Re: A question Re: Level 1 <---> level 2 mappings; arithmetic versus applications
> From: "Nate Hayes" <nh@xxxxxxxxxxxxxxxxx>
> To: "Dan Zuras Intervals" <intervals08@xxxxxxxxxxxxxx>,
> "Ralph Baker Kearfott" <rbk@xxxxxxxxxxxx>
> Cc: "P-1788" <stds-1788@xxxxxxxxxxxxxxxxx>,
> "Dan Zuras Intervals" <intervals08@xxxxxxxxxxxxxx>
> Subject: Re: A question Re: Level 1 <---> level 2 mappings; arithmetic versus applications
> Date: Wed, 30 Jun 2010 15:05:39 -0500
>
> Dan Zuras wrote:
>
> . . .
>
> John has previously made the observation that there is an exact mapping from
> Level 2 mid-rad interval to Level 1 interval. Of course, once at Level 1
> there is then also an exact mapping from mid-rad to inf-sup (or vice-versa).
> So the only conversion that requires care is mapping back from Level 1 to
> Level 2. However, it seems there is some Level 1 mid-rad interval
> corresponding to some Level 2 mid-rad interval that is provably the tightest
> possible Level 2 enclosure, so long as that Level 2 enclosure is represented
> by a midpoint and a radius.
>
> . . .
>
> Nate
>
Nate,
I am going to pass on most of the content of your note
to focus on this one statement because the fact that
you state things in this way means I have not been
clear.
Level 1 is the set of all possible contiguous subsets
of the extended Reals.
Therefore there ARE NO mid-rad or inf-sups at level 1.
Representations have no meaning there.
Level 2 is some finite subset of the intervals that exist
at level 1.
What I am proposing is that the DEFINING characteristic
of that subset be that the bounds be exactly (some say,
losslessly) extractable as elements of some floating-point
type F.
Therefore, there are no mid-rad or inf-sups at level 2
either. Representations have no more meaning here then
they do at level 1.
All the formats live at lower levels.
And I am proposing an approach that never speaks of them
directly while still knowing that they exist & taking
care that some agreeable behavior is possible for them.
That's all.
Dan