boost/math/special_functions/detail/hypergeometric_1F1_large_abz.hpp
/////////////////////////////////////////////////////////////////////////////// // Copyright 2018 John Maddock // 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_LARGE_ABZ_HPP_ #define BOOST_HYPERGEOMETRIC_1F1_LARGE_ABZ_HPP_ #include <boost/math/special_functions/detail/hypergeometric_1F1_bessel.hpp> #include <boost/math/special_functions/detail/hypergeometric_series.hpp> #include <boost/math/special_functions/gamma.hpp> #include <boost/math/special_functions/trunc.hpp> namespace boost { namespace math { namespace detail { template <class T> inline bool is_negative_integer(const T& x) { using std::floor; return (x <= 0) && (floor(x) == x); } template <class T, class Policy> struct hypergeometric_1F1_igamma_series { enum{ cache_size = 64 }; typedef T result_type; hypergeometric_1F1_igamma_series(const T& alpha, const T& delta, const T& x, const Policy& pol) : delta_poch(-delta), alpha_poch(alpha), x(x), k(0), cache_offset(0), pol(pol) { BOOST_MATH_STD_USING T log_term = log(x) * -alpha; log_scaling = lltrunc(log_term - 3 - boost::math::tools::log_min_value<T>() / 50); term = exp(log_term - log_scaling); refill_cache(); } T operator()() { if (k - cache_offset >= cache_size) { cache_offset += cache_size; refill_cache(); } T result = term * gamma_cache[k - cache_offset]; term *= delta_poch * alpha_poch / (++k * x); delta_poch += 1; alpha_poch += 1; return result; } void refill_cache() { typedef typename lanczos::lanczos<T, Policy>::type lanczos_type; gamma_cache[cache_size - 1] = boost::math::gamma_p(alpha_poch + ((int)cache_size - 1), x, pol); for (int i = cache_size - 1; i > 0; --i) { gamma_cache[i - 1] = gamma_cache[i] >= 1 ? T(1) : T(gamma_cache[i] + regularised_gamma_prefix(T(alpha_poch + (i - 1)), x, pol, lanczos_type()) / (alpha_poch + (i - 1))); } } T delta_poch, alpha_poch, x, term; T gamma_cache[cache_size]; int k; long long log_scaling; int cache_offset; Policy pol; }; template <class T, class Policy> T hypergeometric_1F1_igamma(const T& a, const T& b, const T& x, const T& b_minus_a, const Policy& pol, long long& log_scaling) { BOOST_MATH_STD_USING if (b_minus_a == 0) { // special case: M(a,a,z) == exp(z) long long scale = lltrunc(x, pol); log_scaling += scale; return exp(x - scale); } hypergeometric_1F1_igamma_series<T, Policy> s(b_minus_a, a - 1, x, pol); log_scaling += s.log_scaling; std::uintmax_t max_iter = boost::math::policies::get_max_series_iterations<Policy>(); T result = boost::math::tools::sum_series(s, boost::math::policies::get_epsilon<T, Policy>(), max_iter); boost::math::policies::check_series_iterations<T>("boost::math::tgamma<%1%>(%1%,%1%)", max_iter, pol); T log_prefix = x + boost::math::lgamma(b, pol) - boost::math::lgamma(a, pol); long long scale = lltrunc(log_prefix); log_scaling += scale; return result * exp(log_prefix - scale); } template <class T, class Policy> T hypergeometric_1F1_shift_on_a(T h, const T& a_local, const T& b_local, const T& x, int a_shift, const Policy& pol, long long& log_scaling) { BOOST_MATH_STD_USING T a = a_local + a_shift; if (a_shift == 0) return h; else if (a_shift > 0) { // // Forward recursion on a is stable as long as 2a-b+z > 0. // If 2a-b+z < 0 then backwards recursion is stable even though // the function may be strictly increasing with a. Potentially // we may need to split the recurrence in 2 sections - one using // forward recursion, and one backwards. // // We will get the next seed value from the ratio // on b as that's stable and quick to compute. // T crossover_a = (b_local - x) / 2; int crossover_shift = itrunc(crossover_a - a_local); if (crossover_shift > 1) { // // Forwards recursion will start off unstable, but may switch to the stable direction later. // Start in the middle and go in both directions: // if (crossover_shift > a_shift) crossover_shift = a_shift; crossover_a = a_local + crossover_shift; boost::math::detail::hypergeometric_1F1_recurrence_b_coefficients<T> b_coef(crossover_a, b_local, x); std::uintmax_t max_iter = boost::math::policies::get_max_series_iterations<Policy>(); T b_ratio = boost::math::tools::function_ratio_from_backwards_recurrence(b_coef, boost::math::policies::get_epsilon<T, Policy>(), max_iter); boost::math::policies::check_series_iterations<T>("boost::math::hypergeometric_1F1_large_abz<%1%>(%1%,%1%,%1%)", max_iter, pol); // // Convert to a ratio: // (1+a-b)M(a, b, z) - aM(a+1, b, z) + (b-1)M(a, b-1, z) = 0 // // hence: M(a+1,b,z) = ((1+a-b) / a) M(a,b,z) + ((b-1) / a) M(a,b,z)/b_ratio // T first = 1; T second = ((1 + crossover_a - b_local) / crossover_a) + ((b_local - 1) / crossover_a) / b_ratio; // // Recurse down to a_local, compare values and re-normalise first and second: // boost::math::detail::hypergeometric_1F1_recurrence_a_coefficients<T> a_coef(crossover_a, b_local, x); long long backwards_scale = 0; T comparitor = boost::math::tools::apply_recurrence_relation_backward(a_coef, crossover_shift, second, first, &backwards_scale); log_scaling -= backwards_scale; if ((h < 1) && (tools::max_value<T>() * h > comparitor)) { // Need to rescale! long long scale = lltrunc(log(h), pol) + 1; h *= exp(T(-scale)); log_scaling += scale; } comparitor /= h; first /= comparitor; second /= comparitor; // // Now we can recurse forwards for the rest of the range: // if (crossover_shift < a_shift) { boost::math::detail::hypergeometric_1F1_recurrence_a_coefficients<T> a_coef_2(crossover_a + 1, b_local, x); h = boost::math::tools::apply_recurrence_relation_forward(a_coef_2, a_shift - crossover_shift - 1, first, second, &log_scaling); } else h = first; } else { // // Regular case where forwards iteration is stable right from the start: // boost::math::detail::hypergeometric_1F1_recurrence_b_coefficients<T> b_coef(a_local, b_local, x); std::uintmax_t max_iter = boost::math::policies::get_max_series_iterations<Policy>(); T b_ratio = boost::math::tools::function_ratio_from_backwards_recurrence(b_coef, boost::math::policies::get_epsilon<T, Policy>(), max_iter); boost::math::policies::check_series_iterations<T>("boost::math::hypergeometric_1F1_large_abz<%1%>(%1%,%1%,%1%)", max_iter, pol); // // Convert to a ratio: // (1+a-b)M(a, b, z) - aM(a+1, b, z) + (b-1)M(a, b-1, z) = 0 // // hence: M(a+1,b,z) = ((1+a-b) / a) M(a,b,z) + ((b-1) / a) M(a,b,z)/b_ratio // T second = ((1 + a_local - b_local) / a_local) * h + ((b_local - 1) / a_local) * h / b_ratio; boost::math::detail::hypergeometric_1F1_recurrence_a_coefficients<T> a_coef(a_local + 1, b_local, x); h = boost::math::tools::apply_recurrence_relation_forward(a_coef, --a_shift, h, second, &log_scaling); } } else { // // We've calculated h for a larger value of a than we want, and need to recurse down. // However, only forward iteration is stable, so calculate the ratio, compare values, // and normalise. Note that we calculate the ratio on b and convert to a since the // direction is the minimal solution for N->+INF. // // IMPORTANT: this is only currently called for a > b and therefore forwards iteration // is the only stable direction as we will only iterate down until a ~ b, but we // will check this with an assert: // BOOST_MATH_ASSERT(2 * a - b_local + x > 0); boost::math::detail::hypergeometric_1F1_recurrence_b_coefficients<T> b_coef(a, b_local, x); std::uintmax_t max_iter = boost::math::policies::get_max_series_iterations<Policy>(); T b_ratio = boost::math::tools::function_ratio_from_backwards_recurrence(b_coef, boost::math::policies::get_epsilon<T, Policy>(), max_iter); boost::math::policies::check_series_iterations<T>("boost::math::hypergeometric_1F1_large_abz<%1%>(%1%,%1%,%1%)", max_iter, pol); // // Convert to a ratio: // (1+a-b)M(a, b, z) - aM(a+1, b, z) + (b-1)M(a, b-1, z) = 0 // // hence: M(a+1,b,z) = (1+a-b) / a M(a,b,z) + (b-1) / a M(a,b,z)/ (M(a,b,z)/M(a,b-1,z)) // T first = 1; // arbitrary value; T second = ((1 + a - b_local) / a) + ((b_local - 1) / a) * (1 / b_ratio); if (a_shift == -1) h = h / second; else { boost::math::detail::hypergeometric_1F1_recurrence_a_coefficients<T> a_coef(a + 1, b_local, x); T comparitor = boost::math::tools::apply_recurrence_relation_forward(a_coef, -(a_shift + 1), first, second); if (boost::math::tools::min_value<T>() * comparitor > h) { // Ooops, need to rescale h: long long rescale = lltrunc(log(fabs(h))); T scale = exp(T(-rescale)); h *= scale; log_scaling += rescale; } h = h / comparitor; } } return h; } template <class T, class Policy> T hypergeometric_1F1_shift_on_b(T h, const T& a, const T& b_local, const T& x, int b_shift, const Policy& pol, long long& log_scaling) { BOOST_MATH_STD_USING T b = b_local + b_shift; if (b_shift == 0) return h; else if (b_shift > 0) { // // We get here for b_shift > 0 when b > z. We can't use forward recursion on b - it's unstable, // so grab the ratio and work backwards to b - b_shift and normalise. // boost::math::detail::hypergeometric_1F1_recurrence_b_coefficients<T> b_coef(a, b, x); std::uintmax_t max_iter = boost::math::policies::get_max_series_iterations<Policy>(); T first = 1; // arbitrary value; T second = 1 / boost::math::tools::function_ratio_from_backwards_recurrence(b_coef, boost::math::policies::get_epsilon<T, Policy>(), max_iter); boost::math::policies::check_series_iterations<T>("boost::math::hypergeometric_1F1_large_abz<%1%>(%1%,%1%,%1%)", max_iter, pol); if (b_shift == 1) h = h / second; else { // // Reset coefficients and recurse: // boost::math::detail::hypergeometric_1F1_recurrence_b_coefficients<T> b_coef_2(a, b - 1, x); long long local_scale = 0; T comparitor = boost::math::tools::apply_recurrence_relation_backward(b_coef_2, --b_shift, first, second, &local_scale); log_scaling -= local_scale; if (boost::math::tools::min_value<T>() * comparitor > h) { // Ooops, need to rescale h: long long rescale = lltrunc(log(fabs(h))); T scale = exp(T(-rescale)); h *= scale; log_scaling += rescale; } h = h / comparitor; } } else { T second; if (a == b_local) { // recurrence is trivial for a == b and method of ratios fails as the c-term goes to zero: second = -b_local * (1 - b_local - x) * h / (b_local * (b_local - 1)); } else { BOOST_MATH_ASSERT(!is_negative_integer(b - a)); boost::math::detail::hypergeometric_1F1_recurrence_b_coefficients<T> b_coef(a, b_local, x); std::uintmax_t max_iter = boost::math::policies::get_max_series_iterations<Policy>(); second = h / boost::math::tools::function_ratio_from_backwards_recurrence(b_coef, boost::math::policies::get_epsilon<T, Policy>(), max_iter); boost::math::policies::check_series_iterations<T>("boost::math::hypergeometric_1F1_large_abz<%1%>(%1%,%1%,%1%)", max_iter, pol); } if (b_shift == -1) h = second; else { boost::math::detail::hypergeometric_1F1_recurrence_b_coefficients<T> b_coef_2(a, b_local - 1, x); h = boost::math::tools::apply_recurrence_relation_backward(b_coef_2, -(++b_shift), h, second, &log_scaling); } } return h; } template <class T, class Policy> T hypergeometric_1F1_large_igamma(const T& a, const T& b, const T& x, const T& b_minus_a, const Policy& pol, long long& log_scaling) { BOOST_MATH_STD_USING // // We need a < b < z in order to ensure there's at least a chance of convergence, // we can use recurrence relations to shift forwards on a+b or just a to achieve this, // for decent accuracy, try to keep 2b - 1 < a < 2b < z // int b_shift = b * 2 < x ? 0 : itrunc(b - x / 2); int a_shift = a > b - b_shift ? -itrunc(b - b_shift - a - 1) : -itrunc(b - b_shift - a); if (a_shift < 0) { // Might as well do all the shifting on b as scale a downwards: b_shift -= a_shift; a_shift = 0; } T a_local = a - a_shift; T b_local = b - b_shift; T b_minus_a_local = (a_shift == 0) && (b_shift == 0) ? b_minus_a : b_local - a_local; long long local_scaling = 0; T h = hypergeometric_1F1_igamma(a_local, b_local, x, b_minus_a_local, pol, local_scaling); log_scaling += local_scaling; // // Apply shifts on a and b as required: // h = hypergeometric_1F1_shift_on_a(h, a_local, b_local, x, a_shift, pol, log_scaling); h = hypergeometric_1F1_shift_on_b(h, a, b_local, x, b_shift, pol, log_scaling); return h; } template <class T, class Policy> T hypergeometric_1F1_large_series(const T& a, const T& b, const T& z, const Policy& pol, long long& log_scaling) { BOOST_MATH_STD_USING // // We make a small, and b > z: // int a_shift(0), b_shift(0); if (a * z > b) { a_shift = itrunc(a) - 5; b_shift = b < z ? itrunc(b - z - 1) : 0; } // // If a_shift is trivially small, there's really not much point in losing // accuracy to save a couple of iterations: // if (a_shift < 5) a_shift = 0; T a_local = a - a_shift; T b_local = b - b_shift; T h = boost::math::detail::hypergeometric_1F1_generic_series(a_local, b_local, z, pol, log_scaling, "hypergeometric_1F1_large_series<%1%>(a,b,z)"); // // Apply shifts on a and b as required: // if (a_shift && (a_local == 0)) { // // Shifting on a via method of ratios in hypergeometric_1F1_shift_on_a fails when // a_local == 0. However, the value of h calculated was trivial (unity), so // calculate a second 1F1 for a == 1 and recurse as normal: // long long scale = 0; T h2 = boost::math::detail::hypergeometric_1F1_generic_series(T(a_local + 1), b_local, z, pol, scale, "hypergeometric_1F1_large_series<%1%>(a,b,z)"); if (scale != log_scaling) { h2 *= exp(T(scale - log_scaling)); } boost::math::detail::hypergeometric_1F1_recurrence_a_coefficients<T> coef(a_local + 1, b_local, z); h = boost::math::tools::apply_recurrence_relation_forward(coef, a_shift - 1, h, h2, &log_scaling); h = hypergeometric_1F1_shift_on_b(h, a, b_local, z, b_shift, pol, log_scaling); } else { h = hypergeometric_1F1_shift_on_a(h, a_local, b_local, z, a_shift, pol, log_scaling); h = hypergeometric_1F1_shift_on_b(h, a, b_local, z, b_shift, pol, log_scaling); } return h; } template <class T, class Policy> T hypergeometric_1F1_large_13_3_6_series(const T& a, const T& b, const T& z, const Policy& pol, long long& log_scaling) { BOOST_MATH_STD_USING // // A&S 13.3.6 is good only when a ~ b, but isn't too fussy on the size of z. // So shift b to match a (b shifting seems to be more stable via method of ratios). // int b_shift = itrunc(b - a); if ((b_shift < 0) && (b - b_shift != a)) b_shift -= 1; T b_local = b - b_shift; if ((b_local - a - 0.5 <= 0) && (b_local != a)) { // Make sure b_local - a - 0.5 > 0 b_shift -= 1; b_local += 1; } T h = boost::math::detail::hypergeometric_1F1_AS_13_3_6(a, b_local, z, T(b_local - a), pol, log_scaling); return hypergeometric_1F1_shift_on_b(h, a, b_local, z, b_shift, pol, log_scaling); } template <class T, class Policy> T hypergeometric_1F1_large_abz(const T& a, const T& b, const T& z, const Policy& pol, long long& log_scaling) { BOOST_MATH_STD_USING // // This is the selection logic to pick the "best" method. // We have a,b,z >> 0 and need to compute the approximate cost of each method // and then select whichever wins out. // enum method { method_series = 0, method_shifted_series, method_gamma, method_bessel }; // // Cost of direct series, is the approx number of terms required until we hit convergence: // T current_cost = (sqrt(16 * z * (3 * a + z) + 9 * b * b - 24 * b * z) - 3 * b + 4 * z) / 6; method current_method = method_series; // // Cost of shifted series, is the number of recurrences required to move to a zone where // the series is convergent right from the start. // Note that recurrence relations fail for very small b, and too many recurrences on a // will completely destroy all our digits. // Also note that the method fails when b-a is a negative integer unless b is already // larger than z and thus does not need shifting. // T cost = a + ((b < z) ? T(z - b) : T(0)); if((b > 1) && (cost < current_cost) && ((b > z) || !is_negative_integer(b-a))) { current_method = method_shifted_series; current_cost = cost; } // // Cost for gamma function method is the number of recurrences required to move it // into a convergent zone, note that recurrence relations fail for very small b. // Also add on a fudge factor to account for the fact that this method is both // more expensive to compute (requires gamma functions), and less accurate than the // methods above: // T b_shift = fabs(b * 2 < z ? T(0) : T(b - z / 2)); T a_shift = fabs(a > b - b_shift ? T(-(b - b_shift - a - 1)) : T(-(b - b_shift - a))); cost = 1000 + b_shift + a_shift; if((b > 1) && (cost <= current_cost)) { current_method = method_gamma; current_cost = cost; } // // Cost for bessel approximation is the number of recurrences required to make a ~ b, // Note that recurrence relations fail for very small b. We also have issue with large // z: either overflow/numeric instability or else the series goes divergent. We seem to be // OK for z smaller than log_max_value<Quad> though, maybe we can stretch a little further // but that's not clear... // Also need to add on a fudge factor to the cost to account for the fact that we need // to calculate the Bessel functions, this is not quite as high as the gamma function // method above as this is generally more accurate and so preferred if the methods are close: // cost = 50 + fabs(b - a); if((b > 1) && (cost <= current_cost) && (z < tools::log_max_value<T>()) && (z < 11356) && (b - a != 0.5f)) { current_method = method_bessel; current_cost = cost; } switch (current_method) { case method_series: return detail::hypergeometric_1F1_generic_series(a, b, z, pol, log_scaling, "hypergeometric_1f1_large_abz<%1%>(a,b,z)"); case method_shifted_series: return detail::hypergeometric_1F1_large_series(a, b, z, pol, log_scaling); case method_gamma: return detail::hypergeometric_1F1_large_igamma(a, b, z, T(b - a), pol, log_scaling); case method_bessel: return detail::hypergeometric_1F1_large_13_3_6_series(a, b, z, pol, log_scaling); } return 0; // We don't get here! } } } } // namespaces #endif // BOOST_HYPERGEOMETRIC_1F1_LARGE_ABZ_HPP_