Re: [IEEE P1788 er subgroup]: My first cut at the Level 1 list...
> Date: Fri, 27 Mar 2009 12:15:40 +0100
> From: Ulrich Kulisch <Ulrich.Kulisch@xxxxxxxxxxx>
> To: Dan Zuras Intervals <intervals08@xxxxxxxxxxxxxx>
> CC: stds-1788 <stds-1788@xxxxxxxxxxxxxxxxx>
> Subject: Re: [IEEE P1788 er subgroup]: My first cut at the Level 1 list...
>
>
> Dear colleagues:
>
> Please see the attached comments.
>
> Best whishes
> Ulrich Kulisch
>
>
> . . .
> >
> >> >
> >> > OK, from the top...
> >> >
> >> >
> >> > Page 7, 3.1 Level 1 debates, 1a: Should the model be R or R*?
> >> >
> >> > I must admit that both the principles & I are largely agnostic
> >> > on this point. That having been said, I will argue in favor of
> >> > R*.
> >> >
> >> > It seems to me that among the standard intervals the effective
> >> > difference between R & R* boils down to the exclusion or
> >> > inclusion of the two elements [-oo,-oo] & [+oo,+oo] in IR.
> >> >
> >> > While we can argue about the details of things like 1/[0,0],
> >> > I think we can all agree that the function f(xx) = 1/(xx - 1)^2
> >> > should be [+oo,+oo] when evaluated at xx = [1,1].
> >> >
> >> > It is true that (1) & (2) would permit us to return [+max,+oo]
> >> > in this case but this would violate the best practices we want
> >> > out of (3).
> >> >
> >> > So, I conclude we want R* but admit the case is weak.
> >> >
Hi, Ulrich,
While Ulrich's argument ASSUMES that the basis for the interval
domain is R, he is quite correct. If we choose R then both
1/([1,1] - 1)^2 & 1/[0,0] have no answer. Or more precisely,
those arguments are outside the domain of those functions so
the answer is [empty].
But in assuming the answer we are dodging the question which
is: SHOULD the basis for our arithmetic be R or R* (including
-oo & +oo)?
As the distinction between the two seems to be whether or not
we include the two elements [-oo,-oo] & [+oo,+oo] in our set
of intervals, I was exploring what would happen if we made it
R*.
There is also the fact, which I failed to mention, that the
basis for our underlying floating-point arithmetic is R*.
Still, Ulrich is correct to point out that including the two
infinity singletons doesn't answer all our questions. Namely
we still must decide what to do with odd order singularities
such as 1/xx. I don't know the answer. Maybe choosing R
instead is the correct answer.
I thought that if we allow ourselves to use R* in designing
the underpinnings of intervals we should also allow our users
the same benifits.
So I chose R* but, as I said above, I admit that the case is
a weak one.
It is not a result I'm married to so I would be happy either
way we go so long as any inconsistencies are ironed out first.
OK, ironed out eventually. :-)
Nice to hear from you Ulrich. Its been a long time.
Dan