Re: A question Re: Level 1 <---> level 2 mappings; arithmetic versus applications
Nate,
On 6/30/2010 16:42, Nate Hayes wrote:
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.
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It all hangs together for me until mention of extracting bounds of the
Level 2 interval losslessly as a floating-point number. I don't see how
a mid-rad implementaiton (or even some of the other examples you gave)
could conform to that.
It seems to hinge on what objects we are talking about. Take complex
intervals (which probably are outside the scope of 1788, at
least for now, but which illustrate the point): The mid-rad representation
gives a different set of objects than the inf-sup representation.
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. 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.)
Sorry to be a buzzkill, but I guess I'm a little lost.
Did I clarify?
I think I'm just going to shut-up and listen...
Nate
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."
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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