Re: Do we have implicit semi-infinites...?
On 2012-03-14 10:22:51 -0700, Dan Zuras Intervals wrote:
> Folks,
>
> I have been contemplating that split function we discussed
> earlier. In so far as is possible, I would like to write
> it independent of the underlying interval form. In effect,
> not caring whether one uses explicit (inf-sup or [a,b] form)
> or implicit (mid-rad or <m,r> form).
>
> But I am running up against the differences in the set of
> intervals these two forms can represent. Specifically, I
> need to know something about how (or whether) implicit forms
> will represet semi-infinite intervals.
>
> How does one (or can one) represent a semi-infinite interval
> of the explicit form [a,+inf] as an implicit <m,r>?
My point of view is that mid-rad is mostly useful when the radius is
small, in order to compute approximations with a valid (and small)
error bound. Thus semi-unbounded intervals should not occur, or they
mean that there is a problem with the code (e.g. not accurate enough,
or the range is too small...). So, I think that Entire could be used
in place of semi-unbounded intervals for the closure of the arithmetic
(therefore semi-unbounded intervals need not be represented). In
practice, this would mean: an accurate approximation to the exact
result could not be determined.
> Can one represent Entire? Perhaps as <0,+inf>?
Or any <m,+inf> with m finite, <0,+inf> being the canonical
representation (that would be a good reason to define
mid_F(Entire) = 0).
> Can one use r = +inf at all?
>
> Can one represent Empty? Say <0,-inf>?
>
> Can one use r < 0 at all?
Yes, and <m,r> where m is infinite or NaN could also represent Empty.
What would the canonical representation be?
<mid_F(Empty),rad_F(Empty)>?
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/>
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Work: CR INRIA - computer arithmetic / AriC project (LIP, ENS-Lyon)