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162 lines
5.4 KiB
Plaintext
162 lines
5.4 KiB
Plaintext
[section:lgamma Log Gamma]
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[h4 Synopsis]
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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 T>
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``__sf_result`` lgamma(T z);
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template <class T, class ``__Policy``>
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``__sf_result`` lgamma(T z, const ``__Policy``&);
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template <class T>
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``__sf_result`` lgamma(T z, int* sign);
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template <class T, class ``__Policy``>
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``__sf_result`` lgamma(T z, int* sign, const ``__Policy``&);
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}} // namespaces
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[h4 Description]
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The [@http://en.wikipedia.org/wiki/Gamma_function lgamma function] is defined by:
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[equation lgamm1]
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The second form of the function takes a pointer to an integer,
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which if non-null is set on output to the sign of tgamma(z).
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[optional_policy]
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[graph lgamma]
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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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the result is of type `double` if T is an integer type, or type T otherwise.
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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
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with various floating point types, along with comparisons to
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various other libraries. Unless otherwise specified any
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floating point type that is narrower than the one shown will have
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__zero_error.
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Note that while the relative errors near the positive roots of lgamma
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are very low, the lgamma function has an infinite number of irrational
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roots for negative arguments: very close to these negative roots only
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a low absolute error can be guaranteed.
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[table_lgamma]
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[h4 Testing]
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The main tests for this function involve comparisons against the logs of
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the factorials which can be independently calculated to very high accuracy.
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Random tests in key problem areas are also used.
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[h4 Implementation]
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The generic version of this function is implemented using Sterling's approximation
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for large arguments:
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[equation gamma6]
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For small arguments, the logarithm of tgamma is used.
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For negative /z/ the logarithm version of the
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reflection formula is used:
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[equation lgamm3]
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For types of known precision, the __lanczos is used, a traits class
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`boost::math::lanczos::lanczos_traits` maps type T to an appropriate
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approximation. The logarithmic version of the __lanczos is:
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[equation lgamm4]
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Where L[sub e,g][space] is the Lanczos sum, scaled by e[super g].
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As before the reflection formula is used for /z < 0/.
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When z is very near 1 or 2, then the logarithmic version of the __lanczos
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suffers very badly from cancellation error: indeed for values sufficiently
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close to 1 or 2, arbitrarily large relative errors can be obtained (even though
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the absolute error is tiny).
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For types with up to 113 bits of precision
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(up to and including 128-bit long doubles), root-preserving
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rational approximations [jm_rationals] are used
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over the intervals [1,2] and [2,3]. Over the interval [2,3] the approximation
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form used is:
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lgamma(z) = (z-2)(z+1)(Y + R(z-2));
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Where Y is a constant, and R(z-2) is the rational approximation: optimised
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so that it's absolute error is tiny compared to Y. In addition
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small values of z greater
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than 3 can handled by argument reduction using the recurrence relation:
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lgamma(z+1) = log(z) + lgamma(z);
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Over the interval [1,2] two approximations have to be used, one for small z uses:
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lgamma(z) = (z-1)(z-2)(Y + R(z-1));
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Once again Y is a constant, and R(z-1) is optimised for low absolute error
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compared to Y. For z > 1.5 the above form wouldn't converge to a
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minimax solution but this similar form does:
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lgamma(z) = (2-z)(1-z)(Y + R(2-z));
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Finally for z < 1 the recurrence relation can be used to move to z > 1:
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lgamma(z) = lgamma(z+1) - log(z);
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Note that while this involves a subtraction, it appears not
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to suffer from cancellation error: as z decreases from 1
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the `-log(z)` term grows positive much more
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rapidly than the `lgamma(z+1)` term becomes negative. So in this
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specific case, significant digits are preserved, rather than cancelled.
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For other types which do have a __lanczos defined for them
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the current solution is as follows: imagine we
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balance the two terms in the __lanczos by dividing the power term by its value
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at /z = 1/, and then multiplying the Lanczos coefficients by the same value.
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Now each term will take the value 1 at /z = 1/ and we can rearrange the power terms
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in terms of log1p. Likewise if we subtract 1 from the Lanczos sum part
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(algebraically, by subtracting the value of each term at /z = 1/), we obtain
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a new summation that can be also be fed into log1p. Crucially, all of the
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terms tend to zero, as /z -> 1/:
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[equation lgamm5]
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The C[sub k][space] terms in the above are the same as in the __lanczos.
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A similar rearrangement can be performed at /z = 2/:
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[equation lgamm6]
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[endsect][/section:lgamma The Log Gamma 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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