Re: Potential well and decorations
On 2011-07-07 09:02:20 -0500, Nate Hayes wrote:
> Dominique Lohez wrote:
> >Ralph Baker Kearfott a écrit :
> >>On 07/04/2011 12:51 PM, Nate Hayes wrote:
> >>>Dominique Lohez wrote:
> >>>>Nate Hayes a écrit :
> >>>>>If for example we have the interval extension
> >>>>>
> >>>>>P(X) = U(X) \union V(X)
> >>>>>
> >>>>>with
> >>>>>
> >>>>>U(X) = sqrt((|X| \intersect [roundDown(sqrt(a)),+Inf])^2-a)
> >>>>>V(X) = -sqrt(a-(|X| \intersect [0,roundUp(a)]))
> >>>>>
> >>
> >>If I encountered a function like Dan's, that is,
> >>
> >>if (abs(x) > sqrt(a))
> >> y = sqrt(x^2-a);
> >>else
> >> y = - sqrt(a-x^2);
> >>end
[...]
> One thing I notice is if at Level 2 the function is implemented:
>
> U(X) = sqrt((|X| \intersect [roundUp(sqrt(a)),+Inf])^2-a)
> V(X) = -sqrt(a-(|X| \intersect [0,roundDown(sqrt(a))])^2)
>
> so that there is a 1 ULP gap between the domains of U(X) and V(X) at
> sqrt(a), then P(X) always gives a valid range enclosure and "continuous"
> decoration whenever sqrt(a) is an element of |X|.
>
> The reason this works is because the potential function is monotonic and
> continuous over the Level 2 interval [roundDown(sqrt(a)),roundUp(sqrt(a))],
> and any Level 2 |X| containing sqrt(a) will be a superset of this interval.
>
> However, this trick doesn't seem to work for implicit data types with
> dynamic precision, e.g., MPFR etc.
Anyway this works only in a particular case. For instance, if a is not
representable exactly by a FP number (or not known exactly), it will
be represented by an interval, and this tricks no longer works in such
a case.
Moreover you can still extend the trick to dynamic precision (as long
as you can entirely control the precision, but this is possible with
MPFR). I mean, the problem is not whether a data type is implicit or
not, but whether the precision can be controlled.
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/>
100% accessible validated (X)HTML - Blog: <http://www.vinc17.net/blog/>
Work: CR INRIA - computer arithmetic / Arénaire project (LIP, ENS-Lyon)