George,
Given the uncertainty about the actual probability distribution function
involved it seems that the best way to estimate the probability of having
no roundoff greater than 0.5 in one iteration is to simply plot one minus
your empirical cumulative distribution function (on a log scale) versus
round off error on a scale that goes out to 0.5, and extrapolate by
"eyeball" to see where the curve would hit 0.5 and if it would be below
1e-10 or 1e-15 or whatever. Wherever the curve hits (or appears it would
hit) 0.5, times the number of needed LL iterations (p), gives an empirical
estimate of the probability that there will be no roundoff greater than 0.5
(for small values of probability, of course).
It is likely that some (few?) individual people who are "into" this could
use the raw Excel data numbers, not just the plots you pointed us to (which
I still can't make my computer see.) I would appreciate it if you could
ship me one or two files with probability histograms (or occurrance density
or whatever) versus roundoff value that could be loaded into Excel 4.0 for
Mac.
It might be interesting to see if the shape of this log-lin plot follows
the one given by the cumulative distribution function a-la Brian Beesley.
This function seems to have one degree of freedom, the number of rounded
off values of which the maximum one gives your function, which in his
example he chose to be 100. You probably know from the code what that
number should be. (Another degree of freedom is the underlying uncertainty
or the "scaling factor" for the x axis.) You might also know from the code
what the "underlying" sigma should be under the assumption that it is
something like (sigma0 * SquareRoot(n)) where n is the number of roundings
that occur in the cumulative calculation and sigma0 is the typical
"elementary" round-off error (scaled properly to be in the same units as
the 0.5 critical roundoff limit). If this is not available a-priori from
the code, it would still be interesting to estimate it by finding the best
fit to your empirical cumulative distribution function. If the resulting
smooth cumulative distribution function is a good match to the empirical
one it could serve as a guide for better extrapolation (or instead of the
extrapolation). If it is not a good match it would serve as a "warning"
that there may be "funny" things going on with the errors (particularly
worrysome if the "tail" is trending up, implying larger errors are more
likely than indicated by theory).
Todd Sauke
________________________________________________________________
Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
