boost/math/special_functions/detail/hypergeometric_1F1_bessel.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_1F1_BESSEL_HPP #define BOOST_MATH_HYPERGEOMETRIC_1F1_BESSEL_HPP #include <boost/math/tools/series.hpp> #include <boost/math/special_functions/bessel.hpp> #include <boost/math/special_functions/laguerre.hpp> #include <boost/math/special_functions/detail/hypergeometric_pFq_checked_series.hpp> #include <boost/math/special_functions/bessel_iterators.hpp> namespace boost { namespace math { namespace detail { template <class T, class Policy> T hypergeometric_1F1_divergent_fallback(const T& a, const T& b, const T& z, const Policy& pol, long long& log_scaling); template <class T> bool hypergeometric_1F1_is_tricomi_viable_positive_b(const T& a, const T& b, const T& z) { BOOST_MATH_STD_USING if ((z < b) && (a > -50)) return false; // might as well fall through to recursion if (b <= 100) return true; // Even though we're in a reasonable domain for Tricomi's approximation, // the arguments to the Bessel functions may be so large that we can't // actually evaluate them: T x = sqrt(fabs(2 * z * b - 4 * a * z)); T v = b - 1; return log(boost::math::constants::e<T>() * x / (2 * v)) * v > tools::log_min_value<T>(); } // // Returns an arbitrarily small value compared to "target" for use as a seed // value for Bessel recurrences. Note that we'd better not make it too small // or underflow may occur resulting in either one of the terms in the // recurrence being zero, or else the result being zero. Using 1/epsilon // as a safety factor ensures that if we do underflow to zero, all of the digits // will have been cancelled out anyway: // template <class T> T arbitrary_small_value(const T& target) { using std::fabs; return (tools::min_value<T>() / tools::epsilon<T>()) * (fabs(target) > 1 ? target : 1); } template <class T, class Policy> struct hypergeometric_1F1_AS_13_3_7_tricomi_series { typedef T result_type; enum { cache_size = 64 }; hypergeometric_1F1_AS_13_3_7_tricomi_series(const T& a, const T& b, const T& z, const Policy& pol_) : A_minus_2(1), A_minus_1(0), A(b / 2), mult(z / 2), term(1), b_minus_1_plus_n(b - 1), bessel_arg((b / 2 - a) * z), two_a_minus_b(2 * a - b), pol(pol_), n(2) { BOOST_MATH_STD_USING term /= pow(fabs(bessel_arg), b_minus_1_plus_n / 2); mult /= sqrt(fabs(bessel_arg)); bessel_cache[cache_size - 1] = bessel_arg > 0 ? boost::math::cyl_bessel_j(b_minus_1_plus_n - 1, 2 * sqrt(bessel_arg), pol) : boost::math::cyl_bessel_i(b_minus_1_plus_n - 1, 2 * sqrt(-bessel_arg), pol); if (fabs(bessel_cache[cache_size - 1]) < tools::min_value<T>() / tools::epsilon<T>()) { // We get very limited precision due to rapid denormalisation/underflow of the Bessel values, raise an exception and try something else: policies::raise_evaluation_error("hypergeometric_1F1_AS_13_3_7_tricomi_series<%1%>", "Underflow in Bessel functions", bessel_cache[cache_size - 1], pol); } if ((term * bessel_cache[cache_size - 1] < tools::min_value<T>() / (tools::epsilon<T>() * tools::epsilon<T>())) || !(boost::math::isfinite)(term) || (!std::numeric_limits<T>::has_infinity && (fabs(term) > tools::max_value<T>()))) { term = -log(fabs(bessel_arg)) * b_minus_1_plus_n / 2; log_scale = lltrunc(term); term -= log_scale; term = exp(term); } else log_scale = 0; #ifndef BOOST_NO_CXX17_IF_CONSTEXPR if constexpr (std::numeric_limits<T>::has_infinity) { if (!(boost::math::isfinite)(bessel_cache[cache_size - 1])) policies::raise_evaluation_error("hypergeometric_1F1_AS_13_3_7_tricomi_series<%1%>", "Expected finite Bessel function result but got %1%", bessel_cache[cache_size - 1], pol); } else if ((boost::math::isnan)(bessel_cache[cache_size - 1]) || (fabs(bessel_cache[cache_size - 1]) >= tools::max_value<T>())) policies::raise_evaluation_error("hypergeometric_1F1_AS_13_3_7_tricomi_series<%1%>", "Expected finite Bessel function result but got %1%", bessel_cache[cache_size - 1], pol); #else if ((std::numeric_limits<T>::has_infinity && !(boost::math::isfinite)(bessel_cache[cache_size - 1])) || (!std::numeric_limits<T>::has_infinity && ((boost::math::isnan)(bessel_cache[cache_size - 1]) || (fabs(bessel_cache[cache_size - 1]) >= tools::max_value<T>())))) policies::raise_evaluation_error("hypergeometric_1F1_AS_13_3_7_tricomi_series<%1%>", "Expected finite Bessel function result but got %1%", bessel_cache[cache_size - 1], pol); #endif cache_offset = -cache_size; refill_cache(); } T operator()() { // // We return the n-2 term, and do 2 terms at once as every other term can be // very small (or zero) when b == 2a: // BOOST_MATH_STD_USING if(n - 2 - cache_offset >= cache_size) refill_cache(); T result = A_minus_2 * term * bessel_cache[n - 2 - cache_offset]; term *= mult; ++n; T A_next = ((b_minus_1_plus_n + 2) * A_minus_1 + two_a_minus_b * A_minus_2) / n; b_minus_1_plus_n += 1; A_minus_2 = A_minus_1; A_minus_1 = A; A = A_next; if (A_minus_2 != 0) { if (n - 2 - cache_offset >= cache_size) refill_cache(); result += A_minus_2 * term * bessel_cache[n - 2 - cache_offset]; } term *= mult; ++n; A_next = ((b_minus_1_plus_n + 2) * A_minus_1 + two_a_minus_b * A_minus_2) / n; b_minus_1_plus_n += 1; A_minus_2 = A_minus_1; A_minus_1 = A; A = A_next; return result; } long long scale()const { return log_scale; } private: T A_minus_2, A_minus_1, A, mult, term, b_minus_1_plus_n, bessel_arg, two_a_minus_b; std::array<T, cache_size> bessel_cache; const Policy& pol; int n, cache_offset; long long log_scale; hypergeometric_1F1_AS_13_3_7_tricomi_series operator=(const hypergeometric_1F1_AS_13_3_7_tricomi_series&); void refill_cache() { BOOST_MATH_STD_USING // // We don't calculate a new bessel I/J value: instead start our iterator off // with an arbitrary small value, then when we get back to the last value in the previous cache // calculate the ratio and use it to renormalise all the new values. This is more efficient, but // also avoids problems with J_v(x) or I_v(x) underflowing to zero. // cache_offset += cache_size; T last_value = bessel_cache.back(); T ratio; if (bessel_arg > 0) { // // We will be calculating Bessel J. // We need a different recurrence strategy for positive and negative orders: // if (b_minus_1_plus_n > 0) { bessel_j_backwards_iterator<T, Policy> i(b_minus_1_plus_n + (int)cache_size - 1, 2 * sqrt(bessel_arg), arbitrary_small_value(last_value)); for (int j = cache_size - 1; j >= 0; --j, ++i) { bessel_cache[j] = *i; // // Depending on the value of bessel_arg, the values stored in the cache can grow so // large as to overflow, if that looks likely then we need to rescale all the // existing terms (most of which will then underflow to zero). In this situation // it's likely that our series will only need 1 or 2 terms of the series but we // can't be sure of that: // if ((j < cache_size - 2) && (tools::max_value<T>() / fabs(64 * bessel_cache[j] / bessel_cache[j + 1]) < fabs(bessel_cache[j]))) { T rescale = pow(fabs(bessel_cache[j] / bessel_cache[j + 1]), j + 1) * 2; if (!((boost::math::isfinite)(rescale))) rescale = tools::max_value<T>(); for (int k = j; k < cache_size; ++k) bessel_cache[k] /= rescale; bessel_j_backwards_iterator<T, Policy> ti(b_minus_1_plus_n + j, 2 * sqrt(bessel_arg), bessel_cache[j + 1], bessel_cache[j]); i = ti; } } ratio = last_value / *i; } else { // // Negative order is difficult: the J_v(x) recurrence relations are unstable // *in both directions* for v < 0, except as v -> -INF when we have // J_-v(x) ~= -sin(pi.v)Y_v(x). // For small v what we can do is compute every other Bessel function and // then fill in the gaps using the recurrence relation. This *is* stable // provided that v is not so negative that the above approximation holds. // bessel_cache[1] = cyl_bessel_j(b_minus_1_plus_n + 1, 2 * sqrt(bessel_arg), pol); bessel_cache[0] = (last_value + bessel_cache[1]) / (b_minus_1_plus_n / sqrt(bessel_arg)); int pos = 2; while ((pos < cache_size - 1) && (b_minus_1_plus_n + pos < 0)) { bessel_cache[pos + 1] = cyl_bessel_j(b_minus_1_plus_n + pos + 1, 2 * sqrt(bessel_arg), pol); bessel_cache[pos] = (bessel_cache[pos-1] + bessel_cache[pos+1]) / ((b_minus_1_plus_n + pos) / sqrt(bessel_arg)); pos += 2; } if (pos < cache_size) { // // We have crossed over into the region where backward recursion is the stable direction // start from arbitrary value and recurse down to "pos" and normalise: // bessel_j_backwards_iterator<T, Policy> i2(b_minus_1_plus_n + (int)cache_size - 1, 2 * sqrt(bessel_arg), arbitrary_small_value(bessel_cache[pos-1])); for (int loc = cache_size - 1; loc >= pos; --loc) bessel_cache[loc] = *i2++; ratio = bessel_cache[pos - 1] / *i2; // // Sanity check, if we normalised to an unusually small value then it was likely // to be near a root and the calculated ratio is garbage, if so perform one // more J_v(x) evaluation at position and normalise again: // if (fabs(bessel_cache[pos] * ratio / bessel_cache[pos - 1]) > 5) ratio = cyl_bessel_j(b_minus_1_plus_n + pos, 2 * sqrt(bessel_arg), pol) / bessel_cache[pos]; while (pos < cache_size) bessel_cache[pos++] *= ratio; } ratio = 1; } } else { // // Bessel I. // We need a different recurrence strategy for positive and negative orders: // if (b_minus_1_plus_n > 0) { bessel_i_backwards_iterator<T, Policy> i(b_minus_1_plus_n + (int)cache_size - 1, 2 * sqrt(-bessel_arg), arbitrary_small_value(last_value)); for (int j = cache_size - 1; j >= 0; --j, ++i) { bessel_cache[j] = *i; // // Depending on the value of bessel_arg, the values stored in the cache can grow so // large as to overflow, if that looks likely then we need to rescale all the // existing terms (most of which will then underflow to zero). In this situation // it's likely that our series will only need 1 or 2 terms of the series but we // can't be sure of that: // if ((j < cache_size - 2) && (tools::max_value<T>() / fabs(64 * bessel_cache[j] / bessel_cache[j + 1]) < fabs(bessel_cache[j]))) { T rescale = pow(fabs(bessel_cache[j] / bessel_cache[j + 1]), j + 1) * 2; if (!((boost::math::isfinite)(rescale))) rescale = tools::max_value<T>(); for (int k = j; k < cache_size; ++k) bessel_cache[k] /= rescale; i = bessel_i_backwards_iterator<T, Policy>(b_minus_1_plus_n + j, 2 * sqrt(-bessel_arg), bessel_cache[j + 1], bessel_cache[j]); } } ratio = last_value / *i; } else { // // Forwards iteration is stable: // bessel_i_forwards_iterator<T, Policy> i(b_minus_1_plus_n, 2 * sqrt(-bessel_arg)); int pos = 0; while ((pos < cache_size) && (b_minus_1_plus_n + pos < 0.5)) { bessel_cache[pos++] = *i++; } if (pos < cache_size) { // // We have crossed over into the region where backward recursion is the stable direction // start from arbitrary value and recurse down to "pos" and normalise: // bessel_i_backwards_iterator<T, Policy> i2(b_minus_1_plus_n + (int)cache_size - 1, 2 * sqrt(-bessel_arg), arbitrary_small_value(last_value)); for (int loc = cache_size - 1; loc >= pos; --loc) bessel_cache[loc] = *i2++; ratio = bessel_cache[pos - 1] / *i2; while (pos < cache_size) bessel_cache[pos++] *= ratio; } ratio = 1; } } if(ratio != 1) for (auto j = bessel_cache.begin(); j != bessel_cache.end(); ++j) *j *= ratio; // // Very occasionally our normalisation fails because the normalisztion value // is sitting right on top of a root (or very close to it). When that happens // best to calculate a fresh Bessel evaluation and normalise again. // if (fabs(bessel_cache[0] / last_value) > 5) { ratio = (bessel_arg < 0 ? cyl_bessel_i(b_minus_1_plus_n, 2 * sqrt(-bessel_arg), pol) : cyl_bessel_j(b_minus_1_plus_n, 2 * sqrt(bessel_arg), pol)) / bessel_cache[0]; if (ratio != 1) for (auto j = bessel_cache.begin(); j != bessel_cache.end(); ++j) *j *= ratio; } } }; template <class T, class Policy> T hypergeometric_1F1_AS_13_3_7_tricomi(const T& a, const T& b, const T& z, const Policy& pol, long long& log_scale) { BOOST_MATH_STD_USING // // Works for a < 0, b < 0, z > 0. // // For convergence we require A * term to be converging otherwise we get // a divergent alternating series. It's actually really hard to analyse this // and the best purely heuristic policy we've found is // z < fabs((2 * a - b) / (sqrt(fabs(a)))) ; b > 0 or: // z < fabs((2 * a - b) / (sqrt(fabs(ab)))) ; b < 0 // T prefix(0); int prefix_sgn(0); bool use_logs = false; long long scale = 0; // // We can actually support the b == 2a case within here, but we haven't // as we appear never to get here in practice. Which means this get out // clause is a bit of defensive programming.... // if(b == 2 * a) return hypergeometric_1F1_divergent_fallback(a, b, z, pol, log_scale); try { prefix = boost::math::tgamma(b, pol); prefix *= exp(z / 2); } catch (const std::runtime_error&) { use_logs = true; } if (use_logs || (prefix == 0) || !(boost::math::isfinite)(prefix) || (!std::numeric_limits<T>::has_infinity && (fabs(prefix) >= tools::max_value<T>()))) { use_logs = true; prefix = boost::math::lgamma(b, &prefix_sgn, pol) + z / 2; scale = lltrunc(prefix); log_scale += scale; prefix -= scale; } T result(0); std::uintmax_t max_iter = boost::math::policies::get_max_series_iterations<Policy>(); bool retry = false; long long series_scale = 0; try { hypergeometric_1F1_AS_13_3_7_tricomi_series<T, Policy> s(a, b, z, pol); series_scale = s.scale(); log_scale += s.scale(); try { T norm = 0; result = 0; if((a < 0) && (b < 0)) result = boost::math::tools::checked_sum_series(s, boost::math::policies::get_epsilon<T, Policy>(), max_iter, result, norm); else result = boost::math::tools::sum_series(s, boost::math::policies::get_epsilon<T, Policy>(), max_iter, result); if (!(boost::math::isfinite)(result) || (result == 0) || (!std::numeric_limits<T>::has_infinity && (fabs(result) >= tools::max_value<T>()))) retry = true; if (norm / fabs(result) > 1 / boost::math::tools::root_epsilon<T>()) retry = true; // fatal cancellation } catch (const std::overflow_error&) { retry = true; } catch (const boost::math::evaluation_error&) { retry = true; } } catch (const std::overflow_error&) { log_scale -= scale; return hypergeometric_1F1_divergent_fallback(a, b, z, pol, log_scale); } catch (const boost::math::evaluation_error&) { log_scale -= scale; return hypergeometric_1F1_divergent_fallback(a, b, z, pol, log_scale); } if (retry) { log_scale -= scale; log_scale -= series_scale; return hypergeometric_1F1_divergent_fallback(a, b, z, pol, log_scale); } boost::math::policies::check_series_iterations<T>("boost::math::hypergeometric_1F1_AS_13_3_7<%1%>(%1%,%1%,%1%)", max_iter, pol); if (use_logs) { int sgn = boost::math::sign(result); prefix += log(fabs(result)); result = sgn * prefix_sgn * exp(prefix); } else { if ((fabs(result) > 1) && (fabs(prefix) > 1) && (tools::max_value<T>() / fabs(result) < fabs(prefix))) { // Overflow: scale = lltrunc(tools::log_max_value<T>()) - 10; log_scale += scale; result /= exp(T(scale)); } result *= prefix; } return result; } template <class T> struct cyl_bessel_i_large_x_sum { typedef T result_type; cyl_bessel_i_large_x_sum(const T& v, const T& x) : v(v), z(x), term(1), k(0) {} T operator()() { T result = term; ++k; term *= -(4 * v * v - (2 * k - 1) * (2 * k - 1)) / (8 * k * z); return result; } T v, z, term; int k; }; template <class T, class Policy> T cyl_bessel_i_large_x_scaled(const T& v, const T& x, long long& log_scaling, const Policy& pol) { BOOST_MATH_STD_USING cyl_bessel_i_large_x_sum<T> s(v, x); 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::cyl_bessel_i_large_x<%1%>(%1%,%1%)", max_iter, pol); long long scale = boost::math::lltrunc(x); log_scaling += scale; return result * exp(x - scale) / sqrt(boost::math::constants::two_pi<T>() * x); } template <class T, class Policy> struct hypergeometric_1F1_AS_13_3_6_series { typedef T result_type; enum { cache_size = 64 }; // // This series is only convergent/useful for a and b approximately equal // (ideally |a-b| < 1). The series can also go divergent after a while // when b < 0, which limits precision to around that of double. In that // situation we return 0 to terminate the series as otherwise the divergent // terms will destroy all the bits in our result before they do eventually // converge again. One important use case for this series is for z < 0 // and |a| << |b| so that either b-a == b or at least most of the digits in a // are lost in the subtraction. Note that while you can easily convince yourself // that the result should be unity when b-a == b, in fact this is not (quite) // the case for large z. // hypergeometric_1F1_AS_13_3_6_series(const T& a, const T& b, const T& z, const T& b_minus_a, const Policy& pol_) : b_minus_a(b_minus_a), half_z(z / 2), poch_1(2 * b_minus_a - 1), poch_2(b_minus_a - a), b_poch(b), term(1), last_result(1), sign(1), n(0), cache_offset(-cache_size), scale(0), pol(pol_) { bessel_i_cache[cache_size - 1] = half_z > tools::log_max_value<T>() ? cyl_bessel_i_large_x_scaled(T(b_minus_a - 1.5f), half_z, scale, pol) : boost::math::cyl_bessel_i(b_minus_a - 1.5f, half_z, pol); refill_cache(); } T operator()() { BOOST_MATH_STD_USING if(n - cache_offset >= cache_size) refill_cache(); T result = term * (b_minus_a - 0.5f + n) * sign * bessel_i_cache[n - cache_offset]; ++n; term *= poch_1; poch_1 = (n == 1) ? T(2 * b_minus_a) : T(poch_1 + 1); term *= poch_2; poch_2 += 1; term /= n; term /= b_poch; b_poch += 1; sign = -sign; if ((fabs(result) > fabs(last_result)) && (n > 100)) return 0; // series has gone divergent! last_result = result; return result; } long long scaling()const { return scale; } private: T b_minus_a, half_z, poch_1, poch_2, b_poch, term, last_result; int sign; int n, cache_offset; long long scale; const Policy& pol; std::array<T, cache_size> bessel_i_cache; void refill_cache() { BOOST_MATH_STD_USING // // We don't calculate a new bessel I value: instead start our iterator off // with an arbitrary small value, then when we get back to the last value in the previous cache // calculate the ratio and use it to renormalise all the values. This is more efficient, but // also avoids problems with I_v(x) underflowing to zero. // cache_offset += cache_size; T last_value = bessel_i_cache.back(); bessel_i_backwards_iterator<T, Policy> i(b_minus_a + cache_offset + (int)cache_size - 1.5f, half_z, tools::min_value<T>() * (fabs(last_value) > 1 ? last_value : 1) / tools::epsilon<T>()); for (int j = cache_size - 1; j >= 0; --j, ++i) { bessel_i_cache[j] = *i; // // Depending on the value of half_z, the values stored in the cache can grow so // large as to overflow, if that looks likely then we need to rescale all the // existing terms (most of which will then underflow to zero). In this situation // it's likely that our series will only need 1 or 2 terms of the series but we // can't be sure of that: // if((j < cache_size - 2) && (bessel_i_cache[j + 1] != 0) && (tools::max_value<T>() / fabs(64 * bessel_i_cache[j] / bessel_i_cache[j + 1]) < fabs(bessel_i_cache[j]))) { T rescale = pow(fabs(bessel_i_cache[j] / bessel_i_cache[j + 1]), j + 1) * 2; if (rescale > tools::max_value<T>()) rescale = tools::max_value<T>(); for (int k = j; k < cache_size; ++k) bessel_i_cache[k] /= rescale; i = bessel_i_backwards_iterator<T, Policy>(b_minus_a -0.5f + cache_offset + j, half_z, bessel_i_cache[j + 1], bessel_i_cache[j]); } } T ratio = last_value / *i; for (auto j = bessel_i_cache.begin(); j != bessel_i_cache.end(); ++j) *j *= ratio; } hypergeometric_1F1_AS_13_3_6_series(); hypergeometric_1F1_AS_13_3_6_series(const hypergeometric_1F1_AS_13_3_6_series&); hypergeometric_1F1_AS_13_3_6_series& operator=(const hypergeometric_1F1_AS_13_3_6_series&); }; template <class T, class Policy> T hypergeometric_1F1_AS_13_3_6(const T& a, const T& b, const T& z, 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(z, pol); log_scaling += scale; return exp(z - scale); } hypergeometric_1F1_AS_13_3_6_series<T, Policy> s(a, b, z, b_minus_a, pol); 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::hypergeometric_1F1_AS_13_3_6<%1%>(%1%,%1%,%1%)", max_iter, pol); result *= boost::math::tgamma(b_minus_a - 0.5f, pol) * pow(z / 4, -b_minus_a + 0.5f); long long scale = lltrunc(z / 2); log_scaling += scale; log_scaling += s.scaling(); result *= exp(z / 2 - scale); return result; } /****************************************************************************************************************/ // // The following are not used at present and are commented out for that reason: // /****************************************************************************************************************/ #if 0 template <class T, class Policy> struct hypergeometric_1F1_AS_13_3_8_series { // // TODO: store and cache Bessel function evaluations via backwards recurrence. // // The C term grows by at least an order of magnitude with each iteration, and // rate of growth is largely independent of the arguments. Free parameter h // seems to give accurate results for small values (almost zero) or h=1. // Convergence and accuracy, only when -a/z > 100, this appears to have no // or little benefit over 13.3.7 as it generally requires more iterations? // hypergeometric_1F1_AS_13_3_8_series(const T& a, const T& b, const T& z, const T& h, const Policy& pol_) : C_minus_2(1), C_minus_1(-b * h), C(b * (b + 1) * h * h / 2 - (2 * h - 1) * a / 2), bessel_arg(2 * sqrt(-a * z)), bessel_order(b - 1), power_term(std::pow(-a * z, (1 - b) / 2)), mult(z / std::sqrt(-a * z)), a_(a), b_(b), z_(z), h_(h), n(2), pol(pol_) { } T operator()() { // we actually return the n-2 term: T result = C_minus_2 * power_term * boost::math::cyl_bessel_j(bessel_order, bessel_arg, pol); bessel_order += 1; power_term *= mult; ++n; T C_next = ((1 - 2 * h_) * (n - 1) - b_ * h_) * C + ((1 - 2 * h_) * a_ - h_ * (h_ - 1) *(b_ + n - 2)) * C_minus_1 - h_ * (h_ - 1) * a_ * C_minus_2; C_next /= n; C_minus_2 = C_minus_1; C_minus_1 = C; C = C_next; return result; } T C, C_minus_1, C_minus_2, bessel_arg, bessel_order, power_term, mult, a_, b_, z_, h_; const Policy& pol; int n; typedef T result_type; }; template <class T, class Policy> T hypergeometric_1F1_AS_13_3_8(const T& a, const T& b, const T& z, const T& h, const Policy& pol) { BOOST_MATH_STD_USING T prefix = exp(h * z) * boost::math::tgamma(b); hypergeometric_1F1_AS_13_3_8_series<T, Policy> s(a, b, z, h, pol); 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::hypergeometric_1F1_AS_13_3_8<%1%>(%1%,%1%,%1%)", max_iter, pol); result *= prefix; return result; } // // This is the series from https://dlmf.nist.gov/13.11 // It appears to be unusable for a,z < 0, and for // b < 0 appears to never be better than the defining series // for 1F1. // template <class T, class Policy> struct hypergeometric_1f1_13_11_1_series { typedef T result_type; hypergeometric_1f1_13_11_1_series(const T& a, const T& b, const T& z, const Policy& pol_) : term(1), two_a_minus_1_plus_s(2 * a - 1), two_a_minus_b_plus_s(2 * a - b), b_plus_s(b), a_minus_half_plus_s(a - 0.5f), half_z(z / 2), s(0), pol(pol_) { } T operator()() { T result = term * a_minus_half_plus_s * boost::math::cyl_bessel_i(a_minus_half_plus_s, half_z, pol); term *= two_a_minus_1_plus_s * two_a_minus_b_plus_s / (b_plus_s * ++s); two_a_minus_1_plus_s += 1; a_minus_half_plus_s += 1; two_a_minus_b_plus_s += 1; b_plus_s += 1; return result; } T term, two_a_minus_1_plus_s, two_a_minus_b_plus_s, b_plus_s, a_minus_half_plus_s, half_z; long long s; const Policy& pol; }; template <class T, class Policy> T hypergeometric_1f1_13_11_1(const T& a, const T& b, const T& z, const Policy& pol, long long& log_scaling) { BOOST_MATH_STD_USING bool use_logs = false; T prefix; int prefix_sgn = 1; if (true/*(a < boost::math::max_factorial<T>::value) && (a > 0)*/) prefix = boost::math::tgamma(a - 0.5f, pol); else { prefix = boost::math::lgamma(a - 0.5f, &prefix_sgn, pol); use_logs = true; } if (use_logs) { prefix += z / 2; prefix += log(z / 4) * (0.5f - a); } else if (z > 0) { prefix *= pow(z / 4, 0.5f - a); prefix *= exp(z / 2); } else { prefix *= exp(z / 2); prefix *= pow(z / 4, 0.5f - a); } hypergeometric_1f1_13_11_1_series<T, Policy> s(a, b, z, pol); 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::hypergeometric_1f1_13_11_1<%1%>(%1%,%1%,%1%)", max_iter, pol); if (use_logs) { long long scaling = lltrunc(prefix); log_scaling += scaling; prefix -= scaling; result *= exp(prefix) * prefix_sgn; } else result *= prefix; return result; } #endif } } } // namespaces #endif // BOOST_MATH_HYPERGEOMETRIC_1F1_BESSEL_HPP