Re: A question Re: Level 1 <---> level 2 mappings; arithmetic versus applications
On 2010-06-30 17:17:56 -0500, Ralph Baker Kearfott wrote:
> For real intervals, we can think of mid-rad at level 2 as
> giving a different set of objects than inf-sup, just as we think
> of binary and decimal floating point data as different sets. We
> can then talk of unique and lossless representation, within the
> particular set of objects. Also, conversion between the different
> sets then takes on the character of conversion between, say, binary
> and decimal, and we could specify, say, that the conversion be
> the tightest possible result, if we wanted.
Just like in base conversion (or other operation on floating-point
numbers), where one can have halfway cases, the "tightest possible
result" doesn't necessarily exist in a unique way at Level 2.
> We could also specify the mid-rad result of an operation in, say,
> mid-rad as being, say, the tightest possible superset of the true
> result within the set of floating point intervals represented in
> mid-rad form. The standard can dictate "smallest superset," (or
> whatever we deem appropriate) independently of whether we the set of
> interval objects is defined by mid-rad or inf-sup over the
> underlying floating point objects. (The underlying objects perhaps
> do not even need to be floating point, but I'm assuming for now that
> their cardinality is finite.)
I think it is sufficient to assume that there are no accumulation
points.
> P.S. Opinion: As someone with mathematical training, I would prefer
> "min-max" to "inf-sup" since the min and the max exist, because
> the mathematical objects to which we refer are closed and
> bounded sets. However, we are definitely stuck with "inf-sup,"
> because that's what practically everyone uses everywhere. In
> any case, a "min" is an "inf" and a "max" is a "sup."
I disagree: intervals at Level 2 can be unbounded. So, IMHO, "inf-sup"
is better.
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/>
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