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111 lines
3.4 KiB
Plaintext
111 lines
3.4 KiB
Plaintext
[section:beta_function Beta]
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[h4 Synopsis]
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``
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#include <boost/math/special_functions/beta.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`` beta(T1 a, T2 b);
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template <class T1, class T2, class ``__Policy``>
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``__sf_result`` beta(T1 a, T2 b, const ``__Policy``&);
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}} // namespaces
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[h4 Description]
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The beta function is defined by:
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[equation beta1]
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[graph beta]
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[optional_policy]
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There are effectively two versions of this function internally: a fully
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generic version that is slow, but reasonably accurate, and a much more
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efficient approximation that is used where the number of digits in the significand
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of T correspond to a certain __lanczos. In practice any built-in
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floating-point type you will encounter has an appropriate __lanczos
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defined for it. It is also possible, given enough machine time, to generate
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further __lanczos's using the program libs/math/tools/lanczos_generator.cpp.
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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.
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[h4 Accuracy]
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The following table shows peak errors for various domains of input arguments,
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along with comparisons to the __gsl and __cephes libraries. Note that
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only results for the widest floating point type on the system are given as
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narrower types have __zero_error.
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[table_beta]
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Note that the worst errors occur when a or b are large, and that
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when this is the case the result is very close to zero, so absolute
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errors will be very small.
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[h4 Testing]
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A mixture of spot tests of exact values, and randomly generated test data are
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used: the test data was computed using
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[@http://shoup.net/ntl/doc/RR.txt NTL::RR] at 1000-bit precision.
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[h4 Implementation]
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Traditional methods of evaluating the beta function either involve evaluating
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the gamma functions directly, or taking logarithms and then
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exponentiating the result. However, the former is prone to overflows
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for even very modest arguments, while the latter is prone to cancellation
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errors. As an alternative, if we regard the gamma function as a white-box
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containing the __lanczos, then we can combine the power terms:
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[equation beta2]
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which is almost the ideal solution, however almost all of the error occurs
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in evaluating the power terms when /a/ or /b/ are large. If we assume that /a > b/
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then the larger of the two power terms can be reduced by a factor of /b/, which
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immediately cuts the maximum error in half:
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[equation beta3]
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This may not be the final solution, but it is very competitive compared to
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other implementation methods.
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The generic implementation - where no __lanczos approximation is available - is
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implemented in a very similar way to the generic version of the gamma function.
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Again in order to avoid numerical overflow the power terms that prefix the series and
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continued fraction parts are collected together into:
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[equation beta8]
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where la, lb and lc are the integration limits used for a, b, and a+b.
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There are a few special cases worth mentioning:
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When /a/ or /b/ are less than one, we can use the recurrence relations:
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[equation beta4]
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[equation beta5]
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to move to a more favorable region where they are both greater than 1.
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In addition:
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[equation beta7]
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[endsect][/section:beta_function The Beta Function]
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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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