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- ///////////////////////////////////////////////////////////////////////////////
- // 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_HPP
- #define BOOST_MATH_HYPERGEOMETRIC_1F1_HPP
- #include <boost/math/tools/config.hpp>
- #include <boost/math/policies/policy.hpp>
- #include <boost/math/policies/error_handling.hpp>
- #include <boost/math/special_functions/detail/hypergeometric_series.hpp>
- #include <boost/math/special_functions/detail/hypergeometric_asym.hpp>
- #include <boost/math/special_functions/detail/hypergeometric_rational.hpp>
- #include <boost/math/special_functions/detail/hypergeometric_1F1_recurrence.hpp>
- #include <boost/math/special_functions/detail/hypergeometric_1F1_by_ratios.hpp>
- #include <boost/math/special_functions/detail/hypergeometric_pade.hpp>
- #include <boost/math/special_functions/detail/hypergeometric_1F1_bessel.hpp>
- #include <boost/math/special_functions/detail/hypergeometric_1F1_scaled_series.hpp>
- #include <boost/math/special_functions/detail/hypergeometric_pFq_checked_series.hpp>
- #include <boost/math/special_functions/detail/hypergeometric_1F1_addition_theorems_on_z.hpp>
- #include <boost/math/special_functions/detail/hypergeometric_1F1_large_abz.hpp>
- #include <boost/math/special_functions/detail/hypergeometric_1F1_small_a_negative_b_by_ratio.hpp>
- #include <boost/math/special_functions/detail/hypergeometric_1F1_negative_b_regions.hpp>
- namespace boost { namespace math { namespace detail {
- // check when 1F1 series can't decay to polynom
- template <class T>
- inline bool check_hypergeometric_1F1_parameters(const T& a, const T& b)
- {
- BOOST_MATH_STD_USING
- if ((b <= 0) && (b == floor(b)))
- {
- if ((a >= 0) || (a < b) || (a != floor(a)))
- return false;
- }
- return true;
- }
- 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)
- {
- BOOST_MATH_STD_USING
- const char* function = "hypergeometric_1F1_divergent_fallback<%1%>(%1%,%1%,%1%)";
- //
- // We get here if either:
- // 1) We decide up front that Tricomi's method won't work, or:
- // 2) We've called Tricomi's method and it's failed.
- //
- if (b > 0)
- {
- // Commented out since recurrence seems to always be better?
- #if 0
- if ((z < b) && (a > -50))
- // Might as well use a recurrence in preference to z-recurrence:
- return hypergeometric_1F1_backward_recurrence_for_negative_a(a, b, z, pol, function, log_scaling);
- T z_limit = fabs((2 * a - b) / (sqrt(fabs(a))));
- int k = 1 + itrunc(z - z_limit);
- // If k is too large we destroy all the digits in the result:
- T convergence_at_50 = (b - a + 50) * k / (z * 50);
- if ((k > 0) && (k < 50) && (fabs(convergence_at_50) < 1) && (z > z_limit))
- {
- return boost::math::detail::hypergeometric_1f1_recurrence_on_z_minus_zero(a, b, T(z - k), k, pol, log_scaling);
- }
- #endif
- if (z < b)
- return hypergeometric_1F1_backward_recurrence_for_negative_a(a, b, z, pol, function, log_scaling);
- else
- return hypergeometric_1F1_backwards_recursion_on_b_for_negative_a(a, b, z, pol, function, log_scaling);
- }
- else // b < 0
- {
- if (a < 0)
- {
- if ((b < a) && (z < -b / 4))
- return hypergeometric_1F1_from_function_ratio_negative_ab(a, b, z, pol, log_scaling);
- else
- {
- //
- // Solve (a+n)z/((b+n)n) == 1 for n, the number of iterations till the series starts to converge.
- // If this is well away from the origin then it's probably better to use the series to evaluate this.
- // Note that if sqr is negative then we have no solution, so assign an arbitrarily large value to the
- // number of iterations.
- //
- bool can_use_recursion = (z - b + 100 < boost::math::policies::get_max_series_iterations<Policy>()) && (100 - a < boost::math::policies::get_max_series_iterations<Policy>());
- T sqr = 4 * a * z + b * b - 2 * b * z + z * z;
- T iterations_to_convergence = sqr > 0 ? T(0.5f * (-sqrt(sqr) - b + z)) : T(-a - b);
- if(can_use_recursion && ((std::max)(a, b) + iterations_to_convergence > -300))
- return hypergeometric_1F1_backwards_recursion_on_b_for_negative_a(a, b, z, pol, function, log_scaling);
- //
- // When a < b and if we fall through to the series, then we get divergent behaviour when b crosses the origin
- // so ideally we would pick another method. Otherwise the terms immediately after b crosses the origin may
- // suffer catastrophic cancellation....
- //
- if((a < b) && can_use_recursion)
- return hypergeometric_1F1_backwards_recursion_on_b_for_negative_a(a, b, z, pol, function, log_scaling);
- }
- }
- else
- {
- //
- // Start by getting the domain of the recurrence relations, we get either:
- // -1 Backwards recursion is stable and the CF will converge to double precision.
- // +1 Forwards recursion is stable and the CF will converge to double precision.
- // 0 No man's land, we're not far enough away from the crossover point to get double precision from either CF.
- //
- // At higher than double precision we need to be further away from the crossover location to
- // get full converge, but it's not clear how much further - indeed at quad precision it's
- // basically impossible to ever get forwards iteration to work. Backwards seems to work
- // OK as long as a > 1 whatever the precision though.
- //
- int domain = hypergeometric_1F1_negative_b_recurrence_region(a, b, z);
- if ((domain < 0) && ((a > 1) || (boost::math::policies::digits<T, Policy>() <= 64)))
- return hypergeometric_1F1_from_function_ratio_negative_b(a, b, z, pol, log_scaling);
- else if (domain > 0)
- {
- if (boost::math::policies::digits<T, Policy>() <= 64)
- return hypergeometric_1F1_from_function_ratio_negative_b_forwards(a, b, z, pol, log_scaling);
- #ifndef BOOST_MATH_NO_EXCEPTIONS
- try
- #endif
- {
- return hypergeometric_1F1_checked_series_impl(a, b, z, pol, log_scaling);
- }
- #ifndef BOOST_MATH_NO_EXCEPTIONS
- catch (const evaluation_error&)
- {
- //
- // The series failed, try the recursions instead and hope we get at least double precision:
- //
- return hypergeometric_1F1_from_function_ratio_negative_b_forwards(a, b, z, pol, log_scaling);
- }
- #endif
- }
- //
- // We could fall back to Tricomi's approximation if we're in the transition zone
- // between the above two regions. However, I've been unable to find any examples
- // where this is better than the series, and there are many cases where it leads to
- // quite grievous errors.
- /*
- else if (allow_tricomi)
- {
- T aa = a < 1 ? T(1) : a;
- if (z < fabs((2 * aa - b) / (sqrt(fabs(aa * b)))))
- return hypergeometric_1F1_AS_13_3_7_tricomi(a, b, z, pol, log_scaling);
- }
- */
- }
- }
- // If we get here, then we've run out of methods to try, use the checked series which will
- // raise an error if the result is garbage:
- return hypergeometric_1F1_checked_series_impl(a, b, z, pol, log_scaling);
- }
- template <class T>
- bool is_convergent_negative_z_series(const T& a, const T& b, const T& z, const T& b_minus_a)
- {
- BOOST_MATH_STD_USING
- //
- // Filter out some cases we don't want first:
- //
- if((b_minus_a > 0) && (b > 0))
- {
- if (a < 0)
- return false;
- }
- //
- // Generic check: we have small initial divergence and are convergent after 10 terms:
- //
- if ((fabs(z * a / b) < 2) && (fabs(z * (a + 10) / ((b + 10) * 10)) < 1))
- {
- // Double check for divergence when we cross the origin on a and b:
- if (a < 0)
- {
- T n = 300 - floor(a);
- if (fabs((a + n) * z / ((b + n) * n)) < 1)
- {
- if (b < 0)
- {
- T m = 3 - floor(b);
- if (fabs((a + m) * z / ((b + m) * m)) < 1)
- return true;
- }
- else
- return true;
- }
- }
- else if (b < 0)
- {
- T n = 3 - floor(b);
- if (fabs((a + n) * z / ((b + n) * n)) < 1)
- return true;
- }
- }
- if ((b > 0) && (a < 0))
- {
- //
- // For a and z both negative, we're OK with some initial divergence as long as
- // it occurs before we hit the origin, as to start with all the terms have the
- // same sign.
- //
- // https://www.wolframalpha.com/input/?i=solve+(a%2Bn)z+%2F+((b%2Bn)n)+%3D%3D+1+for+n
- //
- T sqr = 4 * a * z + b * b - 2 * b * z + z * z;
- T iterations_to_convergence = sqr > 0 ? T(0.5f * (-sqrt(sqr) - b + z)) : T(-a + b);
- if (iterations_to_convergence < 0)
- iterations_to_convergence = 0.5f * (sqrt(sqr) - b + z);
- if (a + iterations_to_convergence < -50)
- {
- // Need to check for divergence when we cross the origin on a:
- if (a > -1)
- return true;
- T n = 300 - floor(a);
- if(fabs((a + n) * z / ((b + n) * n)) < 1)
- return true;
- }
- }
- return false;
- }
- template <class T>
- inline T cyl_bessel_i_shrinkage_rate(const T& z)
- {
- // Approximately the ratio I_10.5(z/2) / I_9.5(z/2), this gives us an idea of how quickly
- // the Bessel terms in A&S 13.6.4 are converging:
- if (z < -160)
- return 1;
- if (z < -40)
- return 0.75f;
- if (z < -20)
- return 0.5f;
- if (z < -7)
- return 0.25f;
- if (z < -2)
- return 0.1f;
- return 0.05f;
- }
- template <class T>
- inline bool hypergeometric_1F1_is_13_3_6_region(const T& a, const T& b, const T& z)
- {
- BOOST_MATH_STD_USING
- if(fabs(a) == 0.5)
- return false;
- if ((z < 0) && (fabs(10 * a / b) < 1) && (fabs(a) < 50))
- {
- T shrinkage = cyl_bessel_i_shrinkage_rate(z);
- // We want the first term not too divergent, and convergence by term 10:
- if ((fabs((2 * a - 1) * (2 * a - b) / b) < 2) && (fabs(shrinkage * (2 * a + 9) * (2 * a - b + 10) / (10 * (b + 10))) < 0.75))
- return true;
- }
- return false;
- }
- template <class T>
- inline bool hypergeometric_1F1_need_kummer_reflection(const T& a, const T& b, const T& z)
- {
- BOOST_MATH_STD_USING
- //
- // Check to see if we should apply Kummer's relation or not:
- //
- if (z > 0)
- return false;
- if (z < -1)
- return true;
- //
- // When z is small and negative, things get more complex.
- // More often than not we do not need apply Kummer's relation and the
- // series is convergent as is, but we do need to check:
- //
- if (a > 0)
- {
- if (b > 0)
- {
- return fabs((a + 10) * z / (10 * (b + 10))) < 1; // Is the 10'th term convergent?
- }
- else
- {
- return true; // Likely to be divergent as b crosses the origin
- }
- }
- else // a < 0
- {
- if (b > 0)
- {
- return false; // Terms start off all positive and then by the time a crosses the origin we *must* be convergent.
- }
- else
- {
- return true; // Likely to be divergent as b crosses the origin, but hard to rationalise about!
- }
- }
- }
-
- template <class T, class Policy>
- T hypergeometric_1F1_imp(const T& a, const T& b, const T& z, const Policy& pol, long long& log_scaling)
- {
- BOOST_MATH_STD_USING // exp, fabs, sqrt
- static const char* const function = "boost::math::hypergeometric_1F1<%1%,%1%,%1%>(%1%,%1%,%1%)";
- if ((z == 0) || (a == 0))
- return T(1);
- // undefined result:
- if (!detail::check_hypergeometric_1F1_parameters(a, b))
- return policies::raise_domain_error<T>(
- function,
- "Function is indeterminate for negative integer b = %1%.",
- b,
- pol);
- // other checks:
- if (a == -1)
- {
- T r = 1 - (z / b);
- if (fabs(r) < 0.5)
- r = (b - z) / b;
- return r;
- }
- const T b_minus_a = b - a;
- // 0f0 a == b case;
- if (b_minus_a == 0)
- {
- if ((a < 0) && (floor(a) == a))
- {
- // Special case, use the truncated series to match what Mathematica does.
- if ((a < -20) && (z > 0) && (z < 1))
- {
- // https://functions.wolfram.com/HypergeometricFunctions/Hypergeometric1F1/03/01/04/02/0002/
- return exp(z) * boost::math::gamma_q(1 - a, z, pol);
- }
- // https://functions.wolfram.com/HypergeometricFunctions/Hypergeometric1F1/03/01/04/02/0003/
- return hypergeometric_1F1_checked_series_impl(a, b, z, pol, log_scaling);
- }
- long long scale = lltrunc(z, pol);
- log_scaling += scale;
- return exp(z - scale);
- }
- // Special case for b-a = -1, we don't use for small a as it throws the digits of a away and leads to large errors:
- if ((b_minus_a == -1) && (fabs(a) > 0.5))
- {
- // for negative small integer a it is reasonable to use truncated series - polynomial
- if ((a < 0) && (a == ceil(a)) && (a > -50))
- return detail::hypergeometric_1F1_generic_series(a, b, z, pol, log_scaling, function);
- log_scaling = lltrunc(floor(z));
- T local_z = z - log_scaling;
- return (b + z) * exp(local_z) / b;
- }
- if ((a == 1) && (b == 2))
- return boost::math::expm1(z, pol) / z;
- if ((b - a == b) && (fabs(z / b) < policies::get_epsilon<T, Policy>()))
- return 1;
- //
- // Special case for A&S 13.3.6:
- //
- if (z < 0)
- {
- if (hypergeometric_1F1_is_13_3_6_region(a, b, z))
- {
- // a is tiny compared to b, and z < 0
- // 13.3.6 appears to be the most efficient and often the most accurate method.
- T r = boost::math::detail::hypergeometric_1F1_AS_13_3_6(b_minus_a, b, T(-z), a, pol, log_scaling);
- long long scale = lltrunc(z, pol);
- log_scaling += scale;
- return r * exp(z - scale);
- }
- if ((b < 0) && (fabs(a) < 1e-2))
- {
- //
- // This is a tricky area, potentially we have no good method at all:
- //
- if (b - ceil(b) == a)
- {
- // Fractional parts of a and b are genuinely equal, we might as well
- // apply Kummer's relation and get a truncated series:
- long long scaling = lltrunc(z);
- T r = exp(z - scaling) * detail::hypergeometric_1F1_imp<T>(b_minus_a, b, -z, pol, log_scaling);
- log_scaling += scaling;
- return r;
- }
- if ((b < -1) && (max_b_for_1F1_small_a_negative_b_by_ratio(z) < b))
- return hypergeometric_1F1_small_a_negative_b_by_ratio(a, b, z, pol, log_scaling);
- if ((b > -1) && (b < -0.5f))
- {
- // Recursion is meta-stable:
- T first = hypergeometric_1F1_imp(a, T(b + 2), z, pol);
- T second = hypergeometric_1F1_imp(a, T(b + 1), z, pol);
- return tools::apply_recurrence_relation_backward(hypergeometric_1F1_recurrence_small_b_coefficients<T>(a, b, z, 1), 1, first, second);
- }
- //
- // We've got nothing left but 13.3.6, even though it may be initially divergent:
- //
- T r = boost::math::detail::hypergeometric_1F1_AS_13_3_6(b_minus_a, b, T(-z), a, pol, log_scaling);
- long long scale = lltrunc(z, pol);
- log_scaling += scale;
- return r * exp(z - scale);
- }
- }
- //
- // Asymptotic expansion for large z
- // TODO: check region for higher precision types.
- // Use recurrence relations to move to this region when a and b are also large.
- //
- if (detail::hypergeometric_1F1_asym_region(a, b, z, pol))
- {
- long long saved_scale = log_scaling;
- #ifndef BOOST_MATH_NO_EXCEPTIONS
- try
- #endif
- {
- return hypergeometric_1F1_asym_large_z_series(a, b, z, pol, log_scaling);
- }
- #ifndef BOOST_MATH_NO_EXCEPTIONS
- catch (const evaluation_error&)
- {
- }
- #endif
- //
- // Very occasionally our convergence criteria don't quite go to full precision
- // and we have to try another method:
- //
- log_scaling = saved_scale;
- }
- if ((fabs(a * z / b) < 3.5) && (fabs(z * 100) < fabs(b)) && ((fabs(a) > 1e-2) || (b < -5)))
- return detail::hypergeometric_1F1_rational(a, b, z, pol);
- if (hypergeometric_1F1_need_kummer_reflection(a, b, z))
- {
- if (a == 1)
- return detail::hypergeometric_1F1_pade(b, z, pol);
- if (is_convergent_negative_z_series(a, b, z, b_minus_a))
- {
- if ((boost::math::sign(b_minus_a) == boost::math::sign(b)) && ((b > 0) || (b < -200)))
- {
- // Series is close enough to convergent that we should be OK,
- // In this domain b - a ~ b and since 1F1[a, a, z] = e^z 1F1[b-a, b, -z]
- // and 1F1[a, a, -z] = e^-z the result must necessarily be somewhere near unity.
- // We have to rule out b small and negative because if b crosses the origin early
- // in the series (before we're pretty much converged) then all bets are off.
- // Note that this can go badly wrong when b and z are both large and negative,
- // in that situation the series goes in waves of large and small values which
- // may or may not cancel out. Likewise the initial part of the series may or may
- // not converge, and even if it does may or may not give a correct answer!
- // For example 1F1[-small, -1252.5, -1043.7] can loose up to ~800 digits due to
- // cancellation and is basically incalculable via this method.
- return hypergeometric_1F1_checked_series_impl(a, b, z, pol, log_scaling);
- }
- }
- if ((b < 0) && (floor(b) == b))
- {
- // Negative integer b, so a must be a negative integer too.
- // Kummer's transformation fails here!
- if(a > -50)
- return detail::hypergeometric_1F1_generic_series(a, b, z, pol, log_scaling, function);
- // Is there anything better than this??
- return hypergeometric_1F1_imp(a, float_next(b), z, pol, log_scaling);
- }
- else
- {
- // Let's otherwise make z positive (almost always)
- // by Kummer's transformation
- // (we also don't transform if z belongs to [-1,0])
- // Also note that Kummer's transformation fails when b is
- // a negative integer, although this seems to be unmentioned
- // in the literature...
- long long scaling = lltrunc(z);
- T r = exp(z - scaling) * detail::hypergeometric_1F1_imp<T>(b_minus_a, b, -z, pol, log_scaling);
- log_scaling += scaling;
- return r;
- }
- }
- //
- // Check for initial divergence:
- //
- bool series_is_divergent = (a + 1) * z / (b + 1) < -1;
- if (series_is_divergent && (a < 0) && (b < 0) && (a > -1))
- series_is_divergent = false; // Best off taking the series in this situation
- //
- // If series starts off non-divergent, and becomes divergent later
- // then it's because both a and b are negative, so check for later
- // divergence as well:
- //
- if (!series_is_divergent && (a < 0) && (b < 0) && (b > a))
- {
- //
- // We need to exclude situations where we're over the initial "hump"
- // in the series terms (ie series has already converged by the time
- // b crosses the origin:
- //
- //T fa = fabs(a);
- //T fb = fabs(b);
- T convergence_point = sqrt((a - 1) * (a - b)) - a;
- if (-b < convergence_point)
- {
- T n = -floor(b);
- series_is_divergent = (a + n) * z / ((b + n) * n) < -1;
- }
- }
- else if (!series_is_divergent && (b < 0) && (a > 0))
- {
- // Series almost always become divergent as b crosses the origin:
- series_is_divergent = true;
- }
- if (series_is_divergent && (b < -1) && (b > -5) && (a > b))
- series_is_divergent = false; // don't bother with divergence, series will be OK
- //
- // Test for alternating series due to negative a,
- // in particular, see if the series is initially divergent
- // If so use the recurrence relation on a:
- //
- if (series_is_divergent)
- {
- if((a < 0) && (floor(a) == a) && (-a < policies::get_max_series_iterations<Policy>()))
- // This works amazingly well for negative integer a:
- return hypergeometric_1F1_backward_recurrence_for_negative_a(a, b, z, pol, function, log_scaling);
- //
- // In what follows we have to set limits on how large z can be otherwise
- // the Bessel series become large and divergent and all the digits cancel out.
- // The criteria are distinctly empiracle rather than based on a firm analysis
- // of the terms in the series.
- //
- if (b > 0)
- {
- T z_limit = fabs((2 * a - b) / (sqrt(fabs(a))));
- if ((z < z_limit) && hypergeometric_1F1_is_tricomi_viable_positive_b(a, b, z))
- return detail::hypergeometric_1F1_AS_13_3_7_tricomi(a, b, z, pol, log_scaling);
- }
- else // b < 0
- {
- if (a < 0)
- {
- T z_limit = fabs((2 * a - b) / (sqrt(fabs(a))));
- //
- // I hate these hard limits, but they're about the best we can do to try and avoid
- // Bessel function internal failures: these will be caught and handled
- // but up the expense of this function call:
- //
- if (((z < z_limit) || (a > -500)) && ((b > -500) || (b - 2 * a > 0)) && (z < -a))
- {
- //
- // Outside this domain we will probably get better accuracy from the recursive methods.
- //
- if(!(((a < b) && (z > -b)) || (z > z_limit)))
- return detail::hypergeometric_1F1_AS_13_3_7_tricomi(a, b, z, pol, log_scaling);
- //
- // When b and z are both very small, we get large errors from the recurrence methods
- // in the fallbacks. Tricomi seems to work well here, as does direct series evaluation
- // at least some of the time. Picking the right method is not easy, and sometimes this
- // is much worse than the fallback. Overall though, it's a reasonable choice that keeps
- // the very worst errors under control.
- //
- if(b > -1)
- return detail::hypergeometric_1F1_AS_13_3_7_tricomi(a, b, z, pol, log_scaling);
- }
- }
- //
- // We previously used Tricomi here, but it appears to be worse than
- // the recurrence-based algorithms in hypergeometric_1F1_divergent_fallback.
- /*
- else
- {
- T aa = a < 1 ? T(1) : a;
- if (z < fabs((2 * aa - b) / (sqrt(fabs(aa * b)))))
- return detail::hypergeometric_1F1_AS_13_3_7_tricomi(a, b, z, pol, log_scaling);
- }*/
- }
- return hypergeometric_1F1_divergent_fallback(a, b, z, pol, log_scaling);
- }
- if (hypergeometric_1F1_is_13_3_6_region(b_minus_a, b, T(-z)))
- {
- // b_minus_a is tiny compared to b, and -z < 0
- // 13.3.6 appears to be the most efficient and often the most accurate method.
- return boost::math::detail::hypergeometric_1F1_AS_13_3_6(a, b, z, b_minus_a, pol, log_scaling);
- }
- #if 0
- if ((a > 0) && (b > 0) && (a * z / b > 2))
- {
- //
- // Series is initially divergent and slow to converge, see if applying
- // Kummer's relation can improve things:
- //
- if (is_convergent_negative_z_series(b_minus_a, b, T(-z), b_minus_a))
- {
- long long scaling = lltrunc(z);
- T r = exp(z - scaling) * detail::hypergeometric_1F1_checked_series_impl(b_minus_a, b, T(-z), pol, log_scaling);
- log_scaling += scaling;
- return r;
- }
- }
- #endif
- if ((a > 0) && (b > 0) && (a * z > 50))
- return detail::hypergeometric_1F1_large_abz(a, b, z, pol, log_scaling);
- if (b < 0)
- return detail::hypergeometric_1F1_checked_series_impl(a, b, z, pol, log_scaling);
-
- return detail::hypergeometric_1F1_generic_series(a, b, z, pol, log_scaling, function);
- }
- template <class T, class Policy>
- inline T hypergeometric_1F1_imp(const T& a, const T& b, const T& z, const Policy& pol)
- {
- BOOST_MATH_STD_USING // exp, fabs, sqrt
- long long log_scaling = 0;
- T result = hypergeometric_1F1_imp(a, b, z, pol, log_scaling);
- //
- // Actual result will be result * e^log_scaling.
- //
- static const thread_local long long max_scaling = lltrunc(boost::math::tools::log_max_value<T>()) - 2;
- static const thread_local T max_scale_factor = exp(T(max_scaling));
- while (log_scaling > max_scaling)
- {
- result *= max_scale_factor;
- log_scaling -= max_scaling;
- }
- while (log_scaling < -max_scaling)
- {
- result /= max_scale_factor;
- log_scaling += max_scaling;
- }
- if (log_scaling)
- result *= exp(T(log_scaling));
- return result;
- }
- template <class T, class Policy>
- inline T log_hypergeometric_1F1_imp(const T& a, const T& b, const T& z, int* sign, const Policy& pol)
- {
- BOOST_MATH_STD_USING // exp, fabs, sqrt
- long long log_scaling = 0;
- T result = hypergeometric_1F1_imp(a, b, z, pol, log_scaling);
- if (sign)
- *sign = result < 0 ? -1 : 1;
- result = log(fabs(result)) + log_scaling;
- return result;
- }
- template <class T, class Policy>
- inline T hypergeometric_1F1_regularized_imp(const T& a, const T& b, const T& z, const Policy& pol)
- {
- BOOST_MATH_STD_USING // exp, fabs, sqrt
- long long log_scaling = 0;
- T result = hypergeometric_1F1_imp(a, b, z, pol, log_scaling);
- //
- // Actual result will be result * e^log_scaling / tgamma(b).
- //
- int result_sign = 1;
- T scale = log_scaling - boost::math::lgamma(b, &result_sign, pol);
- static const thread_local T max_scaling = boost::math::tools::log_max_value<T>() - 2;
- static const thread_local T max_scale_factor = exp(max_scaling);
- while (scale > max_scaling)
- {
- result *= max_scale_factor;
- scale -= max_scaling;
- }
- while (scale < -max_scaling)
- {
- result /= max_scale_factor;
- scale += max_scaling;
- }
- if (scale != 0)
- result *= exp(scale);
- return result * result_sign;
- }
- } // namespace detail
- template <class T1, class T2, class T3, class Policy>
- inline typename tools::promote_args<T1, T2, T3>::type hypergeometric_1F1(T1 a, T2 b, T3 z, const Policy& /* pol */)
- {
- BOOST_FPU_EXCEPTION_GUARD
- typedef typename tools::promote_args<T1, T2, T3>::type result_type;
- typedef typename policies::evaluation<result_type, Policy>::type value_type;
- typedef typename policies::normalise<
- Policy,
- policies::promote_float<false>,
- policies::promote_double<false>,
- policies::discrete_quantile<>,
- policies::assert_undefined<> >::type forwarding_policy;
- return policies::checked_narrowing_cast<result_type, Policy>(
- detail::hypergeometric_1F1_imp<value_type>(
- static_cast<value_type>(a),
- static_cast<value_type>(b),
- static_cast<value_type>(z),
- forwarding_policy()),
- "boost::math::hypergeometric_1F1<%1%>(%1%,%1%,%1%)");
- }
- template <class T1, class T2, class T3>
- inline typename tools::promote_args<T1, T2, T3>::type hypergeometric_1F1(T1 a, T2 b, T3 z)
- {
- return hypergeometric_1F1(a, b, z, policies::policy<>());
- }
- template <class T1, class T2, class T3, class Policy>
- inline typename tools::promote_args<T1, T2, T3>::type hypergeometric_1F1_regularized(T1 a, T2 b, T3 z, const Policy& /* pol */)
- {
- BOOST_FPU_EXCEPTION_GUARD
- typedef typename tools::promote_args<T1, T2, T3>::type result_type;
- typedef typename policies::evaluation<result_type, Policy>::type value_type;
- typedef typename policies::normalise<
- Policy,
- policies::promote_float<false>,
- policies::promote_double<false>,
- policies::discrete_quantile<>,
- policies::assert_undefined<> >::type forwarding_policy;
- return policies::checked_narrowing_cast<result_type, Policy>(
- detail::hypergeometric_1F1_regularized_imp<value_type>(
- static_cast<value_type>(a),
- static_cast<value_type>(b),
- static_cast<value_type>(z),
- forwarding_policy()),
- "boost::math::hypergeometric_1F1<%1%>(%1%,%1%,%1%)");
- }
- template <class T1, class T2, class T3>
- inline typename tools::promote_args<T1, T2, T3>::type hypergeometric_1F1_regularized(T1 a, T2 b, T3 z)
- {
- return hypergeometric_1F1_regularized(a, b, z, policies::policy<>());
- }
- template <class T1, class T2, class T3, class Policy>
- inline typename tools::promote_args<T1, T2, T3>::type log_hypergeometric_1F1(T1 a, T2 b, T3 z, const Policy& /* pol */)
- {
- BOOST_FPU_EXCEPTION_GUARD
- typedef typename tools::promote_args<T1, T2, T3>::type result_type;
- typedef typename policies::evaluation<result_type, Policy>::type value_type;
- typedef typename policies::normalise<
- Policy,
- policies::promote_float<false>,
- policies::promote_double<false>,
- policies::discrete_quantile<>,
- policies::assert_undefined<> >::type forwarding_policy;
- return policies::checked_narrowing_cast<result_type, Policy>(
- detail::log_hypergeometric_1F1_imp<value_type>(
- static_cast<value_type>(a),
- static_cast<value_type>(b),
- static_cast<value_type>(z),
- 0,
- forwarding_policy()),
- "boost::math::hypergeometric_1F1<%1%>(%1%,%1%,%1%)");
- }
- template <class T1, class T2, class T3>
- inline typename tools::promote_args<T1, T2, T3>::type log_hypergeometric_1F1(T1 a, T2 b, T3 z)
- {
- return log_hypergeometric_1F1(a, b, z, policies::policy<>());
- }
- template <class T1, class T2, class T3, class Policy>
- inline typename tools::promote_args<T1, T2, T3>::type log_hypergeometric_1F1(T1 a, T2 b, T3 z, int* sign, const Policy& /* pol */)
- {
- BOOST_FPU_EXCEPTION_GUARD
- typedef typename tools::promote_args<T1, T2, T3>::type result_type;
- typedef typename policies::evaluation<result_type, Policy>::type value_type;
- typedef typename policies::normalise<
- Policy,
- policies::promote_float<false>,
- policies::promote_double<false>,
- policies::discrete_quantile<>,
- policies::assert_undefined<> >::type forwarding_policy;
- return policies::checked_narrowing_cast<result_type, Policy>(
- detail::log_hypergeometric_1F1_imp<value_type>(
- static_cast<value_type>(a),
- static_cast<value_type>(b),
- static_cast<value_type>(z),
- sign,
- forwarding_policy()),
- "boost::math::hypergeometric_1F1<%1%>(%1%,%1%,%1%)");
- }
- template <class T1, class T2, class T3>
- inline typename tools::promote_args<T1, T2, T3>::type log_hypergeometric_1F1(T1 a, T2 b, T3 z, int* sign)
- {
- return log_hypergeometric_1F1(a, b, z, sign, policies::policy<>());
- }
- } } // namespace boost::math
- #endif // BOOST_MATH_HYPERGEOMETRIC_HPP
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