boost/math/special_functions/detail/hypergeometric_1F1_recurrence.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_HYPERGEOMETRIC_1F1_RECURRENCE_HPP_ #define BOOST_HYPERGEOMETRIC_1F1_RECURRENCE_HPP_ #include <boost/math/special_functions/modf.hpp> #include <boost/math/special_functions/next.hpp> #include <boost/math/tools/recurrence.hpp> #include <boost/math/special_functions/detail/hypergeometric_pFq_checked_series.hpp> namespace boost { namespace math { namespace detail { // forward declaration for initial values template <class T, class Policy> inline T hypergeometric_1F1_imp(const T& a, const T& b, const T& z, const Policy& pol); template <class T, class Policy> inline T hypergeometric_1F1_imp(const T& a, const T& b, const T& z, const Policy& pol, long long& log_scaling); template <class T> struct hypergeometric_1F1_recurrence_a_coefficients { using result_type = boost::math::tuple<T, T, T>; hypergeometric_1F1_recurrence_a_coefficients(const T& a, const T& b, const T& z): a(a), b(b), z(z) { } hypergeometric_1F1_recurrence_a_coefficients(const hypergeometric_1F1_recurrence_a_coefficients&) = default; hypergeometric_1F1_recurrence_a_coefficients operator=(const hypergeometric_1F1_recurrence_a_coefficients&) = delete; result_type operator()(std::intmax_t i) const { const T ai = a + i; const T an = b - ai; const T bn = (2 * ai - b + z); const T cn = -ai; return boost::math::make_tuple(an, bn, cn); } private: const T a; const T b; const T z; }; template <class T> struct hypergeometric_1F1_recurrence_b_coefficients { using result_type = boost::math::tuple<T, T, T>; hypergeometric_1F1_recurrence_b_coefficients(const T& a, const T& b, const T& z): a(a), b(b), z(z) { } hypergeometric_1F1_recurrence_b_coefficients(const hypergeometric_1F1_recurrence_b_coefficients&) = default; hypergeometric_1F1_recurrence_b_coefficients& operator=(const hypergeometric_1F1_recurrence_b_coefficients&) = delete; result_type operator()(std::intmax_t i) const { const T bi = b + i; const T an = bi * (bi - 1); const T bn = bi * (1 - bi - z); const T cn = z * (bi - a); return boost::math::make_tuple(an, bn, cn); } private: const T a; const T b; const T z; }; // // for use when we're recursing to a small b: // template <class T> struct hypergeometric_1F1_recurrence_small_b_coefficients { using result_type = boost::math::tuple<T, T, T>; hypergeometric_1F1_recurrence_small_b_coefficients(const T& a, const T& b, const T& z, int N) : a(a), b(b), z(z), N(N) { } hypergeometric_1F1_recurrence_small_b_coefficients(const hypergeometric_1F1_recurrence_small_b_coefficients&) = default; hypergeometric_1F1_recurrence_small_b_coefficients operator=(const hypergeometric_1F1_recurrence_small_b_coefficients&) = delete; result_type operator()(std::intmax_t i) const { const T bi = b + (i + N); const T bi_minus_1 = b + (i + N - 1); const T an = bi * bi_minus_1; const T bn = bi * (-bi_minus_1 - z); const T cn = z * (bi - a); return boost::math::make_tuple(an, bn, cn); } private: const T a; const T b; const T z; int N; }; template <class T> struct hypergeometric_1F1_recurrence_a_and_b_coefficients { using result_type = boost::math::tuple<T, T, T>; hypergeometric_1F1_recurrence_a_and_b_coefficients(const T& a, const T& b, const T& z, int offset = 0): a(a), b(b), z(z), offset(offset) { } hypergeometric_1F1_recurrence_a_and_b_coefficients(const hypergeometric_1F1_recurrence_a_and_b_coefficients&) = default; hypergeometric_1F1_recurrence_a_and_b_coefficients operator=(const hypergeometric_1F1_recurrence_a_and_b_coefficients&) = delete; result_type operator()(std::intmax_t i) const { const T ai = a + (offset + i); const T bi = b + (offset + i); const T an = bi * (b + (offset + i - 1)); const T bn = bi * (z - (b + (offset + i - 1))); const T cn = -ai * z; return boost::math::make_tuple(an, bn, cn); } private: const T a; const T b; const T z; int offset; }; #if 0 // // These next few recurrence relations are archived for future reference, some of them are novel, though all // are trivially derived from the existing well known relations: // // Recurrence relation for double-stepping on both a and b: // - b(b-1)(b-2) / (2-b+z) M(a-2,b-2,z) + [b(a-1)z / (2-b+z) + b(1-b+z) + abz(b+1) /(b+1)(z-b)] M(a,b,z) - a(a+1)z^2 / (b+1)(z-b) M(a+2,b+2,z) // template <class T> struct hypergeometric_1F1_recurrence_2a_and_2b_coefficients { typedef boost::math::tuple<T, T, T> result_type; hypergeometric_1F1_recurrence_2a_and_2b_coefficients(const T& a, const T& b, const T& z, int offset = 0) : a(a), b(b), z(z), offset(offset) { } result_type operator()(std::intmax_t i) const { i *= 2; const T ai = a + (offset + i); const T bi = b + (offset + i); const T an = -bi * (b + (offset + i - 1)) * (b + (offset + i - 2)) / (-(b + (offset + i - 2)) + z); const T bn = bi * (a + (offset + i - 1)) * z / (z - (b + (offset + i - 2))) + bi * (z - (b + (offset + i - 1))) + ai * bi * z * (b + (offset + i + 1)) / ((b + (offset + i + 1)) * (z - bi)); const T cn = -ai * (a + (offset + i + 1)) * z * z / ((b + (offset + i + 1)) * (z - bi)); return boost::math::make_tuple(an, bn, cn); } private: const T a, b, z; int offset; hypergeometric_1F1_recurrence_2a_and_2b_coefficients operator=(const hypergeometric_1F1_recurrence_2a_and_2b_coefficients&); }; // // Recurrence relation for double-stepping on a: // -(b-a)(1 + b - a)/(2a-2-b+z)M(a-2,b,z) + [(b-a)(a-1)/(2a-2-b+z) + (2a-b+z) + a(b-a-1)/(2a+2-b+z)]M(a,b,z) -a(a+1)/(2a+2-b+z)M(a+2,b,z) // template <class T> struct hypergeometric_1F1_recurrence_2a_coefficients { typedef boost::math::tuple<T, T, T> result_type; hypergeometric_1F1_recurrence_2a_coefficients(const T& a, const T& b, const T& z, int offset = 0) : a(a), b(b), z(z), offset(offset) { } result_type operator()(std::intmax_t i) const { i *= 2; const T ai = a + (offset + i); // -(b-a)(1 + b - a)/(2a-2-b+z) const T an = -(b - ai) * (b - (a + (offset + i - 1))) / (2 * (a + (offset + i - 1)) - b + z); const T bn = (b - ai) * (a + (offset + i - 1)) / (2 * (a + (offset + i - 1)) - b + z) + (2 * ai - b + z) + ai * (b - (a + (offset + i + 1))) / (2 * (a + (offset + i + 1)) - b + z); const T cn = -ai * (a + (offset + i + 1)) / (2 * (a + (offset + i + 1)) - b + z); return boost::math::make_tuple(an, bn, cn); } private: const T a, b, z; int offset; hypergeometric_1F1_recurrence_2a_coefficients operator=(const hypergeometric_1F1_recurrence_2a_coefficients&); }; // // Recurrence relation for double-stepping on b: // b(b-1)^2(b-2)/((1-b)(2-b-z)) M(a,b-2,z) + [zb(b-1)(b-1-a)/((1-b)(2-b-z)) + b(1-b-z) + z(b-a)(b+1)b/((b+1)(b+z)) ] M(a,b,z) + z^2(b-a)(b+1-a)/((b+1)(b+z)) M(a,b+2,z) // template <class T> struct hypergeometric_1F1_recurrence_2b_coefficients { typedef boost::math::tuple<T, T, T> result_type; hypergeometric_1F1_recurrence_2b_coefficients(const T& a, const T& b, const T& z, int offset = 0) : a(a), b(b), z(z), offset(offset) { } result_type operator()(std::intmax_t i) const { i *= 2; const T bi = b + (offset + i); const T bi_m1 = b + (offset + i - 1); const T bi_p1 = b + (offset + i + 1); const T bi_m2 = b + (offset + i - 2); const T an = bi * (bi_m1) * (bi_m1) * (bi_m2) / (-bi_m1 * (-bi_m2 - z)); const T bn = z * bi * bi_m1 * (bi_m1 - a) / (-bi_m1 * (-bi_m2 - z)) + bi * (-bi_m1 - z) + z * (bi - a) * bi_p1 * bi / (bi_p1 * (bi + z)); const T cn = z * z * (bi - a) * (bi_p1 - a) / (bi_p1 * (bi + z)); return boost::math::make_tuple(an, bn, cn); } private: const T a, b, z; int offset; hypergeometric_1F1_recurrence_2b_coefficients operator=(const hypergeometric_1F1_recurrence_2b_coefficients&); }; // // Recurrence relation for a+ b-: // -z(b-a)(a-1-b)/(b(a-1+z)) M(a-1,b+1,z) + [(b-a)(a-1)b/(b(a-1+z)) + (2a-b+z) + a(b-a-1)/(a+z)] M(a,b,z) + a(1-b)/(a+z) M(a+1,b-1,z) // // This is potentially the most useful of these novel recurrences. // - - + - + template <class T> struct hypergeometric_1F1_recurrence_a_plus_b_minus_coefficients { typedef boost::math::tuple<T, T, T> result_type; hypergeometric_1F1_recurrence_a_plus_b_minus_coefficients(const T& a, const T& b, const T& z, int offset = 0) : a(a), b(b), z(z), offset(offset) { } result_type operator()(std::intmax_t i) const { const T ai = a + (offset + i); const T bi = b - (offset + i); const T an = -z * (bi - ai) * (ai - 1 - bi) / (bi * (ai - 1 + z)); const T bn = z * ((-1 / (ai + z) - 1 / (ai + z - 1)) * (bi + z - 1) + 3) + bi - 1; const T cn = ai * (1 - bi) / (ai + z); return boost::math::make_tuple(an, bn, cn); } private: const T a, b, z; int offset; hypergeometric_1F1_recurrence_a_plus_b_minus_coefficients operator=(const hypergeometric_1F1_recurrence_a_plus_b_minus_coefficients&); }; #endif template <class T, class Policy> inline T hypergeometric_1F1_backward_recurrence_for_negative_a(const T& a, const T& b, const T& z, const Policy& pol, const char* function, long long& log_scaling) { BOOST_MATH_STD_USING // modf, frexp, fabs, pow std::intmax_t integer_part = 0; T ak = modf(a, &integer_part); // // We need ak-1 positive to avoid infinite recursion below: // if (0 != ak) { ak += 2; integer_part -= 2; } if (ak - 1 == b) { // When ak - 1 == b are recursion coefficients dissappear to zero and // we end up with a NaN result. Reduce the recursion steps by 1 to // avoid this. We rely on |b| small and therefore no infinite recursion. ak -= 1; integer_part += 1; } if (-integer_part > static_cast<std::intmax_t>(policies::get_max_series_iterations<Policy>())) return policies::raise_evaluation_error<T>(function, "1F1 arguments sit in a range with a so negative that we have no evaluation method, got a = %1%", std::numeric_limits<T>::quiet_NaN(), pol); T first {}; T second {}; if(ak == 0) { first = 1; ak -= 1; second = 1 - z / b; if (fabs(second) < 0.5) second = (b - z) / b; // cancellation avoidance } else { long long scaling1 {}; long long scaling2 {}; first = detail::hypergeometric_1F1_imp(ak, b, z, pol, scaling1); ak -= 1; second = detail::hypergeometric_1F1_imp(ak, b, z, pol, scaling2); if (scaling1 != scaling2) { second *= exp(T(scaling2 - scaling1)); } log_scaling += scaling1; } ++integer_part; detail::hypergeometric_1F1_recurrence_a_coefficients<T> s(ak, b, z); return tools::apply_recurrence_relation_backward(s, static_cast<unsigned int>(std::abs(integer_part)), first, second, &log_scaling); } template <class T, class Policy> T hypergeometric_1F1_backwards_recursion_on_b_for_negative_a(const T& a, const T& b, const T& z, const Policy& pol, const char*, long long& log_scaling) { using std::swap; BOOST_MATH_STD_USING // modf, frexp, fabs, pow // // We compute // // M[a + a_shift, b + b_shift; z] // // and recurse backwards on a and b down to // // M[a, b, z] // // With a + a_shift > 1 and b + b_shift > z // // There are 3 distinct regions to ensure stability during the recursions: // // a > 0 : stable for backwards on a // a < 0, b > 0 : stable for backwards on a and b // a < 0, b < 0 : stable for backwards on b (as long as |b| is small). // // We could simplify things by ignoring the middle region, but it's more efficient // to recurse on a and b together when we can. // BOOST_MATH_ASSERT(a < -1); // Not tested nor taken for -1 < a < 0 int b_shift = itrunc(z - b) + 2; int a_shift = itrunc(-a); if (a + a_shift != 0) { a_shift += 2; } // // If the shifts are so large that we would throw an evaluation_error, try the series instead, // even though this will almost certainly throw as well: // if (b_shift > static_cast<std::intmax_t>(boost::math::policies::get_max_series_iterations<Policy>())) return hypergeometric_1F1_checked_series_impl(a, b, z, pol, log_scaling); if (a_shift > static_cast<std::intmax_t>(boost::math::policies::get_max_series_iterations<Policy>())) return hypergeometric_1F1_checked_series_impl(a, b, z, pol, log_scaling); int a_b_shift = b < 0 ? itrunc(b + b_shift) : b_shift; // The max we can shift on a and b together int leading_a_shift = (std::min)(3, a_shift); // Just enough to make a negative if (a_b_shift > a_shift - 3) { a_b_shift = a_shift < 3 ? 0 : a_shift - 3; } else { // Need to ensure that leading_a_shift is large enough that a will reach it's target // after the first 2 phases (-,0) and (-,-) are over: leading_a_shift = a_shift - a_b_shift; } int trailing_b_shift = b_shift - a_b_shift; if (a_b_shift < 5) { // Might as well do things in two steps rather than 3: if (a_b_shift > 0) { leading_a_shift += a_b_shift; trailing_b_shift += a_b_shift; } a_b_shift = 0; --leading_a_shift; } BOOST_MATH_ASSERT(leading_a_shift > 1); BOOST_MATH_ASSERT(a_b_shift + leading_a_shift + (a_b_shift == 0 ? 1 : 0) == a_shift); BOOST_MATH_ASSERT(a_b_shift + trailing_b_shift == b_shift); if ((trailing_b_shift == 0) && (fabs(b) < 0.5) && a_b_shift) { // Better to have the final recursion on b alone, otherwise we lose precision when b is very small: int diff = (std::min)(a_b_shift, 3); a_b_shift -= diff; leading_a_shift += diff; trailing_b_shift += diff; } T first {}; T second {}; long long scale1 {}; long long scale2 {}; first = boost::math::detail::hypergeometric_1F1_imp(T(a + a_shift), T(b + b_shift), z, pol, scale1); // // It would be good to compute "second" from first and the ratio - unfortunately we are right on the cusp // recursion on a switching from stable backwards to stable forwards behaviour and so this is not possible here. // second = boost::math::detail::hypergeometric_1F1_imp(T(a + a_shift - 1), T(b + b_shift), z, pol, scale2); if (scale1 != scale2) second *= exp(T(scale2 - scale1)); log_scaling += scale1; // // Now we have [a + a_shift, b + b_shift, z] and [a + a_shift - 1, b + b_shift, z] // and want to recurse until [a + a_shift - leading_a_shift, b + b_shift, z] and [a + a_shift - leadng_a_shift - 1, b + b_shift, z] // which is leading_a_shift -1 steps. // second = boost::math::tools::apply_recurrence_relation_backward( hypergeometric_1F1_recurrence_a_coefficients<T>(a + a_shift - 1, b + b_shift, z), leading_a_shift, first, second, &log_scaling, &first); if (a_b_shift) { // // Now we need to switch to an a+b shift so that we have: // [a + a_shift - leading_a_shift, b + b_shift, z] and [a + a_shift - leadng_a_shift - 1, b + b_shift - 1, z] // A&S 13.4.3 gives us what we need: // { // local a's and b's: T la = a + a_shift - leading_a_shift - 1; T lb = b + b_shift; second = ((1 + la - lb) * second - la * first) / (1 - lb); } // // Now apply a_b_shift - 1 recursions to get down to // [a + 1, b + trailing_b_shift + 1, z] and [a, b + trailing_b_shift, z] // second = boost::math::tools::apply_recurrence_relation_backward( hypergeometric_1F1_recurrence_a_and_b_coefficients<T>(a, b + b_shift - a_b_shift, z, a_b_shift - 1), a_b_shift - 1, first, second, &log_scaling, &first); // // Now we need to switch to a b shift, a different application of A&S 13.4.3 // will get us there, we leave "second" where it is, and move "first" sideways: // { T lb = b + trailing_b_shift + 1; first = (second * (lb - 1) - a * first) / -(1 + a - lb); } } else { // // We have M[a+1, b+b_shift, z] and M[a, b+b_shift, z] and need M[a, b+b_shift-1, z] for // recursion on b: A&S 13.4.3 gives us what we need. // T third = -(second * (1 + a - b - b_shift) - first * a) / (b + b_shift - 1); swap(first, second); swap(second, third); --trailing_b_shift; } // // Finish off by applying trailing_b_shift recursions: // if (trailing_b_shift) { second = boost::math::tools::apply_recurrence_relation_backward( hypergeometric_1F1_recurrence_small_b_coefficients<T>(a, b, z, trailing_b_shift), trailing_b_shift, first, second, &log_scaling); } return second; } } } } // namespaces #endif // BOOST_HYPERGEOMETRIC_1F1_RECURRENCE_HPP_