Dear Dan,
But this proposal met with little support& none
from the mid-rad folks. So we are trying something else.
As a "mid-rad guy", I am very happy with
your proposal to shift the inf-sup and mid-rad
concepts to lower levels. I congratulate this idea
and all your efforts in this direction.
Now I see that you are retreating from
your initial position, which is a pity. You say:
there is one
abstraction for inf-sup representations& another, quite
different, abstraction for mid-rad forms.
This is not so. The abstractions are exactly the same.
Those who claim that there is a difference probably start
the abstract model with defining first the set of intervals.
This is not the correct approach to an abstract model.
The correct one is the axiomatic approach, where only
the operations and relations are specified and the rules
for them. The specification of the supporting sets comes
later with the specification of the presentations (inf-sup
or mid-rad). To be more specific, in the case of addition
and multiplication by scalars we should state on level 1
that the interval system is a quasilinear space and that is all.
Luckily, interval spaces are much more adopted to computations
than number spaces, so now we should not follow exactly the
standard for fp-numbers.
On level 2 the abstraction for machine number arithmetic should
come within the lines of prof Kulisch theory on computer arithmetic.
Your initial idea, I think, was close to his theory. Only then
on level 3, if still necessary, presentations like inf-sup and
mid-rad should be mentioned. This will make the standard
simple and clear.
But it will likely force us to have one level 2 abstraction
for intervals that are represented at level 3 by inf-sup
forms. And another level 2 abstraction for intervals
represented at level 3 by mid-rad forms.
I am very puzzled to see what these two abstractions would
look like. I beleive that Juergen and Marco will make an useful
proposal in the correct direction.
Svetoslav
On 6 Jul 2010 at 15:43, Dan Zuras Intervals wrote:
To: Marco Nehmeier<nehmeier@xxxxxxxxxxxxxxxxxxxxxxxxxxx>
Copies to: stds-1788@xxxxxxxxxxxxxxxxx,
Dan Zuras Intervals
<intervals08@xxxxxxxxxxxxxx>
From: Dan Zuras Intervals<intervals08@xxxxxxxxxxxxxx>
Send reply to: Dan Zuras Intervals<intervals08@xxxxxxxxxxxxxx>
Subject: Re: A question Re: Level 1<---> level 2 mappings; arithmetic
versus applications
Date sent: Tue, 06 Jul 2010 15:43:15 -0700
Date: Tue, 06 Jul 2010 17:29:03 +0200
From: Marco Nehmeier<nehmeier@xxxxxxxxxxxxxxxxxxxxxxxxxxx>
To: Dan Zuras Intervals<intervals08@xxxxxxxxxxxxxx>, stds-1788@xxxxxxxx
Subject: Re: A question Re: Level 1<---> level 2 mappings; arithmetic versus
Am 30.06.2010 23:11, schrieb Dan Zuras Intervals:
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
Dan,
After having been quiet in the long discussion we want to congratulate
you for clarifying the level structure.
Your new definition of level 2 is completely analogue to the
specification of an interface (abstract data type) in object orientated
programming. Where the methods are specified by pre- and
post-conditions. The internal representation is not yet determined.
For example addition of two intervals A and B
pre: A, B in IR
post: (A + B).inf() == rnd_down(A.inf() + B.inf())
(A + B).sup() == rnd_up(A.sup() + B.sup())
Now implementation (level 3) with inf-sup-intervals is obvious. The
mid-rad-guys have to proof if they can meet the specification.
We will put in a motion that the standard will be written in this manner.
Note that motion 5 already close to this definition.
Best regards
J"urgen& Marco
Gentlemen,
While I thank you for your praise, I fear much of it is
undeserved.
You see, my primary purpose in posting that note was to
criticise Nate for confusing representations with the
(otherwise unrepresented) Real intervals that live at
level 1.
That may have been overly critical of me. I have spoken
to Nate in a more private forum since then& it turns out
that he was merely discussing the Real analogues of
intervals that could be DESCRIBED in inf-sup or mid-rad
terms.
I have apologised to Nate in private& I apologise to him
in public now.
Still, I agree with your point that objects at level 2
should still remain an abstraction divorced from any
representation which may live at level 3. The clarity
of this position is muddied, however, when there is one
abstraction for inf-sup representations& another, quite
different, abstraction for mid-rad forms.
I made a half baked proposal that level 3 representations
be RESTRICTED to representing only those level 2 intervals
with endpoints in a set F of numbers that can be found in
some associated floating-point type. And I showed how
this might be done (primarily by slightly modifying mid-rad
results). But this proposal met with little support& none
from the mid-rad folks. So we are trying something else.
But it will likely force us to have one level 2 abstraction
for intervals that are represented at level 3 by inf-sup
forms. And another level 2 abstraction for intervals
represented at level 3 by mid-rad forms.
Thus, the distinction between levels 2& 3 will be less
useful than might otherwise be the case.
I am not happy with it but it may be a necessary compromise.
Yours,
Dan
Prof. Svetoslav Markov, DSci, PhD
Dept. "Biomathematics", phone: +359-2-979-3704
Inst. of Mathematics and Informatics, fax: +359-2-971-3649
Bulgarian Academy of Sciences, e-mail: smarkov@xxxxxxxxxx
"Acad. G. Bonchev" st., block 8,
BG-1113 Sofia, BULGARIA mobile (gsm): 0885871584
URL: http://www.math.bas.bg/~bio/