Re: "still unclear on infinite intervals"
Dan et al,
Ah, yes. There is a bit of history of grappling with
such issues within the interval community. An answer
based on limits is the (somewhat derided) cset theory
developed by John Pryce. That theory could be based
either on the reals or the extended reals, but
considered the result to be the set of all possible
limits (with corresponding interval enclosures
of such sets). (I am somewhat oversimplifying
here.) Since intervals describe sets, we are not
constrained (as 754) to have a point answer, even
if the arguments are degenerate intervals.
Anyway, 1788 seems to have more or less discarded
cset ("containment set") theory, for whatever reason.
Yes, indeed, [0,eps]*[large,infinity] = [0,infinity]
is a natural consequence of most of the proposed
systems I can think of right off the bat.
Regarding worry, I view our work as producing specifications
for a universal tool by trying to anticipate how it
will be used and specifying in order to make its
use easy. Some uses might not be anticipated, and
some difficulties might be unavoidable, since ease for
one use might result in more difficulty for another use.
One is tempted to try to prove somehow that a difficulty
is unavoidable. In any case, in such cases there is no
unambiguous answer, and our democratic process is the
best way to resolve it. Knowing that the tool exists
and knowing its exact specifications are valuable.
Best regards,
Baker
On 05/25/2011 03:25 PM, Dan Zuras Intervals wrote:
Date: Wed, 25 May 2011 12:07:25 -0500
From: Ralph Baker Kearfott<rbk5287@xxxxxxxxxxxxx>
To: Dan Zuras Intervals<intervals08@xxxxxxxxxxxxxx>
CC: "Corliss, George"<george.corliss@xxxxxxxxxxxxx>,
stds-1788@xxxxxxxxxxxxxxxxx
Subject: Re: "still unclear on infinite intervals"
Dan et al,
Wasn't the "still unclear" addressed when motion 3 passed?
That motion states that intervals are closed and connected
sets of real numbers. Thus, 0 * (any interval) = 0
(and infinite bounds are not to be construed to mean
infinity is included.)
Please correct me if I am wrong or if there are cases not covered
under that motion.
Baker
P.S. Other extended interval arithmetics define 0*\infty, and
can also be used. However, we have decided to standardize
with 0*interval = 0. (personal opinion): it is more important,
at least in this case, that the programmer know how the system
defines it, and that the system is the same across platforms,
than what the actual definition is.
Baker,
Yes, we defined it.
And, yes, I agree that it is more important to have us all
agree than the definition itself.
But my concern is whether or not it is correct in the sense
of inclusive than this particular definition.
This comes up in 754 as well.
The argument is that given a function f(x) such that limit
(x --> 0) f(x) = 0,& a function g(x) such that limit (x --> 0)
g(x) = infinity, we have, of course, that limit (x --> 0)
f(x)*g(x) = anything depending on the details of f()& g().
In 754 we concluded that the only meaningful answer was no
answer at all, namely NaN.
In 1788 we want to return the inclusive interval that is
sure to contain the answer.
I haven't said anything up till now because it seems to me
that any even very narrow interval will have the property
that [0,eps]*[large,infinity] = [0,infinity] which is, no
doubt, correct& inclusive.
And we have defined [+oo,+oo] out of existence so that's
not a problem.
And I shouldn't be worried because [0,0] shouldn't happen
in this case.
But it worries me still.
I guess we have it right& I worry too much.
Never mind.
Its OK.
Dan
--
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Ralph 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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