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