Baker Kearfott wrote:
Ah, yes, now I remember. Even though we are working only
with real numbers, an interval can be bounded or not,
and we can distinguish that with a decoration. However,
a consequence of the fact we interpret [a,Infinity) to
be a set of real numbers is [a,Infinity)*0 = 0. Thus,
the question of the point product Infinity*0 does not
even arise.
Right.
...although it _does_ arise when computing the endpoints of an interval.
So it is relevant to the _implementation_ of interval arithmetic
operations. I think this was Ian's main point (no pun intended).
For example, _if_ IEEE 754 had made a distinction between Infinity and
Overflow, then our interval arithmetic implementations would already have
a fast floating-point operation Overflow*0=0 in hardware, which means
computing the appropriate
interval endpoint of the product:
[1,Overflow]*[0,0]=[1*0,Overflow*0]=[0,0]
would already be performed fast and cheaply in hardware.
This would be good for all of our (currently slow) software
implementations of interval arithmetic.
This is in contrast to how IEEE 754 arithmetic hardware as it _actually_
exists today yields:
[1,Infinity]*[0,0]=[1*0,Infinity*0]=[0,NaN]
which according to Motion 5 is not the correct interval result. So it
requires an expensive software routine to "fix" the [0,NaN] result and
turn it into the correct [0,0] interval arithmetic result.
Too bad for us interval practitioners.
So it is a kind of wishful thinking about what "could have been" if IEEE
754 had done things differently. Perhaps it is then also natural to ask
the question: is this a problem IEEE 1788 wants to fix?
I don't know the answer to that question.
Note that Ian's Overflow can simply be viewed as a decorated Infinity.
Only in this case the "decoration" is for a floating-point value
(Infinity) as opposed to an interval value, which is what we're normally
used to thinking about since
Motion 8.
Nate
P.S. Ian, feel free to jump in if I do not do justice to represent your
point.
Baker
On 9/21/2010 04:25, Arnold Neumaier wrote:
Ian McIntosh wrote:
.
.
.
AN> We introduced in Motion 8 decorations to be able to distinguish the
two
where necessary inninterval bounds by having the decoration IsBounded.
.
.
.
My only claims are that by distinguishing Overflow from Infinity you
can
know whether multiplying whichever by zero gives zero or NaNQ in
floating
point, and that if 754 already had that it would be helpful for
interval
arithmetic.
See my comment above.
The distiction can already be made with decorations.
We do not need dubious mathematically indeterminate nunbers.
--
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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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