Re: A question Re: Level 1 <---> level 2 mappings; arithmetic versus applications
On 6/30/2010 15:05, Nate Hayes wrote:
Dan Zuras wrote:
Date: Wed, 30 Jun 2010 08:23:11 -0500
From: Ralph Baker Kearfott <rbk@xxxxxxxxxxxx>
To: Dan Zuras Intervals <intervals08@xxxxxxxxxxxxxx>
CC: Nate Hayes <nh@xxxxxxxxxxxxxxxxx>, stds-1788@xxxxxxxxxxxxxxxxx
Subject: A question Re: Level 1 <---> level 2 mappings; arithmetic
versus
applications
Dan,
On 6/30/2010 02:38, Dan Zuras Intervals wrote:
>> From: "Nate Hayes"<nh@xxxxxxxxxxxxxxxxx>
>> To: "Dan Zuras Intervals"<intervals08@xxxxxxxxxxxxxx>,
>> "Christian Keil"<c.keil@xxxxxxxxxxxxx>
I think this is hugely attractive, if we can pull it off.
Likewise.
These are a few of my initial (random) thoughts:
.
.
.
This makes me think it should be feasible to not even loosen the "tightest
possible subset" restriction, if the definitions are made carefully. It
does
seem that "tightest possible subset" might depend on the Level 2 format,
though.
It would be nice, from the point of view of reproducibility and
predictability to have a "tightest possible subset" restriction,
and it is encouraging that it might be possible to have that
with the mid-rad representation.
However, I now recall an earlier discussion and an earlier vote
regarding the dot product. In particular, there are issues of
implementation of "exact" dot product, "accurate" dot product,
or "faithful" dot product. I think only the "exact" dot product
would satisfy the "tightest possible subset" restriction if we
are returning the result of a dot product operation. In any case,
this issue would also need to be revisited, if we have both
tightest possible subset and required conforming dot product.
For example, any given Level 1 interval [a,b] has the property:
[a,b] = (m;r)
where (m;r) is a mid-rad interval and m=(a+b)/2, r=(b-a)/2. There is then a
tightest possible Level 2 inf-sup interval [A,B] such that A and B are
floating-point numbers and [a,b] \subset [A,B]. Likewise there is (I
believe) a tightest-possible Level 2 mid-rad interval (M;R) such that M and
R are floating-point numbers and (m;r) \subset (M;R). So everything is
fine,
just that if [A,B] and (M;R) are both mapped _exactly_ back to Level 1, it
will not necissarily be the same Level 1 interval. HOWEVER, both will be
guaranteed to be enclosures of the original Level 1 interval [a,b].
I don't see this should be a problem, as I don't expect any user or vendor
mixing inf-sup and mid-rad in a computation would expect otherwise. In
fact,
vendors and users will probably for the most part try to do as much of a
calculation as possible in one interval type or the other, minimizing
conversions between the two.
However I apologize for not being able to be more specific yet, if none of
this makes too much sense. This is all a bit new, and I'm still absorbing.
So although I may or may not eventually be proved wrong about this
intuition, my general impression is that it will probably work (i.e., no
compromise to "tightest possible subset" would be necessary). In any case,
its worth further study and investigation, I think.
All of that is very encouraging.
Best regards,
Baker
.
.
.
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R. Baker Kearfott, rbk@xxxxxxxxxxxxx (337) 482-5346 (fax)
(337) 482-5270 (work) (337) 993-1827 (home)
URL: http://interval.louisiana.edu/kearfott.html
Department of Mathematics, University of Louisiana at Lafayette
(Room 217 Maxim D. Doucet Hall, 1403 Johnston Street)
Box 4-1010, Lafayette, LA 70504-1010, USA
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