168 lines · plain
1///////////////////////////////////////////////////////////////////////////////2// Copyright 2014 Anton Bikineev3// Copyright 2014 Christopher Kormanyos4// Copyright 2014 John Maddock5// Copyright 2014 Paul Bristow6// Distributed under the Boost7// Software License, Version 1.0. (See accompanying file8// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)9//10#ifndef BOOST_MATH_HYPERGEOMETRIC_RATIONAL_HPP11#define BOOST_MATH_HYPERGEOMETRIC_RATIONAL_HPP12 13 #include <array>14 15 namespace boost{ namespace math{ namespace detail{16 17 // Luke: C ------- SUBROUTINE R1F1P(AP, CP, Z, A, B, N) ---------18 // Luke: C --- RATIONAL APPROXIMATION OF 1F1( AP ; CP ; -Z ) ----19 template <class T, class Policy>20 inline T hypergeometric_1F1_rational(const T& ap, const T& cp, const T& zp, const Policy& )21 {22 BOOST_MATH_STD_USING23 24 static const T zero = T(0), one = T(1), two = T(2), three = T(3);25 26 // Luke: C ------------- INITIALIZATION -------------27 const T z = -zp;28 const T z2 = z / two;29 30 T ct1 = ap * (z / cp);31 T ct2 = z2 / (one + cp);32 T xn3 = zero;33 T xn2 = one;34 T xn1 = two;35 T xn0 = three;36 37 T b1 = one;38 T a1 = one;39 T b2 = one + ((one + ap) * (z2 / cp));40 T a2 = b2 - ct1;41 T b3 = one + ((two + b2) * (((two + ap) / three) * ct2));42 T a3 = b3 - ((one + ct2) * ct1);43 ct1 = three;44 45 const unsigned max_iterations = boost::math::policies::get_max_series_iterations<Policy>();46 47 T a4 = T(0), b4 = T(0);48 T result = T(0), prev_result = a3 / b3;49 50 for (unsigned k = 2; k < max_iterations; ++k)51 {52 // Luke: C ----- CALCULATION OF THE MULTIPLIERS -----53 // Luke: C ----------- FOR THE RECURSION ------------54 ct2 = (z2 / ct1) / (cp + xn1);55 const T g1 = one + (ct2 * (xn2 - ap));56 ct2 *= ((ap + xn1) / (cp + xn2));57 const T g2 = ct2 * ((cp - xn1) + (((ap + xn0) / (ct1 + two)) * z2));58 const T g3 = ((ct2 * z2) * (((z2 / ct1) / (ct1 - two)) * ((ap + xn2)) / (cp + xn3))) * (ap - xn2);59 60 // Luke: C ------- THE RECURRENCE RELATIONS ---------61 // Luke: C ------------ ARE AS FOLLOWS --------------62 b4 = (g1 * b3) + (g2 * b2) + (g3 * b1);63 a4 = (g1 * a3) + (g2 * a2) + (g3 * a1);64 65 prev_result = result;66 result = a4 / b4;67 68 // condition for interruption69 if ((fabs(result) * boost::math::tools::epsilon<T>()) > fabs(result - prev_result) / fabs(result))70 break;71 72 b1 = b2; b2 = b3; b3 = b4;73 a1 = a2; a2 = a3; a3 = a4;74 75 xn3 = xn2; 76 xn2 = xn1; 77 xn1 = xn0; 78 xn0 += 1;79 ct1 += two;80 }81 82 return result;83 }84 85 // Luke: C ----- SUBROUTINE R2F1P(AB, BP, CP, Z, A, B, N) -------86 // Luke: C -- RATIONAL APPROXIMATION OF 2F1( AB , BP; CP ; -Z ) -87 template <class T, class Policy>88 inline T hypergeometric_2F1_rational(const T& ap, const T& bp, const T& cp, const T& zp, const unsigned n, const Policy& )89 {90 BOOST_MATH_STD_USING91 92 static const T one = T(1), two = T(2), three = T(3), four = T(4),93 six = T(6), half_7 = T(3.5), half_3 = T(1.5), forth_3 = T(0.75);94 95 // Luke: C ------------- INITIALIZATION -------------96 const T z = -zp;97 const T z2 = z / two;98 99 T sabz = (ap + bp) * z;100 const T ab = ap * bp;101 const T abz = ab * z;102 const T abz1 = z + (abz + sabz);103 const T abz2 = abz1 + (sabz + (three * z));104 const T cp1 = cp + one;105 const T ct1 = cp1 + cp1;106 107 T b1 = one;108 T a1 = one;109 T b2 = one + (abz1 / (cp + cp));110 T a2 = b2 - (abz / cp);111 T b3 = one + ((abz2 / ct1) * (one + (abz1 / ((-six) + (three * ct1)))));112 T a3 = b3 - ((abz / cp) * (one + ((abz2 - abz1) / ct1)));113 sabz /= four;114 115 const T abz1_div_4 = abz1 / four;116 const T cp1_inc = cp1 + one;117 const T cp1_mul_cp1_inc = cp1 * cp1_inc;118 119 std::array<T, 9u> d = {{120 ((half_7 - ab) * z2) - sabz,121 abz1_div_4,122 abz1_div_4 - (two * sabz),123 cp1_inc,124 cp1_mul_cp1_inc,125 cp * cp1_mul_cp1_inc,126 half_3,127 forth_3,128 forth_3 * z129 }};130 131 T xi = three;132 T a4 = T(0), b4 = T(0);133 for (unsigned k = 2; k < n; ++k)134 {135 // Luke: C ----- CALCULATION OF THE MULTIPLIERS -----136 // Luke: C ----------- FOR THE RECURSION ------------137 T g3 = (d[2] / d[7]) * (d[1] / d[5]);138 d[1] += d[8] + sabz;139 d[2] += d[8] - sabz;140 g3 *= d[1] / d[6];141 T g1 = one + (((d[1] + d[0]) / d[6]) / d[3]);142 T g2 = (d[1] / d[4]) / d[6];143 d[7] += two * d[6];144 ++d[6];145 g2 *= cp1 - (xi + ((d[2] + d[0]) / d[6]));146 147 // Luke: C ------- THE RECURRENCE RELATIONS ---------148 // Luke: C ------------ ARE AS FOLLOWS --------------149 b4 = (g1 * b3) + (g2 * b2) + (g3 * b1);150 a4 = (g1 * a3) + (g2 * a2) + (g3 * a1);151 b1 = b2; b2 = b3; b3 = b4;152 a1 = a2; a2 = a3; a3 = a4;153 154 d[8] += z2;155 d[0] += two * d[8];156 d[5] += three * d[4];157 d[4] += two * d[3];158 ++d[3];159 ++xi;160 }161 162 return a4 / b4;163 }164 165 } } } // namespaces166 167#endif // BOOST_MATH_HYPERGEOMETRIC_RATIONAL_HPP168