ((-six) + (three * ct1)))));
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T a3 = b3 - ((abz / cp) * (one + ((abz2 - abz1) / ct1)));
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sabz /= four;
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const T abz1_div_4 = abz1 / four;
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const T cp1_inc = cp1 + one;
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const T cp1_mul_cp1_inc = cp1 * cp1_inc;
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boost::array<T, 9u> d = {{
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((half_7 - ab) * z2) - sabz,
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abz1_div_4,
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abz1_div_4 - (two * sabz),
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cp1_inc,
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cp1_mul_cp1_inc,
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cp * cp1_mul_cp1_inc,
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half_3,
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forth_3,
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forth_3 * z
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}};
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T xi = three;
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T a4 = T(0), b4 = T(0);
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for (unsigned k = 2; k < n; ++k)
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{
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// Luke: C ----- CALCULATION OF THE MULTIPLIERS -----
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// Luke: C ----------- FOR THE RECURSION ------------
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T g3 = (d[2] / d[7]) * (d[1] / d[5]);
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d[1] += d[8] + sabz;
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d[2] += d[8] - sabz;
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g3 *= d[1] / d[6];
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T g1 = one + (((d[1] + d[0]) / d[6]) / d[3]);
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T g2 = (d[1] / d[4]) / d[6];
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d[7] += two * d[6];
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++d[6];
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g2 *= cp1 - (xi + ((d[2] + d[0]) / d[6]));
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// Luke: C ------- THE RECURRENCE RELATIONS ---------
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// Luke: C ------------ ARE AS FOLLOWS --------------
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b4 = (g1 * b3) + (g2 * b2) + (g3 * b1);
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a4 = (g1 * a3) + (g2 * a2) + (g3 * a1);
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b1 = b2; b2 = b3; b3 = b4;
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a1 = a2; a2 = a3; a3 = a4;
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d[8] += z2;
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d[0] += two * d[8];
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d[5] += three * d[4];
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d[4] += two * d[3];
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++d[3];
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++xi;
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}
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return a4 / b4;
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}
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} } } // namespaces
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#endif // BOOST_MATH_HYPERGEOMETRIC_RATIONAL_HPP
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