Re: on-line calculator

David W. Cantrell wrote:

Julio Di Egidio <julio@xxxxxxxxxxxxx> wrote:
David W. Cantrell wrote:

Julio Di Egidio <julio@xxxxxxxxxxxxx> wrote:
troymius wrote:


[snip]

In any case, you make me curious: would I be right
in thinking that a
correct answer might have been (implying we are in
IR*):

tan(pi/2) = [-oo,-owf] u [+owf,+oo] c [-oo,+oo]

so, by containment:

tan(pi/2) = [-oo,+oo]

Yes, you are correct if you insist that the answer be
a single interval
which is a subset (which, in this case, is improper)
of R*, the two-point
extension of the reals. But realize that saying
tan(pi/2) = R* is "not that
useful" in very much the same way as you said,
correctly, that tan(pi/2)
giving NaN is "not that useful."


Right, and I was more or less aware of that kind of potential objection as soon as I had written my "correct" answer, although - as you of course know - there is a hierarchy of closed systems from the simple to the full ones, whatever "full" might mean: see below.



Far more useful here, IMO, would be to use intervals
in the _one_-point
extension of the reals. There, having merged -oo and
+oo into a single
unsigned infinity, your [-oo,-owf] U [+owf,+oo] has
become a single
interval running from +owf through unsigned infinity
and ending at -owf.
Such an interval is sometimes notated in the
generalized form [+owf,-owf].
Being far narrower than R*, it gives far more useful
information about
tan(pi/2).


Yep, agreed, but I still have doubts about the "usefulness" of the affine infinity vs the projective infinity in other cases. No example comes to my mind at the moment, but I am generally thinking about the irreversibility of such operations where you basically lose the sign of the original operands.

I'd then be maybe more interested in a sharpening of interval arithmetic where the two above intervals do *not* get hulled, although I suppose that might lead to some sort of binary explosion in the complexity of interval computations...

Still, in such case, I would think it leads to very straightforward recursive implementations (a la divide-and-conquer again, so inherently parallelizable, as well as easily made non-recursive), but - again - I might be missing something.

Your thoughts on this?

-LV



David
.

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