> > The LLtest sounds like a practical pseudo random generator to me.
>
> The residual may appear pseudorandom - though the period may be shorter than
> you think - in particular if 2^p-1 is a Mersenne prime, the residual becomes
> rather predictable after the first p-2 iterations. There has also been
> discussion in the past on the possibility of using the recurrence period of
> the residual for extraction of actual factors of composite Mersenne numbers
> in a manner similar to Pollard's rho method.
It does seem a pity that we can't use the output of a LL test proving
a number composite to help in some way with finding factors.
>
> However what matters for roundoff error theory is the randomness of otherwise
> of the individual elements before and after the Fourier transformation is
> applied.
These are the numbers I take to be "pseudo random" although I don't think
this is necessary to produce normally distributed roundoff errors.
>We've been through this before, but any similarity of the
> distribution of these numbers to a uniform normal distribution appears only
> after taking the logarithm several times i.e. there is a "power law" which
> generates a distribution of actual errors which is strongly skewed, having a
> very, very long tail on the high side.
>
> Looking at this in detail would be a worthwhile exercise.
>
> Regards
> Brian Beesley
I wish someone would step in and help here. We seem to be
at loggerheads due to some mutual misunderstanding.
It sounds to me as if you think you have found a counter example
to the Central Limit Theorem :-)
It is the signed actual roundoff errors which I maintain are
normally distributed with mean zero.
The probability of a given bit in the FP representation being in
error is bell shaped centred on the standard deviation of the
actual errors, and is not normal, but equally well defined.
David Eddy
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