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Jacob Hass
2026-01-01 10:44:48 -08:00
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doc/sf/igamma.qbk
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@@ -20,6 +20,12 @@
template <class T1, class T2, class ``__Policy``> template <class T1, class T2, class ``__Policy``>
BOOST_MATH_GPU_ENABLED ``__sf_result`` gamma_q(T1 a, T2 z, const ``__Policy``&); BOOST_MATH_GPU_ENABLED ``__sf_result`` gamma_q(T1 a, T2 z, const ``__Policy``&);
template <class T1, class T2>
BOOST_MATH_GPU_ENABLED ``__sf_result`` lgamma_q(T1 a, T2 z);
template <class T1, class T2, class ``__Policy``>
BOOST_MATH_GPU_ENABLED ``__sf_result`` lgamma_q(T1 a, T2 z, const ``__Policy``&);
template <class T1, class T2> template <class T1, class T2>
BOOST_MATH_GPU_ENABLED ``__sf_result`` tgamma_lower(T1 a, T2 z); BOOST_MATH_GPU_ENABLED ``__sf_result`` tgamma_lower(T1 a, T2 z);
@@ -80,6 +86,15 @@ This function changes rapidly from 1 to 0 around the point z == a:
[graph gamma_q] [graph gamma_q]
template <class T1, class T2>
BOOST_MATH_GPU_ENABLED ``__sf_result`` lgamma_q(T1 a, T2 z);
template <class T1, class T2, class ``__Policy``>
BOOST_MATH_GPU_ENABLED ``__sf_result`` lgamma_q(T1 a, T2 z, const ``__Policy``&);
Returns the natural log of the normalized upper incomplete gamma function
of a and z.
template <class T1, class T2> template <class T1, class T2>
BOOST_MATH_GPU_ENABLED ``__sf_result`` tgamma_lower(T1 a, T2 z); BOOST_MATH_GPU_ENABLED ``__sf_result`` tgamma_lower(T1 a, T2 z);
@@ -263,6 +278,16 @@ large a and x the errors will still get you eventually, although this does
delay the inevitable much longer than other methods. Use of /log(1+x)-x/ here delay the inevitable much longer than other methods. Use of /log(1+x)-x/ here
is inspired by Temme (see references below). is inspired by Temme (see references below).
The natural log of the normalized upper incomplete gamma function is computed
as expected except when the normalized upper incomplete gamma function
begins to underflow. This approximately occurs at
((x > 1000) && ((a < x) || (fabs(a - 50) / x < 1))) || ((x > log_max_value<T>() - 10) && (x > a))
in which case an expansion, for large x, of the (non-normalised) upper
incomplete gamma function is used. The return is then normalised by subtracting
the log of the gamma function and adding /a log(x)-x-log(x)/.
[h4 References] [h4 References]
* N. M. Temme, A Set of Algorithms for the Incomplete Gamma Functions, * N. M. Temme, A Set of Algorithms for the Incomplete Gamma Functions,