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143 lines
4.9 KiB
Plaintext
143 lines
4.9 KiB
Plaintext
[section:digamma Digamma]
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[h4 Synopsis]
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``
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#include <boost/math/special_functions/digamma.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`` digamma(T z);
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template <class T, class ``__Policy``>
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``__sf_result`` digamma(T z, const ``__Policy``&);
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}} // namespaces
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[h4 Description]
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Returns the digamma or psi function of /x/. Digamma is defined as the logarithmic
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derivative of the gamma function:
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[equation digamma1]
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[graph digamma]
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[optional_policy]
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The return type of this function is computed using the __arg_promotion_rules:
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the result is of type `double` when T is an integer type, and 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 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_digamma]
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As shown above, error rates for positive arguments are generally very low.
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For negative arguments there are an infinite number of irrational roots:
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relative errors very close to these can be arbitrarily large, although
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absolute error will remain very low.
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[h4 Testing]
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There are two sets of tests: spot values are computed using
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the online calculator at functions.wolfram.com, while random test values
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are generated using the high-precision reference implementation (a
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differentiated __lanczos see below).
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[h4 Implementation]
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The implementation is divided up into the following domains:
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For Negative arguments the reflection formula:
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digamma(1-x) = digamma(x) + pi/tan(pi*x);
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is used to make /x/ positive.
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For arguments in the range [0,1] the recurrence relation:
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digamma(x) = digamma(x+1) - 1/x
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is used to shift the evaluation to [1,2].
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For arguments in the range [1,2] a rational approximation [jm_rationals] is used (see below).
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For arguments in the range [2,BIG] the recurrence relation:
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digamma(x+1) = digamma(x) + 1/x;
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is used to shift the evaluation to the range [1,2].
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For arguments > BIG the asymptotic expansion:
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[equation digamma2]
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can be used. However, this expansion is divergent after a few terms:
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exactly how many terms depends on the size of /x/. Therefore the value
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of /BIG/ must be chosen so that the series can be truncated at a term
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that is too small to have any effect on the result when evaluated at /BIG/.
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Choosing BIG=10 for up to 80-bit reals, and BIG=20 for 128-bit reals allows
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the series to truncated after a suitably small number of terms and evaluated
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as a polynomial in `1/(x*x)`.
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The arbitrary precision version of this function uses recurrence relations until
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x > BIG, and then evaluation via the asymptotic expansion above. As special cases
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integer and half integer arguments are handled via:
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[equation digamma4]
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[equation digamma5]
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The rational approximation [jm_rationals] in the range [1,2] is derived as follows.
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First a high precision approximation to digamma was constructed using a 60-term
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differentiated __lanczos, the form used is:
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[equation digamma3]
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Where P(x) and Q(x) are the polynomials from the rational form of the Lanczos sum,
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and P'(x) and Q'(x) are their first derivatives. The Lanzos part of this
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approximation has a theoretical precision of ~100 decimal digits. However,
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cancellation in the above sum will reduce that to around `99-(1/y)` digits
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if /y/ is the result. This approximation was used to calculate the positive root
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of digamma, and was found to agree with the value used by
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Cody to 25 digits (See Math. Comp. 27, 123-127 (1973) by Cody, Strecok and Thacher)
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and with the value used by Morris to 35 digits (See TOMS Algorithm 708).
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Likewise a few spot tests agreed with values calculated using
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functions.wolfram.com to >40 digits.
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That's sufficiently precise to insure that the approximation below is
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accurate to double precision. Achieving 128-bit long double precision requires that
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the location of the root is known to ~70 digits, and it's not clear whether
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the value calculated by this method meets that requirement: the difficulty
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lies in independently verifying the value obtained.
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The rational approximation [jm_rationals] was optimised for absolute error using the form:
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digamma(x) = (x - X0)(Y + R(x - 1));
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Where X0 is the positive root of digamma, Y is a constant, and R(x - 1) is the
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rational approximation. Note that since X0 is irrational, we need twice as many
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digits in X0 as in x in order to avoid cancellation error during the subtraction
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(this assumes that /x/ is an exact value, if it's not then all bets are off). That
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means that even when x is the value of the root rounded to the nearest
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representable value, the result of digamma(x) ['[*will not be zero]].
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[endsect][/section:digamma The Digamma 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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