Re: Motion P1788/M0014.01: 6.1_and_6.2_of_document: up for discussion
> Subject: Re: Motion P1788/M0014.01: 6.1_and_6.2_of_document: up for discussion
> From: John Pryce <j.d.pryce@xxxxxxxxxxxx>
> Date: Tue, 27 Apr 2010 16:51:30 +0100
> To: stds-1788 <stds-1788@xxxxxxxxxxxxxxxxx>
>
> Folks
>
> Dan has raised a point that worried me when I wrote the text
> in question. I did send a query round at the time but don't
> recall anyone answering.
>
> On 22 Apr 2010, at 11:43, Dan Zuras Intervals wrote:
> > One subtle point that is not made clear by this is that
> > it requires that a representation of the form (mid,del1,
> > del2) which represents the interval [mid+del1,mid+del2]
> > must do so exactly. That is lowerBound = mid + del1 &
> > upperBound = mid + del2 EXACTLY, with no rounding errors.
> >
> > This is not so much of a problem but it does mean that
> > any system choosing to use this mid-rad form in an
> > attempt to speed things up in the |rad| << |mid| case
> > must find some other way of representing semi-infinite
> > intervals such as [2,+oo] & [3,+oo] which don't fit in
> > this format.
>
> PLEASE, Multi-precision interval computation folk:
>
> (a) What does "FP Format" and corresponding "Interval Format" MEAN
> in your context?
> Is there a potentially infinite sequence of them, like "binary_N"
> where N is an arbitrary multiple of 32, say?
> Or just ONE, the union of all these (regarding each one as a
> subset of the extended reals)?
> (b) If your implementation uses something like (mid,del1,del2)
> does it achieve what Dan says:
> > lowerBound = mid + del1 & upperBound = mid + del2 EXACTLY ?
> The answer almost surely depends on the answer to (a).
Actually, that may not be sufficient. Except for
possibly unreasonably large precision, that is.
>
> Basically I'm asking whether you can map what 6.1, 6.2 say into
> what you implement
> - easily
> - in a tortured and unnatural way
> - not at all.
>
> This didn't matter so much until this motion, but now it does.
> Probably a lot.
>
> John
The point is subtle in several ways.
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.)
But the semi infinite intervals, [2,+oo], [-oo,3], &
the like, live in the inf-sup form & not the mid-rad
form.
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.
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.
The mid-rad folks may have a similar criticism about
semi-infinite intervals. After all, the mid-rad form
allows for less precision to be used in the rad part
which becomes a substantial computational advantage
in some applications.
What does this all mean?
Well, even in the common subset of finite intervals,
the requirement that m-r & m+r both be exactly
representable might mean that mid-rad arithmetic ends
up being "tortured and unnatural" as John suggests.
I'm not sure.
Then there must be some break out for the case when
semi infinite intervals are part of the problem.
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.
But that will be considered OK in the case where you
get the answer today rather than 10^10 years from now.
In short, while no mid-rad form that exists today
likely meets these conditions, I think we will be
motivated to figure out how to do it.
IMHO, of course. As always...
Dan