///////////////////////////////////////////////////////////////////////////////
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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_1F1_BESSEL_HPP
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#define BOOST_MATH_HYPERGEOMETRIC_1F1_BESSEL_HPP
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#include <boost/math/tools/series.hpp>
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#include <boost/math/special_functions/bessel.hpp>
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#include <boost/math/special_functions/laguerre.hpp>
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#include <boost/math/special_functions/detail/hypergeometric_pFq_checked_series.hpp>
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#include <boost/math/special_functions/bessel_iterators.hpp>
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namespace boost { namespace math { namespace detail {
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template <class T, class Policy>
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T hypergeometric_1F1_divergent_fallback(const T& a, const T& b, const T& z, const Policy& pol, int& log_scaling);
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template <class T>
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bool hypergeometric_1F1_is_tricomi_viable_positive_b(const T& a, const T& b, const T& z)
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{
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BOOST_MATH_STD_USING
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if ((z < b) && (a > -50))
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return false; // might as well fall through to recursion
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if (b <= 100)
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return true;
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// Even though we're in a reasonable domain for Tricomi's approximation,
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// the arguments to the Bessel functions may be so large that we can't
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// actually evaluate them:
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T x = sqrt(fabs(2 * z * b - 4 * a * z));
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T v = b - 1;
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return log(boost::math::constants::e<T>() * x / (2 * v)) * v > tools::log_min_value<T>();
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}
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//
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// Returns an arbitrarily small value compared to "target" for use as a seed
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// value for Bessel recurrences. Note that we'd better not make it too small
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// or underflow may occur resulting in either one of the terms in the
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// recurrence being zero, or else the result being zero. Using 1/epsilon
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// as a safety factor ensures that if we do underflow to zero, all of the digits
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// will have been cancelled out anyway:
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//
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template <class T>
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T arbitrary_small_value(const T& target)
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{
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using std::fabs;
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return (tools::min_value<T>() / tools::epsilon<T>()) * (fabs(target) > 1 ? target : 1);
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}
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template <class T, class Policy>
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struct hypergeometric_1F1_AS_13_3_7_tricomi_series
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{
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typedef T result_type;
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enum { cache_size = 64 };
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hypergeometric_1F1_AS_13_3_7_tricomi_series(const T& a, const T& b, const T& z, const Policy& pol_)
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: A_minus_2(1), A_minus_1(0), A(b / 2), mult(z / 2), term(1), b_minus_1_plus_n(b - 1),
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bessel_arg((b / 2 - a) * z),
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two_a_minus_b(2 * a - b), pol(pol_), n(2)
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{
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BOOST_MATH_STD_USING
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term /= pow(fabs(bessel_arg), b_minus_1_plus_n / 2);
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mult /= sqrt(fabs(bessel_arg));
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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);
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if (fabs(bessel_cache[cache_size - 1]) < tools::min_value<T>() / tools::epsilon<T>())
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{
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// We get very limited precision due to rapid denormalisation/underflow of the Bessel values, raise an exception and try something else:
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policies::raise_evaluation_error("hypergeometric_1F1_AS_13_3_7_tricomi_series<%1%>", "Underflow in Bessel functions", bessel_cache[cache_size - 1], pol);
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}
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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>())))
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{
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term = -log(fabs(bessel_arg)) * b_minus_1_plus_n / 2;
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log_scale = itrunc(term);
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term -= log_scale;
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term = exp(term);
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}
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else
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log_scale = 0;
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#ifndef BOOST_NO_CXX17_IF_CONSTEXPR
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if constexpr (std::numeric_limits<T>::has_infinity)
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{
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if (!(boost::math::isfinite)(bessel_cache[cache_size - 1]))
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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);
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}
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else
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if ((boost::math::isnan)(bessel_cache[cache_size - 1]) || (fabs(bessel_cache[cache_size - 1]) >= tools::max_value<T>()))
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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);
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#else
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if ((std::numeric_limits<T>::has_infinity && !(boost::math::isfinite)(bessel_cache[cache_size - 1]))
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|| (!std::numeric_limits<T>::has_infinity && ((boost::math::isnan)(bessel_cache[cache_size - 1]) || (fabs(bessel_cache[cache_size - 1]) >= tools::max_value<T>()))))
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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);
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#endif
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cache_offset = -cache_size;
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refill_cache();
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}
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T operator()()
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{
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//
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// We return the n-2 term, and do 2 terms at once as every other term can be
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// very small (or zero) when b == 2a:
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//
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BOOST_MATH_STD_USING
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if(n - 2 - cache_offset >= cache_size)
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refill_cache();
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T result = A_minus_2 * term * bessel_cache[n - 2 - cache_offset];
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term *= mult;
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++n;
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T A_next = ((b_minus_1_plus_n + 2) * A_minus_1 + two_a_minus_b * A_minus_2) / n;
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b_minus_1_plus_n += 1;
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A_minus_2 = A_minus_1;
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A_minus_1 = A;
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A = A_next;
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if (A_minus_2 != 0)
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{
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if (n - 2 - cache_offset >= cache_size)
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refill_cache();
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result += A_minus_2 * term * bessel_cache[n - 2 - cache_offset];
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}
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term *= mult;
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++n;
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A_next = ((b_minus_1_plus_n + 2) * A_minus_1 + two_a_minus_b * A_minus_2) / n;
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b_minus_1_plus_n += 1;
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A_minus_2 = A_minus_1;
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A_minus_1 = A;
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A = A_next;
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return result;
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}
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int scale()const
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{
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return log_scale;
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}
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private:
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T A_minus_2, A_minus_1, A, mult, term, b_minus_1_plus_n, bessel_arg, two_a_minus_b;
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std::array<T, cache_size> bessel_cache;
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const Policy& pol;
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int n, log_scale, cache_offset;
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hypergeometric_1F1_AS_13_3_7_tricomi_series operator=(const hypergeometric_1F1_AS_13_3_7_tricomi_series&);
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void refill_cache()
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{
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BOOST_MATH_STD_USING
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//
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// We don't calculate a new bessel I/J value: instead start our iterator off
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// with an arbitrary small value, then when we get back to the last value in the previous cache
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// calculate the ratio and use it to renormalise all the new values. This is more efficient, but
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// also avoids problems with J_v(x) or I_v(x) underflowing to zero.
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//
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cache_offset += cache_size;
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T last_value = bessel_cache.back();
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T ratio;
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if (bessel_arg > 0)
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{
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//
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// We will be calculating Bessel J.
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// We need a different recurrence strategy for positive and negative orders:
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//
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if (b_minus_1_plus_n > 0)
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{
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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));
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for (int j = cache_size - 1; j >= 0; --j, ++i)
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{
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bessel_cache[j] = *i;
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//
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// Depending on the value of bessel_arg, the values stored in the cache can grow so
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// large as to overflow, if that looks likely then we need to rescale all the
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// existing terms (most of which will then underflow to zero). In this situation
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// it's likely that our series will only need 1 or 2 terms of the series but we
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// can't be sure of that:
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//
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if ((j < cache_size - 2) && (tools::max_value<T>() / fabs(64 * bessel_cache[j] / bessel_cache[j + 1]) < fabs(bessel_cache[j])))
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{
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T rescale = pow(fabs(bessel_cache[j] / bessel_cache[j + 1]), j + 1) * 2;
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if (!((boost::math::isfinite)(rescale)))
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rescale = tools::max_value<T>();
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for (int k = j; k < cache_size; ++k)
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bessel_cache[k] /= rescale;
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bessel_j_backwards_iterator<T, Policy> ti(b_minus_1_plus_n + j, 2 * sqrt(bessel_arg), bessel_cache[j + 1], bessel_cache[j]);
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i = ti;
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}
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}
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ratio = last_value / *i;
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}
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else
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{
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//
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// Negative order is difficult: the J_v(x) recurrence relations are unstable
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// *in both directions* for v < 0, except as v -> -INF when we have
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// J_-v(x) ~= -sin(pi.v)Y_v(x).
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// For small v what we can do is compute every other Bessel function and
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// then fill in the gaps using the recurrence relation. This *is* stable
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// provided that v is not so negative that the above approximation holds.
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//
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bessel_cache[1] = cyl_bessel_j(b_minus_1_plus_n + 1, 2 * sqrt(bessel_arg), pol);
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bessel_cache[0] = (last_value + bessel_cache[1]) / (b_minus_1_plus_n / sqrt(bessel_arg));
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int pos = 2;
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while ((pos < cache_size - 1) && (b_minus_1_plus_n + pos < 0))
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{
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bessel_cache[pos + 1] = cyl_bessel_j(b_minus_1_plus_n + pos + 1, 2 * sqrt(bessel_arg), pol);
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bessel_cache[pos] = (bessel_cache[pos-1] + bessel_cache[pos+1]) / ((b_minus_1_plus_n + pos) / sqrt(bessel_arg));
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pos += 2;
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}
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if (pos < cache_size)
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{
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//
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// We have crossed over into the region where backward recursion is the stable direction
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// start from arbitrary value and recurse down to "pos" and normalise:
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//
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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]));
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for (int loc = cache_size - 1; loc >= pos; --loc)
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bessel_cache[loc] = *i2++;
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ratio = bessel_cache[pos - 1] / *i2;
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//
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// Sanity check, if we normalised to an unusually small value then it was likely
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// to be near a root and the calculated ratio is garbage, if so perform one
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// more J_v(x) evaluation at position and normalise again:
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//
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if (fabs(bessel_cache[pos] * ratio / bessel_cache[pos - 1]) > 5)
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ratio = cyl_bessel_j(b_minus_1_plus_n + pos, 2 * sqrt(bessel_arg), pol) / bessel_cache[pos];
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while (pos < cache_size)
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bessel_cache[pos++] *= ratio;
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}
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ratio = 1;
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}
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}
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else
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{
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//
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// Bessel I.
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// We need a different recurrence strategy for positive and negative orders:
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//
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if (b_minus_1_plus_n > 0)
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{
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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));
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for (int j = cache_size - 1; j >= 0; --j, ++i)
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{
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bessel_cache[j] = *i;
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//
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// Depending on the value of bessel_arg, the values stored in the cache can grow so
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// large as to overflow, if that looks likely then we need to rescale all the
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// existing terms (most of which will then underflow to zero). In this situation
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// it's likely that our series will only need 1 or 2 terms of the series but we
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// can't be sure of that:
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//
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if ((j < cache_size - 2) && (tools::max_value<T>() / fabs(64 * bessel_cache[j] / bessel_cache[j + 1]) < fabs(bessel_cache[j])))
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{
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T rescale = pow(fabs(bessel_cache[j] / bessel_cache[j + 1]), j + 1) * 2;
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if (!((boost::math::isfinite)(rescale)))
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rescale = tools::max_value<T>();
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for (int k = j; k < cache_size; ++k)
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bessel_cache[k] /= rescale;
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i = bessel_i_backwards_iterator<T, Policy>(b_minus_1_plus_n + j, 2 * sqrt(-bessel_arg), bessel_cache[j + 1], bessel_cache[j]);
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}
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}
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ratio = last_value / *i;
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}
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else
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{
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//
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// Forwards iteration is stable:
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//
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bessel_i_forwards_iterator<T, Policy> i(b_minus_1_plus_n, 2 * sqrt(-bessel_arg));
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int pos = 0;
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while ((pos < cache_size) && (b_minus_1_plus_n + pos < 0.5))
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{
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bessel_cache[pos++] = *i++;
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}
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if (pos < cache_size)
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{
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//
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// We have crossed over into the region where backward recursion is the stable direction
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// start from arbitrary value and recurse down to "pos" and normalise:
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//
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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));
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for (int loc = cache_size - 1; loc >= pos; --loc)
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bessel_cache[loc] = *i2++;
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ratio = bessel_cache[pos - 1] / *i2;
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while (pos < cache_size)
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bessel_cache[pos++] *= ratio;
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}
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ratio = 1;
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}
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}
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if(ratio != 1)
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for (auto j = bessel_cache.begin(); j != bessel_cache.end(); ++j)
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*j *= ratio;
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//
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// Very occasionally our normalisation fails because the normalisztion value
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// is sitting right on top of a root (or very close to it). When that happens
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// best to calculate a fresh Bessel evaluation and normalise again.
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//
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if (fabs(bessel_cache[0] / last_value) > 5)
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{
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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];
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if (ratio != 1)
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for (auto j = bessel_cache.begin(); j != bessel_cache.end(); ++j)
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*j *= ratio;
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}
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}
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};
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template <class T, class Policy>
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T hypergeometric_1F1_AS_13_3_7_tricomi(const T& a, const T& b, const T& z, const Policy& pol, int& log_scale)
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{
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BOOST_MATH_STD_USING
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//
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// Works for a < 0, b < 0, z > 0.
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//
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// For convergence we require A * term to be converging otherwise we get
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// a divergent alternating series. It's actually really hard to analyse this
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// and the best purely heuristic policy we've found is
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// z < fabs((2 * a - b) / (sqrt(fabs(a)))) ; b > 0 or:
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// z < fabs((2 * a - b) / (sqrt(fabs(ab)))) ; b < 0
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//
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T prefix(0);
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int prefix_sgn(0);
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bool use_logs = false;
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int scale = 0;
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//
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// We can actually support the b == 2a case within here, but we haven't
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// as we appear never to get here in practice. Which means this get out
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// clause is a bit of defensive programming....
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//
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if(b == 2 * a)
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return hypergeometric_1F1_divergent_fallback(a, b, z, pol, log_scale);
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try
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{
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prefix = boost::math::tgamma(b, pol);
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prefix *= exp(z / 2);
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}
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catch (const std::runtime_error&)
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{
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use_logs = true;
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}
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if (use_logs || (prefix == 0) || !(boost::math::isfinite)(prefix) || (!std::numeric_limits<T>::has_infinity && (fabs(prefix) >= tools::max_value<T>())))
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{
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use_logs = true;
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prefix = boost::math::lgamma(b, &prefix_sgn, pol) + z / 2;
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scale = itrunc(prefix);
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log_scale += scale;
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prefix -= scale;
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}
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T result(0);
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boost::uintmax_t max_iter = boost::math::policies::get_max_series_iterations<Policy>();
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bool retry = false;
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int series_scale = 0;
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try
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{
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hypergeometric_1F1_AS_13_3_7_tricomi_series<T, Policy> s(a, b, z, pol);
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series_scale = s.scale();
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log_scale += s.scale();
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try
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{
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T norm = 0;
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result = 0;
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if((a < 0) && (b < 0))
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result = boost::math::tools::checked_sum_series(s, boost::math::policies::get_epsilon<T, Policy>(), max_iter, result, norm);
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else
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result = boost::math::tools::sum_series(s, boost::math::policies::get_epsilon<T, Policy>(), max_iter, result);
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if (!(boost::math::isfinite)(result) || (result == 0) || (!std::numeric_limits<T>::has_infinity && (fabs(result) >= tools::max_value<T>())))
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retry = true;
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if (norm / fabs(result) > 1 / boost::math::tools::root_epsilon<T>())
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retry = true; // fatal cancellation
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}
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catch (const std::overflow_error&)
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{
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retry = true;
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}
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catch (const boost::math::evaluation_error&)
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{
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retry = true;
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}
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}
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catch (const std::overflow_error&)
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{
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log_scale -= scale;
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return hypergeometric_1F1_divergent_fallback(a, b, z, pol, log_scale);
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}
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catch (const boost::math::evaluation_error&)
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{
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log_scale -= scale;
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return hypergeometric_1F1_divergent_fallback(a, b, z, pol, log_scale);
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}
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if (retry)
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{
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log_scale -= scale;
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log_scale -= series_scale;
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return hypergeometric_1F1_divergent_fallback(a, b, z, pol, log_scale);
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}
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boost::math::policies::check_series_iterations<T>("boost::math::hypergeometric_1F1_AS_13_3_7<%1%>(%1%,%1%,%1%)", max_iter, pol);
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if (use_logs)
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{
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int sgn = boost::math::sign(result);
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prefix += log(fabs(result));
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result = sgn * prefix_sgn * exp(prefix);
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}
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else
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{
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if ((fabs(result) > 1) && (fabs(prefix) > 1) && (tools::max_value<T>() / fabs(result) < fabs(prefix)))
|
{
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// Overflow:
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scale = itrunc(tools::log_max_value<T>()) - 10;
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log_scale += scale;
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result /= exp(T(scale));
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}
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result *= prefix;
|
}
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return result;
|
}
|
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template <class T>
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struct cyl_bessel_i_large_x_sum
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{
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typedef T result_type;
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cyl_bessel_i_large_x_sum(const T& v, const T& x) : v(v), z(x), term(1), k(0) {}
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T operator()()
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{
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T result = term;
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++k;
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term *= -(4 * v * v - (2 * k - 1) * (2 * k - 1)) / (8 * k * z);
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return result;
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}
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T v, z, term;
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int k;
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};
|
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template <class T, class Policy>
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T cyl_bessel_i_large_x_scaled(const T& v, const T& x, int& log_scaling, const Policy& pol)
|
{
|
BOOST_MATH_STD_USING
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cyl_bessel_i_large_x_sum<T> s(v, x);
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boost::uintmax_t max_iter = boost::math::policies::get_max_series_iterations<Policy>();
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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);
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int scale = boost::math::itrunc(x);
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log_scaling += scale;
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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
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{
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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;
|
}
|
|
int 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, 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, int& log_scaling)
|
{
|
BOOST_MATH_STD_USING
|
if(b_minus_a == 0)
|
{
|
// special case: M(a,a,z) == exp(z)
|
int scale = itrunc(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);
|
boost::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);
|
int scale = itrunc(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);
|
boost::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;
|
int 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, int& 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);
|
boost::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)
|
{
|
int scaling = itrunc(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
|
