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99 lines
2.9 KiB
Plaintext
99 lines
2.9 KiB
Plaintext
[section:gamma_ratios Ratios of Gamma Functions]
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``
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#include <boost/math/special_functions/gamma.hpp>
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``
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namespace boost{ namespace math{
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template <class T1, class T2>
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``__sf_result`` tgamma_ratio(T1 a, T2 b);
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template <class T1, class T2, class ``__Policy``>
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``__sf_result`` tgamma_ratio(T1 a, T2 b, const ``__Policy``&);
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template <class T1, class T2>
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``__sf_result`` tgamma_delta_ratio(T1 a, T2 delta);
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template <class T1, class T2, class Policy>
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``__sf_result`` tgamma_delta_ratio(T1 a, T2 delta, const ``__Policy``&);
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}} // namespaces
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[h4 Description]
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template <class T1, class T2>
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``__sf_result`` tgamma_ratio(T1 a, T2 b);
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template <class T1, class T2, class ``__Policy``>
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``__sf_result`` tgamma_ratio(T1 a, T2 b, const ``__Policy``&);
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Returns the ratio of gamma functions:
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[equation gamma_ratio0]
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[optional_policy]
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Internally this just calls `tgamma_delta_ratio(a, b-a)`.
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template <class T1, class T2>
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``__sf_result`` tgamma_delta_ratio(T1 a, T2 delta);
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template <class T1, class T2, class ``__Policy``>
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``__sf_result`` tgamma_delta_ratio(T1 a, T2 delta, const ``__Policy``&);
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Returns the ratio of gamma functions:
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[equation gamma_ratio1]
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[optional_policy]
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Note that the result is calculated accurately even when /delta/ is
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small compared to /a/: indeed even if /a+delta ~ a/. The function is
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typically used when /a/ is large and /delta/ is very small.
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The return type of these functions is computed using the __arg_promotion_rules
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when T1 and T2 are different types, otherwise the result type is simple T1.
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[graph tgamma_delta_ratio]
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[h4 Accuracy]
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The following table shows the peak errors (in units of epsilon)
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found on various platforms with various floating point types.
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Unless otherwise specified any floating point type that is narrower
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than the one shown will have __zero_error.
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[table_tgamma_delta_ratio]
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[table_tgamma_ratio]
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[h4 Testing]
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Accuracy tests use data generated at very high precision
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(with [@http://shoup.net/ntl/doc/RR.txt NTL RR class]
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set at 1000-bit precision: about 300 decimal digits)
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and a deliberately naive calculation of [Gamma](x)/[Gamma](y).
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[h4 Implementation]
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The implementation of these functions is very similar to that of
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__beta, and is based on combining similar power terms
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to improve accuracy and avoid spurious overflow/underflow.
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In addition there are optimisations for the situation where /delta/
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is a small integer: in which case this function is basically
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the reciprocal of a rising factorial, or where both arguments
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are smallish integers: in which case table lookup of factorials
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can be used to calculate the ratio.
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[endsect][/section:gamma_ratios Ratios of Gamma Functions]
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[/
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Copyright 2006 John Maddock and Paul A. Bristow.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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