projects / ceph.git / blame
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|---|---|---|
| 7c673cae FG |
1 | [section:gamma_ratios Ratios of Gamma Functions] |
| 2 | ||
| 3 | `` | |
| 4 | #include <boost/math/special_functions/gamma.hpp> | |
| 5 | `` | |
| 6 | ||
| 7 | namespace boost{ namespace math{ | |
| 8 | ||
| 9 | template <class T1, class T2> | |
| 10 | ``__sf_result`` tgamma_ratio(T1 a, T2 b); | |
| 11 | ||
| 12 | template <class T1, class T2, class ``__Policy``> | |
| 13 | ``__sf_result`` tgamma_ratio(T1 a, T2 b, const ``__Policy``&); | |
| 14 | ||
| 15 | template <class T1, class T2> | |
| 16 | ``__sf_result`` tgamma_delta_ratio(T1 a, T2 delta); | |
| 17 | ||
| 18 | template <class T1, class T2, class Policy> | |
| 19 | ``__sf_result`` tgamma_delta_ratio(T1 a, T2 delta, const ``__Policy``&); | |
| 20 | ||
| 21 | }} // namespaces | |
| 22 | ||
| 23 | [h4 Description] | |
| 24 | ||
| 25 | template <class T1, class T2> | |
| 26 | ``__sf_result`` tgamma_ratio(T1 a, T2 b); | |
| 27 | ||
| 28 | template <class T1, class T2, class ``__Policy``> | |
| 29 | ``__sf_result`` tgamma_ratio(T1 a, T2 b, const ``__Policy``&); | |
| 30 | ||
| 31 | Returns the ratio of gamma functions: | |
| 32 | ||
| 33 | [equation gamma_ratio0] | |
| 34 | ||
| 35 | [optional_policy] | |
| 36 | ||
| 37 | Internally this just calls `tgamma_delta_ratio(a, b-a)`. | |
| 38 | ||
| 39 | template <class T1, class T2> | |
| 40 | ``__sf_result`` tgamma_delta_ratio(T1 a, T2 delta); | |
| 41 | ||
| 42 | template <class T1, class T2, class ``__Policy``> | |
| 43 | ``__sf_result`` tgamma_delta_ratio(T1 a, T2 delta, const ``__Policy``&); | |
| 44 | ||
| 45 | Returns the ratio of gamma functions: | |
| 46 | ||
| 47 | [equation gamma_ratio1] | |
| 48 | ||
| 49 | [optional_policy] | |
| 50 | ||
| 51 | Note that the result is calculated accurately even when /delta/ is | |
| 52 | small compared to /a/: indeed even if /a+delta ~ a/. The function is | |
| 53 | typically used when /a/ is large and /delta/ is very small. | |
| 54 | ||
| 55 | The return type of these functions is computed using the __arg_promotion_rules | |
| 56 | when T1 and T2 are different types, otherwise the result type is simple T1. | |
| 57 | ||
| 58 | [graph tgamma_delta_ratio] | |
| 59 | ||
| 60 | [h4 Accuracy] | |
| 61 | ||
| 62 | The following table shows the peak errors (in units of epsilon) | |
| 63 | found on various platforms with various floating point types. | |
| 64 | Unless otherwise specified any floating point type that is narrower | |
| 65 | than the one shown will have __zero_error. | |
| 66 | ||
| 67 | [table_tgamma_delta_ratio] | |
| 68 | ||
| 69 | [table_tgamma_ratio] | |
| 70 | ||
| 71 | [h4 Testing] | |
| 72 | ||
| 73 | Accuracy tests use data generated at very high precision | |
| 74 | (with [@http://shoup.net/ntl/doc/RR.txt NTL RR class] | |
| 75 | set at 1000-bit precision: about 300 decimal digits) | |
| 76 | and a deliberately naive calculation of [Gamma](x)/[Gamma](y). | |
| 77 | ||
| 78 | [h4 Implementation] | |
| 79 | ||
| 80 | The implementation of these functions is very similar to that of | |
| 81 | __beta, and is based on combining similar power terms | |
| 82 | to improve accuracy and avoid spurious overflow/underflow. | |
| 83 | ||
| 84 | In addition there are optimisations for the situation where /delta/ | |
| 85 | is a small integer: in which case this function is basically | |
| 86 | the reciprocal of a rising factorial, or where both arguments | |
| 87 | are smallish integers: in which case table lookup of factorials | |
| 88 | can be used to calculate the ratio. | |
| 89 | ||
| 90 | [endsect][/section:gamma_ratios Ratios of Gamma Functions] | |
| 91 | ||
| 92 | [/ | |
| 93 | Copyright 2006 John Maddock and Paul A. Bristow. | |
| 94 | Distributed under the Boost Software License, Version 1.0. | |
| 95 | (See accompanying file LICENSE_1_0.txt or copy at | |
| 96 | http://www.boost.org/LICENSE_1_0.txt). | |
| 97 | ] | |
| 98 |