If you have access (I do not) check
Journal article by Gary Tietjen; The American Statistician, Vol. 52, 1998
Recursive schemes for calculating common cumulative distributions.
Estimating the cumulative distribution function (cdf) for various
distributions is at the heart of statistical computing. Both speed and
accuracy are necessary. Although 3 dp accuracy is usually sufficient for
a cdf, the use of cumulative values in other computations makes 5 dp
accuracy highly desirable. Approximations using alternating series are
usually plagued with round off problems. Series in which all the terms
are positive avoid roundoff...
My be useful, perhaps not...
I checked a few different taylor series expansions, and the results were
almost identical with respect to poor convergence in the area of
interest. I even tried Aitken's method to speed convergence
p = (p_{n+1}-p_{n})^2/(p_{n+2} - 2p_{n+1}+p_{n})
This of course leads into Steffensen's algorithm, which essentially uses
Aitken's recursively.
The other thought that I had, is that at some point, you can switch to
an interpolation algorithm for suitable values of x.
Most people seem to quote Bosten and Battiste (1974) and their methods.
or use Didonato/Morris Algortihm 708 [1] as in Boost
Boost is good, I like boost...
I fear, that there is no solution in principle. You lost the accuracy
while subtracting. OOo should have "warning" notes like Excel, where
you can bring such information to the user.
Interval Arithmetic would be useful here... You have a guaranteed bounds
on the solution. Bill Walster at Sun has worked with Interval Arithmetic
(in 1998). If they had high precision routines just sitting around that
you can share between departments.
Kind regards
Regina
[1] A.R.DiDonato, A.H.Morris: Algorithm 708. Significant Digit
Computation if the Incomplete Beta Function Ratios
in ACM Transactions on Mathematical Software, Vol. 18, No 3, Sept.
1992, Pages 360-373
Unfortunately no free access, you need a public library.
--
Andrew Pitonyak
My Macro Document: http://www.pitonyak.org/AndrewMacro.odt
My Book: http://www.hentzenwerke.com/catalog/oome.htm
Info: http://www.pitonyak.org/oo.php
See Also: http://documentation.openoffice.org/HOW_TO/index.html
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