Re: Multi-precision (was...Please give me advice)
On 2010-10-15 12:44:15 +0200, Arnold Neumaier wrote:
> Vincent Lefevre wrote:
> >On 2010-10-14 17:43:19 +0200, Arnold Neumaier wrote:
> >>For me, the criterion distinguishing between good and bad
> >>representations is whether they can represent highly asymmetric
> >>intervals such as [1,1e6] or [-1,inf] without much overestimation.
> >
> >Why? If one uses valid mode, an implementation could still return the
> >same result for f([-1,inf]) and f(Entire). So, requiring [-1,inf] to
> >be representable exactly would be useless.
>
>
> Of course, one can implement all operations to give the result Entire.
> Then requiring any interval besides Entire to be representable exactly
> would be useless.
This is not what I've said. Only unbounded intervals (and maybe some
other large intervals) would be seen as Entire, and this wouldn't be
a problem for applications that use IA to certify a result that could
be computed in FP arithmetic otherwise.
> But I assumed that a reasonable standard would impose at least some
> sensible requirements on accuracy.
Well, [-1,inf] is far from being an accurate representation of a real
number!
> In particular, one can and should require that all arithmetic
> operations +-*/ preserve the sign of the optimal result.
This is something very arbitrary. For instance, a mid-rad implementation
(that doesn't have this property) could give a better accurary than
an inf-sup implementation that uses a format of the same size (because
a mid-rad implementation could use a higher precision for mid). Thus
I disagree.
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/>
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Work: CR INRIA - computer arithmetic / Arénaire project (LIP, ENS-Lyon)