On Feb 10, 6:48 pm, David Harvey <[EMAIL PROTECTED]> wrote:
> On Feb 10, 2007, at 9:40 PM, Carl Witty wrote:
>
> > Some IEEE doubles are exact -- you can't tell just by looking at it
> > whether a value of 0.5 is intended to be exact or approximate.
>
> True.
>
> Can I check I understand the point of real interval arithmetic. I've
> never done any computational work with such objects, and generally I
> even stay away from floating point like the plague :-). I presume the
> idea is that you take some algorithm which normally operates on
> reals, and you would like to be able to just "run it" on real
> intervals instead, so that if you have some idea of the error in your
> input data then the system automatically gives you proven error
> bounds in the output. Is that the basic idea? I take it the whole
> real interval thing is not just some academic curiosity.
That's one of the uses of interval arithmetic. My own use for
interval arithmetic is dealing with algebraic numbers. It is possible
to do exact computations with algebraic numbers, but if the question
you are asking is not too difficult then it is often far more
efficient to compute with interval approximations of the numbers in
question. Here "difficult" means "requires computing the sign of an
algebraic number which is very close to 0"; so checking that
(sqrt(2)+sqrt(3))^5 >= 300 is easy, since the true value of the left
hand side is about 308.302, but checking that phi^150 <
22291846172619859445381409012498 is hard, since the two sides differ
by less than 10^{-31}. (You need intervals with at least 217-bit
mantissas to prove the latter fact; for proving the former, 8-bit
mantissas suffice.)
Carl Witty
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