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Re: A question Re: Level 1 <---> level 2 mappings; arithmetic versus applications

Dan Zuras wrote:

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.
It all hangs together for me until mention of extracting bounds of the Level 2 interval losslessly as a floating-point number. I don't see how a mid-rad implementaiton (or even some of the other examples you gave) could conform to that. 

Sorry to be a buzzkill, but I guess I'm a little lost.

I think I'm just going to shut-up and listen...

Nate