Re: Motion P1788/M0014.01: 6.1_and_6.2_of_document: up for discussion
On 2010-04-27 11:36:39 -0700, Dan Zuras Intervals wrote:
> First, there is the point that mid-rad intervals form
> a different subset of real intervals than do inf-sup
> intervals.
>
> I believe that all inf-sup intervals of finite width
> can be represented in mid-rad form of sufficient
> precision. That is the common subset.
>
> (There is a bit of a problem when the sum of the ULPs
> of the inf & the sup is odd but let's sweep that under
> the term 'sufficient precision' for now & ignore it.)
I think that the most important point is to have a way to represent
and compute intervals with a single full-precision value. The midrad
format would not be a goal, but just a mean to implement efficient
multiple-precision interval arithmetic. The (mid,del1,del2) format
allows that too (applications could also use alternative formats
such as (mid,del) with del = del2 = -del1 if this is interesting).
However, in the former case, I wouldn't say "mid" but "approx".
Moreover it is not obvious that one would have
lowerBound = mid + del1 & upperBound = mid + del2 EXACTLY
In fixed-point arithmetic, this may bo OK. But if one requires that
lowerBound, upperBound and mid are floating-point numbers in some
format (possibly multiple precision) and if the implementation
represents del1 and del2 as floating-point numbers in some other
floating-point format (with less precision, for efficiency), then
there's a problem.
> But the semi infinite intervals, [2,+oo], [-oo,3], &
> the like, live in the inf-sup form & not the mid-rad
> form.
In most cases where midrad would be used, I'd say that all
practical information is lost when one of the bounds is
infinite.
> On the other hand, mid-rad intervals of the form
> veryLarge +/- verySmall exist even in precisions
> substantially less precise than verySmall/veryLarge.
> These have no counterpart in the inf-sup world.
You mean intervals where verySmall is significantly smaller than
the ulp of veryLarge?
> Still, since such intervals form a non-overlapping
> set surrounding only the representable veryLarge
> numbers, I think they are of limited utility in
> converging on representable solutions.
Yes.
> But the computational advantage of mid-rad forms is
> so great in some applications that I believe it will
> be used anyway.
>
> After all, it is sufficient if inf-sup intervals are
> converted on input, the entire calculation performed
> in mid-rad form, & the results are converted back to
> inf-sup form. The final result may be more or less
> accurate than if it were done all in inf-sup form.
Yes.
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/>
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