Accuracy and error analysis (was Re: [Maxima] primes)

Allan Adler wrote:
> Maybe an implementation of interval arithmetic would do the job,
> with paranoid rounding to compensate for truncation errors. Then
> evaluate numerical functions on interval inputs and give interval
> answers.

This is possible and could be done. For example it is done in
Mathematica's software floats and also Interval. And in Maple with range
arithmetic.

It is good for some things but not for important things.
A convergent Newton iteration, viewed crudely via intervals,
never converges, but diverges. The necessity for human intervention
crops up pretty often, unless you are happy with very pessimistic
results. I think baysian analysis might be slightly less pessimistic
at the cost of sometimes being wrong.

People who have written about automated error analysis (and its
difficulty) include W. Kahan and Bruce Char.
RJF