mirror of
https://github.com/saitohirga/WSJT-X.git
synced 2024-11-07 09:44:16 -05:00
143 lines
4.9 KiB
Plaintext
143 lines
4.9 KiB
Plaintext
[section:digamma Digamma]
|
|
|
|
[h4 Synopsis]
|
|
|
|
``
|
|
#include <boost/math/special_functions/digamma.hpp>
|
|
``
|
|
|
|
namespace boost{ namespace math{
|
|
|
|
template <class T>
|
|
``__sf_result`` digamma(T z);
|
|
|
|
template <class T, class ``__Policy``>
|
|
``__sf_result`` digamma(T z, const ``__Policy``&);
|
|
|
|
}} // namespaces
|
|
|
|
[h4 Description]
|
|
|
|
Returns the digamma or psi function of /x/. Digamma is defined as the logarithmic
|
|
derivative of the gamma function:
|
|
|
|
[equation digamma1]
|
|
|
|
[graph digamma]
|
|
|
|
[optional_policy]
|
|
|
|
The return type of this function is computed using the __arg_promotion_rules:
|
|
the result is of type `double` when T is an integer type, and type T otherwise.
|
|
|
|
[h4 Accuracy]
|
|
|
|
The following table shows the peak errors (in units of epsilon)
|
|
found on various platforms with various floating point types.
|
|
Unless otherwise specified any floating point type that is narrower
|
|
than the one shown will have __zero_error.
|
|
|
|
[table_digamma]
|
|
|
|
As shown above, error rates for positive arguments are generally very low.
|
|
For negative arguments there are an infinite number of irrational roots:
|
|
relative errors very close to these can be arbitrarily large, although
|
|
absolute error will remain very low.
|
|
|
|
[h4 Testing]
|
|
|
|
There are two sets of tests: spot values are computed using
|
|
the online calculator at functions.wolfram.com, while random test values
|
|
are generated using the high-precision reference implementation (a
|
|
differentiated __lanczos see below).
|
|
|
|
[h4 Implementation]
|
|
|
|
The implementation is divided up into the following domains:
|
|
|
|
For Negative arguments the reflection formula:
|
|
|
|
digamma(1-x) = digamma(x) + pi/tan(pi*x);
|
|
|
|
is used to make /x/ positive.
|
|
|
|
For arguments in the range [0,1] the recurrence relation:
|
|
|
|
digamma(x) = digamma(x+1) - 1/x
|
|
|
|
is used to shift the evaluation to [1,2].
|
|
|
|
For arguments in the range [1,2] a rational approximation [jm_rationals] is used (see below).
|
|
|
|
For arguments in the range [2,BIG] the recurrence relation:
|
|
|
|
digamma(x+1) = digamma(x) + 1/x;
|
|
|
|
is used to shift the evaluation to the range [1,2].
|
|
|
|
For arguments > BIG the asymptotic expansion:
|
|
|
|
[equation digamma2]
|
|
|
|
can be used. However, this expansion is divergent after a few terms:
|
|
exactly how many terms depends on the size of /x/. Therefore the value
|
|
of /BIG/ must be chosen so that the series can be truncated at a term
|
|
that is too small to have any effect on the result when evaluated at /BIG/.
|
|
Choosing BIG=10 for up to 80-bit reals, and BIG=20 for 128-bit reals allows
|
|
the series to truncated after a suitably small number of terms and evaluated
|
|
as a polynomial in `1/(x*x)`.
|
|
|
|
The arbitrary precision version of this function uses recurrence relations until
|
|
x > BIG, and then evaluation via the asymptotic expansion above. As special cases
|
|
integer and half integer arguments are handled via:
|
|
|
|
[equation digamma4]
|
|
|
|
[equation digamma5]
|
|
|
|
The rational approximation [jm_rationals] in the range [1,2] is derived as follows.
|
|
|
|
First a high precision approximation to digamma was constructed using a 60-term
|
|
differentiated __lanczos, the form used is:
|
|
|
|
[equation digamma3]
|
|
|
|
Where P(x) and Q(x) are the polynomials from the rational form of the Lanczos sum,
|
|
and P'(x) and Q'(x) are their first derivatives. The Lanzos part of this
|
|
approximation has a theoretical precision of ~100 decimal digits. However,
|
|
cancellation in the above sum will reduce that to around `99-(1/y)` digits
|
|
if /y/ is the result. This approximation was used to calculate the positive root
|
|
of digamma, and was found to agree with the value used by
|
|
Cody to 25 digits (See Math. Comp. 27, 123-127 (1973) by Cody, Strecok and Thacher)
|
|
and with the value used by Morris to 35 digits (See TOMS Algorithm 708).
|
|
|
|
Likewise a few spot tests agreed with values calculated using
|
|
functions.wolfram.com to >40 digits.
|
|
That's sufficiently precise to insure that the approximation below is
|
|
accurate to double precision. Achieving 128-bit long double precision requires that
|
|
the location of the root is known to ~70 digits, and it's not clear whether
|
|
the value calculated by this method meets that requirement: the difficulty
|
|
lies in independently verifying the value obtained.
|
|
|
|
The rational approximation [jm_rationals] was optimised for absolute error using the form:
|
|
|
|
digamma(x) = (x - X0)(Y + R(x - 1));
|
|
|
|
Where X0 is the positive root of digamma, Y is a constant, and R(x - 1) is the
|
|
rational approximation. Note that since X0 is irrational, we need twice as many
|
|
digits in X0 as in x in order to avoid cancellation error during the subtraction
|
|
(this assumes that /x/ is an exact value, if it's not then all bets are off). That
|
|
means that even when x is the value of the root rounded to the nearest
|
|
representable value, the result of digamma(x) ['[*will not be zero]].
|
|
|
|
|
|
[endsect][/section:digamma The Digamma Function]
|
|
|
|
[/
|
|
Copyright 2006 John Maddock and Paul A. Bristow.
|
|
Distributed under the Boost Software License, Version 1.0.
|
|
(See accompanying file LICENSE_1_0.txt or copy at
|
|
http://www.boost.org/LICENSE_1_0.txt).
|
|
]
|
|
|