///////////////////////////////////////////////////////////////////////////////
|
// 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/config.hpp>
|
|
#if defined(BOOST_NO_CXX11_AUTO_DECLARATIONS) || defined(BOOST_NO_CXX11_LAMBDAS) || defined(BOOST_NO_CXX11_UNIFIED_INITIALIZATION_SYNTAX)
|
# error "hypergeometric_1F1 requires a C++11 compiler"
|
#endif
|
|
#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, int& 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 tbough.
|
//
|
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);
|
try
|
{
|
return hypergeometric_1F1_checked_series_impl(a, b, z, pol, log_scaling);
|
}
|
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);
|
}
|
}
|
//
|
// 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, int& 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)
|
return 1 - (z / b);
|
|
const T b_minus_a = b - a;
|
|
// 0f0 a == b case;
|
if (b_minus_a == 0)
|
{
|
int scale = itrunc(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);
|
|
return (b + z) * exp(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);
|
int scale = itrunc(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:
|
int scaling = itrunc(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);
|
int scale = itrunc(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))
|
{
|
int saved_scale = log_scaling;
|
try
|
{
|
return hypergeometric_1F1_asym_large_z_series(a, b, z, pol, log_scaling);
|
}
|
catch (const evaluation_error&)
|
{
|
}
|
//
|
// 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);
|
}
|
}
|
// Let's otherwise make z positive (almost always)
|
// by Kummer's transformation
|
// (we also don't transform if z belongs to [-1,0])
|
int scaling = itrunc(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))
|
{
|
int scaling = itrunc(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
|
int log_scaling = 0;
|
T result = hypergeometric_1F1_imp(a, b, z, pol, log_scaling);
|
//
|
// Actual result will be result * e^log_scaling.
|
//
|
#ifndef BOOST_NO_CXX11_THREAD_LOCAL
|
static const thread_local int max_scaling = itrunc(boost::math::tools::log_max_value<T>()) - 2;
|
static const thread_local T max_scale_factor = exp(T(max_scaling));
|
#else
|
int max_scaling = itrunc(boost::math::tools::log_max_value<T>()) - 2;
|
T max_scale_factor = exp(T(max_scaling));
|
#endif
|
|
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
|
int 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
|
int 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);
|
#ifndef BOOST_NO_CXX11_THREAD_LOCAL
|
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);
|
#else
|
T max_scaling = boost::math::tools::log_max_value<T>() - 2;
|
T max_scale_factor = exp(max_scaling);
|
#endif
|
|
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
|
