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Date: Wed, 01 May 2002 10:58:57 -0500
To: reliable_computing [at] interval [dot] louisiana.edu
From: "R. Baker Kearfott" <rbk [at] louisiana [dot] edu>
Subject: New preprint available: verification of minimax
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Colleagues,

I would like to call your attention to a new short preprint, 

"On Verification of Solutions to Nonlinear Minimax Problems"

that you can download from


http://interval.louisiana.edu/preprints/2002_nonlinear_minimax.ps

or

http://interval.louisiana.edu/preprints/2002_nonlinear_minimax.dvi

or

http://interval.louisiana.edu/preprints/2002_nonlinear_minimax.pdf

In it, I show that, if Lemarechal's suggestions are used to convert
the problem into a smooth problem with constraints, the corresponding
linear problem must have Fritz-John matrix that is multiply singular
at the solution.  Thus, that avenue cannot be used for solution
verification.  I propose an alternate technique, based on 
Jansson's work.

If anyone knows of similar work in the literature, I would appreciate
knowing of it.

We're presently investigating the "alternate technique" :-)

Best regards,

Baker

---------------------------------------------------------------
R. Baker Kearfott,    rbk [at] louisiana [dot] edu   (337) 482-5346 (fax)
(337) 482-5270 (work)                     (337) 981-9744 (home)
URL: http://interval.louisiana.edu/kearfott.html
Department of Mathematics, University of Louisiana at Lafayette
Box 4-1010, Lafayette, LA 70504-1010, USA
---------------------------------------------------------------


From owner-reliable_computing [at] interval [dot] louisiana.edu Thu May  2 06:50:14 2002
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Date: Thu, 02 May 2002 13:49:59 +0200
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Subject: Re: verification of minimax
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"R. Baker Kearfott" wrote:
> 
> I would like to call your attention to a new short preprint,
> "On Verification of Solutions to Nonlinear Minimax Problems"
> If anyone knows of similar work in the literature, I would appreciate
> knowing of it.



Shen Zuhe, A. Neumaier and M.C. Eiermann, Solving
minimax problems by interval methods, BIT 30 (1990), 742-751. 



L. Jaulin (2001). Reliable minimax parameter estimation.
Reliable Computing, issue 3, volume 7. 231-246. 

preprint from:
http://www.istia.univ-angers.fr/~jaulin/publications.html



Jaulin L., M. Kieffer, O. Didrit and E. Walter, 
Applied Interval Analysis,
Springer, 2001

also contains some material on minimax problems.

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Date: Thu, 02 May 2002 09:47:30 -0700 (PDT)
From: William Walster <Bill.Walster [at] Eng [dot] Sun.COM>
Subject: Re: Manifolds
To: tupper [at] peda [dot] com, Arnold.Neumaier [at] univie [dot] ac.at
Cc: reliable_computing [at] interval [dot] louisiana.edu
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Friends,

According to James & James Mathematics Dictionary, Fourth Edition, there
appears to be room for the word "manifold" to be used in a variety of
ways.  They say in part:

	Manifold: In general, manifold may mean any collection or set of
	objects.  E.g., a Riemannian space is also called a Riemannian
	manifold; a subset of a vector space is said to be a *linear set* or
	a *linear manifold* if it contains all linear combinations of its
	members.  However, manifold frequently has technical meaning beyond
	being a mere set, as illustrated by the following definitions:
	
They go on to list a number, which I will not copy, but just list:

	topological manifold;
	pseudomanifold;
	orientable manifold
	non-orientable manifold.
	
My interpretation is that as long as we are clear how we are using the
term, we should not feel inhibited from using it in the way George and
others have done in the past.

Cheers,

Bill

  >Date: Thu, 18 Apr 2002 10:21:24 +0200
  >From: Arnold Neumaier <Arnold.Neumaier [at] univie [dot] ac.at>
  >Subject: Re: Manifolds
  >To: Jeff Tupper / Pedagoguery Software Inc <tupper [at] peda [dot] com>
  >Cc: reliable_computing [at] interval [dot] louisiana.edu
  >MIME-version: 1.0
  >Content-transfer-encoding: 8BIT
  >X-Accept-Language: en, de
  >
  >Jeff Tupper wrote:
  >
  >> >When verifying solutions, one can verify only the nonsingular part
  >> >of the solution set (its relative interior), and this always forms a
  >> >manifold. But sometimes (outside theorems), there is a sloppy use of
  >> >manifold as a synonym for solution set.
  >> 
  >> It seems to me that if, for example, Bolzano's theorem, or the
  >> (Poincaré-)Miranda theorem, is used by an interval method, that what
  >> is known is that the portion of the solution set just verified is
  >> known to be closed (not necessarily a manifold).
  >
  >To be 100% precise, what is verified is always a closed subset of a
  >manifold (without boundary).
  >
  >> More abstractly, I don't see why a special-purpose verifier couldn't
  >> be built for one particular (non-manifold) solution set. (I.e. look
  >> up the formula in a table and give back the exact solution set.)
  >
  >Unless the singular points are exactly representable and represented 
  >(which is the case only in *very* few problems from applications),
  >and no rounding error occurs in the verification phase (which excludes
  >practically all interval techniques and necessitates symbolic methods)
  >they cannot be verified, since they are unstable under perturbations.
  >(Look at double roots or bifurcation points...)
  >One can produce valid enclosures for the region, but no associated
  >existence statement. 
  >
  >Only generic situations can be verified by general purpose 
  >interval techniques.
  >
  >Arnold


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From: Vladik Kreinovich <vladik [at] cs [dot] utep.edu>
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Subject: Re: Manifolds
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> Date: Thu, 02 May 2002 09:47:30 -0700 (PDT)
> From: William Walster <Bill.Walster [at] Eng [dot] Sun.COM>
> Subject: Re: Manifolds
> To: tupper [at] peda [dot] com, Arnold.Neumaier [at] univie [dot] ac.at
> Cc: reliable_computing [at] interval [dot] louisiana.edu
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id g42Gm1h12834
> 
Dear Bill,

Thanks for your thoughtful comments.

> Manifold: In general, manifold may mean any collection or set of
> 	objects. 

...

> My interpretation is that as long as we are clear how we are using the
> term, we should not feel inhibited from using it in the way George and
> others have done in the past.

I think there is no disagreement between what you are saying here and what Jeff 
was saying. 

It is very important to spell out what exactly is meant by a manifold. Jeff's 
point was that is a paper says "manifold" without any further explanation of 
what is meant, then readers will assume that it is a topological or smooth 
manifold (because these are the most frequent usages of this word in math), and 
become confused if this is not what was meant. 

>From this viewpoint, if the intent is to say a set of solutions, it is better 
not to use the word manifold without explaining this non-standard meaning 
because otherwise, readers may erroneously assume that the word is taken in its 
technical meaning as opposed to commonsense meaning. 

My understanding is that you are not objecting to making such clarifications, 
so I really do not see any disagreement. 

Vladik




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This topic seems to be a bit of a sleeper.


At 9:47 AM -0700 5/2/02, William Walster wrote:
>My interpretation is that as long as we are clear how we are using the
>term, we should not feel inhibited from using it in the way George and
>others have done in the past.

I'm not trying to make others feel inhibited!



At 11:00 AM -0600 5/2/02, Vladik Kreinovich wrote:
>Jeff's point was that is a paper says "manifold" without any further 
>explanation of
>what is meant, then readers will assume that it is a topological or smooth
>manifold (because these are the most frequent usages of this word in 
>math), and
>become confused if this is not what was meant.

I'm not sure if my communiques had much of a point other than to 
distract readers from the mundane details of day-to-day life.



At 11:00 AM -0600 5/2/02, Vladik Kreinovich wrote:
>I think there is no disagreement between what you are saying here 
>and what Jeff
>was saying.

I tend to agree with this.



At 9:47 AM -0700 5/2/02, William Walster wrote:
>According to James & James Mathematics Dictionary, Fourth Edition, there
>appears to be room for the word "manifold" to be used in a variety of
>ways.

Out of interest, is there a reason you ascribe such authority to this 
text? I ask since I've noticed that you've referenced it in at least 
one of your papers.



At 9:47 AM -0700 5/2/02, William Walster wrote:
>They say in part:
>
>	Manifold: In general, manifold may mean any collection or set of
>	objects.

I would argue that this shows the age of the dictionary. Manifolds 
are now, and have been for some time, a standard part of proper 
upper-level undergraduate university mathematics curricula around the 
world (courses designed for students going on to further study / work 
in mathematics) and are not the somewhat new idea they were many 
decades ago. I think you'd have a hard time finding a mathematician 
comfortable with using the term in this very general use.



At 9:47 AM -0700 5/2/02, William Walster wrote:
>They go on to list a number, which I will not copy, but just list:
>
>	topological manifold;
>	pseudomanifold;
>	orientable manifold
>	non-orientable manifold.

Solution sets need not fit into any of these types of manifold. I am 
not aware of any technical definition for any type of manifold that 
is loose enough to include general solutions sets. Keep in mind that 
solution sets do not even have to have a uniform dimension: for 
example, consider the solution set of x(y-|y|)=0.

By the way, there are some other terms available as well. One 
example: the set of solutions to a set of real polynomials over R^n 
is a real algebraic variety. (And these do admit sets of non-uniform 
dimension.) Using these terms correctly may help bridge the gap 
between various disciplines. Misusing them may help keep this web 
site more lively.



At 4:21 AM -0400 4/18/02, Arnold Neumaier wrote:
>Unless the singular points are exactly representable and represented
>(which is the case only in *very* few problems from applications),
>and no rounding error occurs in the verification phase (which excludes
>practically all interval techniques and necessitates symbolic methods)
>they cannot be verified, since they are unstable under perturbations.
>(Look at double roots or bifurcation points...)
>One can produce valid enclosures for the region, but no associated
>existence statement.

It seems that we have a somewhat different view on this. I see more 
and more symbolic reasoning being folded in to interval methods. (Or 
I may just become more and more of an outcast of the interval 
community.) Furthermore, one of the applications that I am interested 
in is mathematical visualization, where the structure of solutions 
and their representation can be somewhat different than with 
scientific applications.




Please keep in mind that I'm but a lowly Ph.D. student that doesn't 
know much about anything! But I'm hoping to become a Ph.D. candidate 
in the not-too-distant future (i.e. get past my depth exam). No 
disrespect is meant to anyone. :-)

Best wishes,
Jeff
-- 

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Date: Fri, 03 May 2002 07:52:44 -0700 (PDT)
From: William Walster <Bill.Walster [at] Eng [dot] Sun.COM>
Subject: Re: Manifolds
To: reliable_computing [at] interval [dot] louisiana.edu, tupper [at] peda [dot] com
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  >
  >
  >At 9:47 AM -0700 5/2/02, William Walster wrote:
  >>According to James & James Mathematics Dictionary, Fourth Edition, there
  >>appears to be room for the word "manifold" to be used in a variety of
  >>ways.
  
Jeff Tupper responded:

  >
  >Out of interest, is there a reason you ascribe such authority to this 
  >text? I ask since I've noticed that you've referenced it in at least 
  >one of your papers.
  >
  >
  >

Jeff,

No particular reason.  It is handy.  I should probably purchase a
more recent edition.  I will be surprised if general usage (the way
interval researchers have used the word) has changed.

My only point was to to reinforce the idea that the word manifold has
in the past, at least, been used in a general way.  BTW, The Merrian
Webster's Collegiate Dictionay, Tenth Edition 1997, lists for the
mathematical definition of manifold: A topological space in which every
point has a neighborhood that is homemorphic to the interior of
a sphere in Euclidean space of the same number of dimensions.

My other citation to the James and James dictionary was regarding 
the definition of an indeterminate form, which I found to be very 
helpful and provided me with a metaphor to use in thinking about 
what their containment sets should be.

I have always been very impressed by your work, and look forward to
you finishing you Ph.D.  As I am sure is the case with other members 
of the interval research community, if I can do anything to help, do not
hesitate to contact me.

Best regards,

Bill


	G. William (Bill) Walster, Ph.D. 
	Interval Technology Engineering Manager
	Sun Microsystems, Inc.
	16 Network Circle, MS UMPK16-304
	Menlo Park, CA 94025
	(650) 786-9004 Direct
	(650) 786-9551 Fax

	bill.walster [at] eng [dot] sun.com



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Jeff Tupper / Pedagoguery Software Inc <tupper [at] peda [dot] com> writes:

[...]

| At 9:47 AM -0700 5/2/02, William Walster wrote:
| >They say in part:
| >
| >	Manifold: In general, manifold may mean any collection or set of
| >	objects.
| 
| I would argue that this shows the age of the dictionary. Manifolds 
| are now, and have been for some time, a standard part of proper 
| upper-level undergraduate university mathematics curricula around the 
| world (courses designed for students going on to further study / work 
| in mathematics) and are not the somewhat new idea they were many 
| decades ago. I think you'd have a hard time finding a mathematician 
| comfortable with using the term in this very general use.

I would like to second Jeff here.  I don't think that liberal (if not
uncommon) usage of "manifold" is appropriate in such a technical paper. 
I must confess that as a Differential Geometer, I'm uncomfortable with
it :-) 

Best,

-- Gaby
Gabriel Dos Reis                   |  Centre de Mathématiques et de 
dosreis [at] cmla [dot] ens-cachan.fr         |         Leurs Applications
Fax : (33) 01 47 40 59 01          |   ENS de Cachan -- CNRS (URA 1611)
               61, Avenue du Pdt Wilson, 94235 Cachan - Cedex

From owner-reliable_computing [at] interval [dot] louisiana.edu Thu May  9 13:07:24 2002
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Date: Thu, 9 May 2002 12:07:13 -0600 (MDT)
From: Vladik Kreinovich <vladik [at] cs [dot] utep.edu>
Reply-To: Vladik Kreinovich <vladik [at] cs [dot] utep.edu>
Subject: is there ready software for exact range estimation of quadratic functions?
To: reliable_computing [at] interval [dot] louisiana.edu
Cc: scott [at] ramas [dot] com, vladik [at] cs [dot] utep.edu
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Dear Friends,

It is well known that for a quadratic function f of a reasonably small number 
of variables (e.g., 5) defined on a box [a1,b1] x ... x [an,bn] with rational 
endpoints ai and bi, it is possible to compute its exact range. 

For example, we can use the fact that if the maximum of f on the box is 
attained at a point x1,...,xn, then for each variable xi, either xi=ai or 
xi=bi, or xi is inside the interval in which case the partial derivative df/dxi 
is equal to 0. Since f is quadratic, the equation df/dxi=0 is a linear 
equation.

Thus, for each variable i, one of the three linear equation holds. Overall, we 
have a quite tractable number of 3^5=243 systems of 5 x 5 linear equations. We 
can solve all these systems, dismiss solutions which are outside the box, and 
find the larget value for all solutions that are inside. 

There are also probably better algorithms as well. 

The question that we have is as follows: 

*****is there available software that can compute this exact range - whether 
using the above algorithm or some better one?*********

Such software will help us - and others - in testing different techniques and 
algorithms for computing the range of quadratic functions. 

Please reply to both of us, to vladik [at] cs [dot] utep.edu and scott [at] ramas [dot] com

Thanks a lot.

Scott and Vladik

 


From owner-reliable_computing [at] interval [dot] louisiana.edu Thu May  9 14:10:11 2002
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Date: Thu, 09 May 2002 21:09:39 +0200
From: Arnold Neumaier <Arnold.Neumaier [at] univie [dot] ac.at>
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To: Vladik Kreinovich <vladik [at] cs [dot] utep.edu>
CC: reliable_computing [at] interval [dot] louisiana.edu, scott [at] ramas [dot] com
Subject: Re: is there ready software for exact range estimation of quadratic 
 functions?
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Vladik Kreinovich wrote:
> 
> Dear Friends,
> 
> It is well known that for a quadratic function f of a reasonably small number
> of variables (e.g., 5) defined on a box [a1,b1] x ... x [an,bn] with rational
> endpoints ai and bi, it is possible to compute its exact range.
> 
> For example, we can use the fact that if the maximum of f on the box is
> attained at a point x1,...,xn, then for each variable xi, either xi=ai or
> xi=bi, or xi is inside the interval in which case the partial derivative df/dxi
> is equal to 0. Since f is quadratic, the equation df/dxi=0 is a linear
> equation.
> 
> Thus, for each variable i, one of the three linear equation holds. Overall, we
> have a quite tractable number of 3^5=243 systems of 5 x 5 linear equations. We
> can solve all these systems, dismiss solutions which are outside the box, and
> find the larget value for all solutions that are inside.


This is the peeling process described in Section 5.2.3 of
Baker Kearfott's book, and rediscovered in Makino's thesis.
Probably some version of it is implemented in GLOBSOL.

Arnold

From owner-reliable_computing [at] interval [dot] louisiana.edu Thu May  9 14:25:15 2002
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From: Vladik Kreinovich <vladik [at] cs [dot] utep.edu>
Reply-To: Vladik Kreinovich <vladik [at] cs [dot] utep.edu>
Subject: Re: is there ready software for exact range estimation of quadratic  functions?
To: vladik [at] cs [dot] utep.edu, Arnold.Neumaier [at] univie [dot] ac.at
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Dear Arnold, Thanks for the prompt reply. Let us see what Globsol people 
answer. 

Vladik

We described this algorithm (as a "textbook method") on pp. 9-10 of our 
complexity book 

Vladik Kreinovich, Anatoly Lakeyev, Jiri Rohn, and Patrick Kahl,
  "Computational complexity and feasibility of 
  data processing and interval computations", 
  Kluwer, Dordrecht, 1997. 
  
(actually, we did cite Baker Kearfott's 1996 book). 

I apologize for not citing it in my query, I definitely did not mean to 
indicate that we invented this idea.





> Date: Thu, 09 May 2002 21:09:39 +0200
> From: Arnold Neumaier <Arnold.Neumaier [at] univie [dot] ac.at>
> X-Accept-Language: en, de
> MIME-Version: 1.0
> To: Vladik Kreinovich <vladik [at] cs [dot] utep.edu>
> CC: reliable_computing [at] interval [dot] louisiana.edu, scott [at] ramas [dot] com
> Subject: Re: is there ready software for exact range estimation of quadratic  
functions?
> Content-Transfer-Encoding: 7bit
> 
> Vladik Kreinovich wrote:
> > 
> > Dear Friends,
> > 
> > It is well known that for a quadratic function f of a reasonably small 
number
> > of variables (e.g., 5) defined on a box [a1,b1] x ... x [an,bn] with 
rational
> > endpoints ai and bi, it is possible to compute its exact range.
> > 
> > For example, we can use the fact that if the maximum of f on the box is
> > attained at a point x1,...,xn, then for each variable xi, either xi=ai or
> > xi=bi, or xi is inside the interval in which case the partial derivative 
df/dxi
> > is equal to 0. Since f is quadratic, the equation df/dxi=0 is a linear
> > equation.
> > 
> > Thus, for each variable i, one of the three linear equation holds. Overall, 
we
> > have a quite tractable number of 3^5=243 systems of 5 x 5 linear equations. 
We
> > can solve all these systems, dismiss solutions which are outside the box, 
and
> > find the larget value for all solutions that are inside.
> 
> 
> This is the peeling process described in Section 5.2.3 of
> Baker Kearfott's book, and rediscovered in Makino's thesis.
> Probably some version of it is implemented in GLOBSOL.
> 
> Arnold


From owner-reliable_computing [at] interval [dot] louisiana.edu Thu May  9 15:50:54 2002
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Message-ID: <3CDAE127.1E126035 [at] cs [dot] berkeley.edu>
Date: Thu, 09 May 2002 13:50:47 -0700
From: James Demmel <demmel [at] cs [dot] berkeley.edu>
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To: interval <reliable_computing [at] interval [dot] louisiana.edu>
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You  may be interested in a paper by Yozo Hida and myself
on how to exploit the arithmetic formats in the IEEE 754 floating point standard
to get guaranteed accurate sums and dot products in an efficient manner.

You can find the paper on my web page www.cs.berkeley.edu/~demmel under

   Research Projects and Books/BeBOP/Selected Papers and Reports

or directly at

   www.cs.berkeley.edu/~demmel/AccurateSummation.{ps,pdf}

Regards,
Jim Demmel




From owner-reliable_computing [at] interval [dot] louisiana.edu Thu May  9 17:53:16 2002
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To: reliable_computing [at] interval [dot] louisiana.edu
Subject: Re: is there ready software for exact range estimation of quadratic  functions?
References: <200205091807.g49I7DM24723 [at] cs [dot] utep.edu>
	<3CDAC973.A5D150BD [at] univie [dot] ac.at>
From: Arrigo Benedetti <arrigo [at] bologna [dot] vision.caltech.edu>
Date: 09 May 2002 15:53:05 -0700
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I am actually working at a polynomial (2nd order) range bounder that is
based on this "peeling" process described by Kearfott. The method that I am
using for generating the lower dimension boundary elements is a bit different
from what Kearfott described in that it is not a recursive algorithm.
I should have results very soon, and I am looking forward to post them to
the list.

Best
-Arrigo


Arnold Neumaier <Arnold.Neumaier [at] univie [dot] ac.at> writes:

> Vladik Kreinovich wrote:
> > 
> > Dear Friends,
> > 
> > It is well known that for a quadratic function f of a reasonably small number
> > of variables (e.g., 5) defined on a box [a1,b1] x ... x [an,bn] with rational
> > endpoints ai and bi, it is possible to compute its exact range.
> > 
> > For example, we can use the fact that if the maximum of f on the box is
> > attained at a point x1,...,xn, then for each variable xi, either xi=ai or
> > xi=bi, or xi is inside the interval in which case the partial derivative df/dxi
> > is equal to 0. Since f is quadratic, the equation df/dxi=0 is a linear
> > equation.
> > 
> > Thus, for each variable i, one of the three linear equation holds. Overall, we
> > have a quite tractable number of 3^5=243 systems of 5 x 5 linear equations. We
> > can solve all these systems, dismiss solutions which are outside the box, and
> > find the larget value for all solutions that are inside.
> 
> 
> This is the peeling process described in Section 5.2.3 of
> Baker Kearfott's book, and rediscovered in Makino's thesis.
> Probably some version of it is implemented in GLOBSOL.
> 
> Arnold
> 

From owner-reliable_computing [at] interval [dot] louisiana.edu Thu May  9 18:20:09 2002
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Date: Thu, 09 May 2002 18:19:59 -0500
To: Arnold Neumaier <Arnold.Neumaier [at] univie [dot] ac.at>,
   Vladik Kreinovich <vladik [at] cs [dot] utep.edu>
From: "R. Baker Kearfott" <rbk [at] louisiana [dot] edu>
Subject: Re: is there ready software for exact range estimation of
  quadratic  functions?
Cc: reliable_computing [at] interval [dot] louisiana.edu, scott [at] ramas [dot] com
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Arnold,

Yes, the peeling process is implemented in GlobSol.  
GlobSol finds the min of a function.  We can call GlobSol twice to
find the range of a quadratic over a box, provided the box is 
sufficiently low-dimensional.  

Best regards,

Baker

At 09:09 PM 5/9/02 +0200, Arnold Neumaier wrote:
>Vladik Kreinovich wrote:
>> 
>> Dear Friends,
>> 
>> It is well known that for a quadratic function f of a reasonably small number
>> of variables (e.g., 5) defined on a box [a1,b1] x ... x [an,bn] with rational
>> endpoints ai and bi, it is possible to compute its exact range.
>> 
>> For example, we can use the fact that if the maximum of f on the box is
>> attained at a point x1,...,xn, then for each variable xi, either xi=ai or
>> xi=bi, or xi is inside the interval in which case the partial derivative df/dxi
>> is equal to 0. Since f is quadratic, the equation df/dxi=0 is a linear
>> equation.
>> 
>> Thus, for each variable i, one of the three linear equation holds. Overall, we
>> have a quite tractable number of 3^5=243 systems of 5 x 5 linear equations. We
>> can solve all these systems, dismiss solutions which are outside the box, and
>> find the larget value for all solutions that are inside.
>
>
>This is the peeling process described in Section 5.2.3 of
>Baker Kearfott's book, and rediscovered in Makino's thesis.
>Probably some version of it is implemented in GLOBSOL.
>
>Arnold
>
>
---------------------------------------------------------------
R. Baker Kearfott,    rbk [at] louisiana [dot] edu   (337) 482-5346 (fax)
(337) 482-5270 (work)                     (337) 981-9744 (home)
URL: http://interval.louisiana.edu/kearfott.html
Department of Mathematics, University of Louisiana at Lafayette
Box 4-1010, Lafayette, LA 70504-1010, USA
---------------------------------------------------------------


From owner-reliable_computing [at] interval [dot] louisiana.edu Fri May 10 10:45:17 2002
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Date: Fri, 10 May 2002 09:45:02 -0600 (MDT)
From: Vladik Kreinovich <vladik [at] cs [dot] utep.edu>
Reply-To: Vladik Kreinovich <vladik [at] cs [dot] utep.edu>
Subject: Issac'2002 Call For Posters 
To: reliable_computing [at] interval [dot] louisiana.edu
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deadline for posters is June 29. 

Issac stands for International Symposium on Symbolic and Algebraic Computation, 
they would welcome interval-related papers.

I am adding this link to the Forthcoming Conferences part of the interval 
computations webpage http://www.cs.utep.edu/interval-comp

------------- Begin Forwarded Message -------------

From: Conference.Issac2002 [at] lifl [dot] fr
Date: Fri, 10 May 2002 02:44:39 +0200 (MET DST)
TO: vladik [at] cs [dot] utep.edu
Subject: Issac'2002 Call For Posters 
Mime-Version: 1.0

The Issac'02 Call for Posters has started. You
will find information on the Issac'02 website at
the following address: 

	http://www.lifl.fr/issac2002/callposters.html

Best regards,
Fran\c{c}ois Lemaire.


------------- End Forwarded Message -------------



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Date: Sat, 11 May 2002 11:34:11 -0500 (CDT)
From: Hans Schneider <hans [at] math [dot] wisc.edu>
To: NETS -- at-net <at-net [at] mi [dot] uni-erlangen.de>,
   "Hershkowitz, Danny -- Hershkowitz Daniel" <hershkow [at] aluf [dot] technion.ac.il>,
   Danny Hershkowitz <hershkow [at] techunix [dot] technion.ac.il>,
   E-LETTER <pmisra [at] cs [dot] wright.edu>, "na.digest" <na.digest@na-net.ornl.gov>,
   ipnet-digest [at] math [dot] msu.edu, wim@bell-labs.com, hjt [at] eos [dot] ncsu.edu,
   vkm [at] eedsp [dot] gatech.edu, reliable_computing [at] interval [dot] louisiana.edu
Subject: LAA contents
Message-ID: <Pine.GSO.4.10.10205111132570.10502-100000 [at] ultra10 [dot] math.wisc.edu>
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Dear Net Organizer:

Please circulate the attached LAA contents over your net.

Thanks

hans



------------------------------------------------------------------------------
Hans Schneider                                 hans [at] math [dot] wisc.edu.
Department of Mathematics                      608-262-1402 (Work)
Van Vleck Hall                                 608-271-7252 (Home)
480 Lincoln Drive                              608-263-8891 (Work FAX)
University of Wisconsin-Madison                No Home FAX at present
Madison WI 53706 USA             http://www.math.wisc.edu/~hans (URL)
------------------------------------------------------------------------------





Journal: Linear Algebra and its Applications
ISSN   : 0024-3795
Volume : 350
Issue  : 1-3
Date   : 15-Jul-2002

Visit the journal at http://www.elsevier.nl/locate/jnlnr/07738
The full text of articles is available via ScienceDirect :
http://www.sciencedirect.com/


The numerical range of a nonnegative matrix
C.-K. Li, B.-S. Tam, P. Yuan Wu
pp 1-23

A weighted Drazin inverse and applications
V. Rakocevic, Y. Wei
pp 25-39

Commutativity preserving linear maps and Lie automorphisms of strictly
triangular matrix space
Y. Cao, Z. Chen, C. Huang
pp 41-66

State space calculations for discrete probability densities
B. Hanzon, R.J. Ober
pp 67-87

A priori error bounds on invariant subspace approximations by block Krylov
subspaces
M. Robbe, M. Sadkane
pp 89-103

Covering numbers under small perturbations
R. Fernandes
pp 105-124

Containment regions for zeros of polynomials from numerical ranges of
companion matrices
H. Linden
pp 125-145

The polytope of degree sequences of hypergraphs
N.L. Bhanu Murthy, M.K. Srinivasan
pp 147-170

Some explicit formulas for the polynomial decomposition of the matrix
exponential and applications
R. Ben Taher, M. Rachidi
pp 171-184

Intersections of nest algebras in finite dimensions
P.A. Fillmore, W.E. Longstaff, G.W.M. H. Radjavi, Y. Zhong
pp 185-197

On the reconstruction of Toeplitz matrix inverses from columns
G. Heinig
pp 199-212

The Lie algebra structure and controllability of spin systems
F. Albertini, D. D'Alessandro
pp 213-235

Condition numbers of algebraic Riccati equations in the Frobenius norm
J.-g. Sun
pp 237-261
Full text via ScienceDirect :
http://www.sciencedirect.com/science?_ob=GatewayURL&_origin=CONTENTS&_method=citationSearch&_piikey=S002437950200294X&_version=1&md5=b26f613add034e5972d5a4c203f57324

Linear operators that preserve pairs of matrices which satisfy extreme rank
properties
L.B. Beasley, S.-G. Lee, S.-Z. Song
pp 263-272

A new proof of Mayer's theorem
C.-T. Pang, Y.-Y. Lur, S.-M. Guu
pp 273-278

Total positivity and Toda flow
G.M.L. Gladwell
pp 279-284

A note on Krylov subspace methods for singular systems
Z.-H. Cao, M. Wang
pp 285-288

Convexoid and generalized derivations
M. Barraa
pp 289-292

Author index
pp 293




From owner-reliable_computing [at] interval [dot] louisiana.edu Mon May 13 02:40:42 2002
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From: "Josep Vehi" <vehi [at] silver [dot] udg.es>
To: <reliable_computing [at] interval [dot] louisiana.edu>
Subject: Promoting interval computations
Date: Mon, 13 May 2002 09:40:30 +0200
Organization: Universitat de Girona
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Dear all,
 
The technical committee on Smart Adaptive Systems of the European
network of excellence EUNITE (http://www.eunite.org
<http://www.eunite.org/> ) is preparing a bibliography on methods ans
techniques for smart adaptive systems. It will not be a huge database
but just a collection of sellected (key) papers for each technique.
 
As intervals is a way to deal with uncertainty and to manage such
uncertainty is crucial in chanching environements and I'm sure that the
are many applications of intervals to adaptive systems, I proposed to
include a section on intervals in the bibliography and it was accepted.
 
Please, would you be so kind to send me references on applications of
intervals with the stress on adaptivity?
 
You will find bellow the details of the bibliographic database
organisation.
 
Josep
 
--------------
 
Scope: Soft computing techniques that can be seen as adaptive.
 

Adaptation Levels:

1.	Changing environment.
2.	Similar Setting without explicit porting (the system should
adapt itself, for a similar application).
3.	Adaptation to a new/unknown application.
 
 
Other definitions: from the task force on taxonomy on the EUNITE web
site.
 
Learning levels (Dourado):
1.	Parameter learning
2.	Structure learning
3.	Trajectory learning
4.	Task learning
5.	Goal learning
6.	Learning organisations
 
Methods:
*	Fuzzy
*	Neural
*	Evolutionary
*	Hybrid
*	Multi-modular
*	Multi-agent
*	Universal learning
*	Symbolic Machine learning
*	Probabilistic
*	Rough sets 
*	Heuristic methods
*	Chaotic systems
*	Interval computations
*	.
 

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<p class=3DMsoNormal><font size=3D3 face=3D"Times New Roman"><span =
lang=3DEN-US
style=3D'font-size:12.0pt;mso-ansi-language:EN-US'>Dear =
all,<o:p></o:p></span></font></p>

<p class=3DMsoNormal><font size=3D3 face=3D"Times New Roman"><span =
lang=3DEN-US
style=3D'font-size:12.0pt;mso-ansi-language:EN-US'><o:p>&nbsp;</o:p></spa=
n></font></p>

<p class=3DMsoNormal><font size=3D3 face=3D"Times New Roman"><span =
lang=3DEN-US
style=3D'font-size:12.0pt;mso-ansi-language:EN-US'>The technical =
committee on
Smart Adaptive Systems of the European network of excellence EUNITE (<a
href=3D"http://www.eunite.org/">http://www.eunite.org</a>) is preparing =
a bibliography
on methods ans techniques for smart adaptive systems. It will not be a =
huge
database but just a collection of sellected (key) papers for each =
technique.<o:p></o:p></span></font></p>

<p class=3DMsoNormal><font size=3D3 face=3D"Times New Roman"><span =
lang=3DEN-US
style=3D'font-size:12.0pt;mso-ansi-language:EN-US'><o:p>&nbsp;</o:p></spa=
n></font></p>

<p class=3DMsoNormal><font size=3D3 face=3D"Times New Roman"><span =
lang=3DEN-US
style=3D'font-size:12.0pt;mso-ansi-language:EN-US'>As intervals is a way =
to deal
with uncertainty and to manage such uncertainty is crucial in chanching
environements and I&#8217;m sure that the are many applications of =
intervals to
adaptive systems, I proposed to include a section on intervals in the
bibliography and it was accepted.<o:p></o:p></span></font></p>

<p class=3DMsoNormal><font size=3D3 face=3D"Times New Roman"><span =
lang=3DEN-US
style=3D'font-size:12.0pt;mso-ansi-language:EN-US'><o:p>&nbsp;</o:p></spa=
n></font></p>

<p class=3DMsoNormal><font size=3D3 face=3D"Times New Roman"><span =
lang=3DEN-US
style=3D'font-size:12.0pt;mso-ansi-language:EN-US'>Please, would you be =
so kind
to send me references on applications of intervals with the stress on
adaptivity?<o:p></o:p></span></font></p>

<p class=3DMsoNormal><font size=3D3 face=3D"Times New Roman"><span =
lang=3DEN-US
style=3D'font-size:12.0pt;mso-ansi-language:EN-US'><o:p>&nbsp;</o:p></spa=
n></font></p>

<p class=3DMsoNormal><font size=3D3 face=3D"Times New Roman"><span =
lang=3DEN-US
style=3D'font-size:12.0pt;mso-ansi-language:EN-US'>You will find bellow =
the
details of the b</span></font><span lang=3DEN-GB =
style=3D'mso-ansi-language:EN-GB'>ibliographic
database organisation.<o:p></o:p></span></p>

<p class=3DMsoNormal><font size=3D3 face=3D"Times New Roman"><span =
lang=3DEN-GB
style=3D'font-size:12.0pt;mso-ansi-language:EN-GB'><o:p>&nbsp;</o:p></spa=
n></font></p>

<p class=3DMsoNormal><font size=3D3 face=3D"Times New Roman"><span =
lang=3DEN-GB
style=3D'font-size:12.0pt;mso-ansi-language:EN-GB'>Josep<o:p></o:p></span=
></font></p>

<p class=3DMsoNormal><font size=3D3 face=3D"Times New Roman"><span =
lang=3DEN-GB
style=3D'font-size:12.0pt;mso-ansi-language:EN-GB'><o:p>&nbsp;</o:p></spa=
n></font></p>

<p class=3DMsoNormal><font size=3D3 face=3D"Times New Roman"><span =
lang=3DEN-GB
style=3D'font-size:12.0pt;mso-ansi-language:EN-GB'>--------------<o:p></o=
:p></span></font></p>

<p class=3DMsoNormal><font size=3D3 face=3D"Times New Roman"><span =
lang=3DEN-GB
style=3D'font-size:12.0pt;mso-ansi-language:EN-GB'><o:p>&nbsp;</o:p></spa=
n></font></p>

<p class=3DMsoNormal><b><font size=3D3 face=3D"Times New Roman"><span =
lang=3DEN-GB
style=3D'font-size:12.0pt;mso-ansi-language:EN-GB;font-weight:bold'>Scope=
:</span></font></b><span
lang=3DEN-GB style=3D'mso-ansi-language:EN-GB'> Soft computing =
techniques that can
be seen as <b><span =
style=3D'font-weight:bold'>adaptive</span></b>.<o:p></o:p></span></p>

<p class=3DMsoNormal><font size=3D3 face=3D"Times New Roman"><span =
lang=3DEN-GB
style=3D'font-size:12.0pt;mso-ansi-language:EN-GB'><o:p>&nbsp;</o:p></spa=
n></font></p>

<h1><b><font size=3D3 face=3D"Times New Roman"><span lang=3DEN-GB =
style=3D'font-size:
12.0pt;mso-ansi-language:EN-GB'>Adaptation =
Levels:<o:p></o:p></span></font></b></h1>

<ol style=3D'mso-margin-top-alt:0cm' start=3D1 type=3D1>
 <li class=3DMsoNormal style=3D'mso-list:l2 level1 lfo1;tab-stops:list =
36.0pt'><font
     size=3D3 face=3D"Times New Roman"><span lang=3DEN-GB =
style=3D'font-size:12.0pt;
     mso-ansi-language:EN-GB'>Changing =
environment.<o:p></o:p></span></font></li>
 <li class=3DMsoNormal style=3D'mso-list:l2 level1 lfo1;tab-stops:list =
36.0pt'><font
     size=3D3 face=3D"Times New Roman"><span lang=3DEN-GB =
style=3D'font-size:12.0pt;
     mso-ansi-language:EN-GB'>Similar Setting without explicit porting =
(the
     system should adapt itself, for a similar =
application).<o:p></o:p></span></font></li>
 <li class=3DMsoNormal style=3D'mso-list:l2 level1 lfo1;tab-stops:list =
36.0pt'><font
     size=3D3 face=3D"Times New Roman"><span lang=3DEN-GB =
style=3D'font-size:12.0pt;
     mso-ansi-language:EN-GB'>Adaptation to a new/unknown =
application.<o:p></o:p></span></font></li>
</ol>

<p class=3DMsoNormal><font size=3D3 face=3D"Times New Roman"><span =
lang=3DEN-GB
style=3D'font-size:12.0pt;mso-ansi-language:EN-GB'><o:p>&nbsp;</o:p></spa=
n></font></p>

<p class=3DMsoNormal><font size=3D3 face=3D"Times New Roman"><span =
lang=3DEN-GB
style=3D'font-size:12.0pt;mso-ansi-language:EN-GB'><o:p>&nbsp;</o:p></spa=
n></font></p>

<p class=3DMsoNormal><font size=3D3 face=3D"Times New Roman"><span =
lang=3DEN-GB
style=3D'font-size:12.0pt;mso-ansi-language:EN-GB'>Other definitions: =
from the
task force on taxonomy on the EUNITE web =
site.<o:p></o:p></span></font></p>

<p class=3DMsoNormal><font size=3D3 face=3D"Times New Roman"><span =
lang=3DEN-GB
style=3D'font-size:12.0pt;mso-ansi-language:EN-GB'><o:p>&nbsp;</o:p></spa=
n></font></p>

<p class=3DMsoNormal><b><font size=3D3 face=3D"Times New Roman"><span =
lang=3DEN-GB
style=3D'font-size:12.0pt;mso-ansi-language:EN-GB;font-weight:bold'>Learn=
ing
levels (Dourado):<o:p></o:p></span></font></b></p>

<ol style=3D'mso-margin-top-alt:0cm' start=3D1 type=3D1>
 <li class=3DMsoNormal style=3D'mso-list:l1 level1 lfo2;tab-stops:list =
36.0pt'><font
     size=3D3 face=3D"Times New Roman"><span lang=3DEN-GB =
style=3D'font-size:12.0pt;
     mso-ansi-language:EN-GB'>Parameter =
learning<o:p></o:p></span></font></li>
 <li class=3DMsoNormal style=3D'mso-list:l1 level1 lfo2;tab-stops:list =
36.0pt'><font
     size=3D3 face=3D"Times New Roman"><span lang=3DEN-GB =
style=3D'font-size:12.0pt;
     mso-ansi-language:EN-GB'>Structure =
learning<o:p></o:p></span></font></li>
 <li class=3DMsoNormal style=3D'mso-list:l1 level1 lfo2;tab-stops:list =
36.0pt'><font
     size=3D3 face=3D"Times New Roman"><span lang=3DEN-GB =
style=3D'font-size:12.0pt;
     mso-ansi-language:EN-GB'>Trajectory =
learning<o:p></o:p></span></font></li>
 <li class=3DMsoNormal style=3D'mso-list:l1 level1 lfo2;tab-stops:list =
36.0pt'><font
     size=3D3 face=3D"Times New Roman"><span lang=3DEN-GB =
style=3D'font-size:12.0pt;
     mso-ansi-language:EN-GB'>Task =
learning<o:p></o:p></span></font></li>
 <li class=3DMsoNormal style=3D'mso-list:l1 level1 lfo2;tab-stops:list =
36.0pt'><font
     size=3D3 face=3D"Times New Roman"><span lang=3DEN-GB =
style=3D'font-size:12.0pt;
     mso-ansi-language:EN-GB'>Goal =
learning<o:p></o:p></span></font></li>
 <li class=3DMsoNormal style=3D'mso-list:l1 level1 lfo2;tab-stops:list =
36.0pt'><font
     size=3D3 face=3D"Times New Roman"><span lang=3DEN-GB =
style=3D'font-size:12.0pt;
     mso-ansi-language:EN-GB'>Learning =
organisations<o:p></o:p></span></font></li>
</ol>

<p class=3DMsoNormal><font size=3D3 face=3D"Times New Roman"><span =
lang=3DEN-GB
style=3D'font-size:12.0pt;mso-ansi-language:EN-GB'><o:p>&nbsp;</o:p></spa=
n></font></p>

<p class=3DMsoNormal><b><font size=3D3 face=3D"Times New Roman"><span =
lang=3DEN-GB
style=3D'font-size:12.0pt;mso-ansi-language:EN-GB;font-weight:bold'>Metho=
ds:<o:p></o:p></span></font></b></p>

<ul style=3D'mso-margin-top-alt:0cm' type=3Ddisc>
 <li class=3DMsoNormal style=3D'mso-list:l0 level1 lfo3;tab-stops:list =
36.0pt'><font
     size=3D3 face=3D"Times New Roman"><span lang=3DEN-GB =
style=3D'font-size:12.0pt;
     mso-ansi-language:EN-GB'>Fuzzy<o:p></o:p></span></font></li>
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36.0pt'><font
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36.0pt'><font
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style=3D'font-size:12.0pt;
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36.0pt'><font
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style=3D'font-size:12.0pt;
     mso-ansi-language:EN-GB'>Hybrid<o:p></o:p></span></font></li>
 <li class=3DMsoNormal style=3D'mso-list:l0 level1 lfo3;tab-stops:list =
36.0pt'><font
     size=3D3 face=3D"Times New Roman"><span lang=3DEN-GB =
style=3D'font-size:12.0pt;
     =
mso-ansi-language:EN-GB'>Multi-modular<o:p></o:p></span></font></li>
 <li class=3DMsoNormal style=3D'mso-list:l0 level1 lfo3;tab-stops:list =
36.0pt'><font
     size=3D3 face=3D"Times New Roman"><span lang=3DEN-GB =
style=3D'font-size:12.0pt;
     mso-ansi-language:EN-GB'>Multi-agent<o:p></o:p></span></font></li>
 <li class=3DMsoNormal style=3D'mso-list:l0 level1 lfo3;tab-stops:list =
36.0pt'><font
     size=3D3 face=3D"Times New Roman"><span lang=3DEN-GB =
style=3D'font-size:12.0pt;
     mso-ansi-language:EN-GB'>Universal =
learning<o:p></o:p></span></font></li>
 <li class=3DMsoNormal style=3D'mso-list:l0 level1 lfo3;tab-stops:list =
36.0pt'><font
     size=3D3 face=3D"Times New Roman"><span lang=3DEN-GB =
style=3D'font-size:12.0pt;
     mso-ansi-language:EN-GB'>Symbolic Machine =
learning<o:p></o:p></span></font></li>
 <li class=3DMsoNormal style=3D'mso-list:l0 level1 lfo3;tab-stops:list =
36.0pt'><font
     size=3D3 face=3D"Times New Roman"><span lang=3DEN-GB =
style=3D'font-size:12.0pt;
     =
mso-ansi-language:EN-GB'>Probabilistic<o:p></o:p></span></font></li>
 <li class=3DMsoNormal style=3D'mso-list:l0 level1 lfo3;tab-stops:list =
36.0pt'><font
     size=3D3 face=3D"Times New Roman"><span lang=3DEN-GB =
style=3D'font-size:12.0pt;
     mso-ansi-language:EN-GB'>Rough sets <o:p></o:p></span></font></li>
 <li class=3DMsoNormal style=3D'mso-list:l0 level1 lfo3;tab-stops:list =
36.0pt'><font
     size=3D3 face=3D"Times New Roman"><span lang=3DEN-GB =
style=3D'font-size:12.0pt;
     mso-ansi-language:EN-GB'>Heuristic =
methods<o:p></o:p></span></font></li>
 <li class=3DMsoNormal style=3D'mso-list:l0 level1 lfo3;tab-stops:list =
36.0pt'><font
     size=3D3 face=3D"Times New Roman"><span lang=3DEN-GB =
style=3D'font-size:12.0pt;
     mso-ansi-language:EN-GB'>Chaotic =
systems<o:p></o:p></span></font></li>
 <li class=3DMsoNormal style=3D'mso-list:l0 level1 lfo3;tab-stops:list =
36.0pt'><font
     size=3D3 face=3D"Times New Roman"><span lang=3DEN-GB =
style=3D'font-size:12.0pt;
     mso-ansi-language:EN-GB'>Interval =
computations<o:p></o:p></span></font></li>
 <li class=3DMsoNormal style=3D'mso-list:l0 level1 lfo3;tab-stops:list =
36.0pt'><font
     size=3D3 face=3D"Times New Roman"><span lang=3DEN-GB =
style=3D'font-size:12.0pt;
     mso-ansi-language:EN-GB'>&#8230;<o:p></o:p></span></font></li>
</ul>

<p class=3DMsoNormal><font size=3D3 face=3D"Times New Roman"><span =
style=3D'font-size:
12.0pt'><o:p>&nbsp;</o:p></span></font></p>

</div>

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From owner-reliable_computing [at] interval [dot] louisiana.edu Wed May 15 09:42:10 2002
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Message-ID: <002201c1fc07$d0891560$66ae1841 [at] columbus [dot] rr.com>
From: "Ramon Moore" <rmoore17 [at] columbus [dot] rr.com>
To: "interval" <reliable_computing [at] interval [dot] louisiana.edu>
Subject: Fw: Difficulty in your paper
Date: Wed, 15 May 2002 07:58:21 -0400
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Dear Colleagues,

Is there a kind soul among you who will help the gentleman who sent me the
plea below?
I suspect he has a bug in his program, but I do not have the energy to help
him find it.

Ray (Ramon Moore)

----- Original Message -----
From: "N. S. Kubal _00423002_" <nskubal [at] ee [dot] iitb.ac.in>
To: "Ramon Moore" <rmoore17 [at] columbus [dot] rr.com>
Sent: Wednesday, May 15, 2002 3:44 AM
Subject: Re: Difficulty in your paper


> On Wed, 15 May 2002, Ramon Moore wrote:
>
> > Unfortunately I am unable to open the document you attached. Perhaps you
can
> > just explain your question in text in an email.
> >
> > R. Moore
>
> Dear Sir,
>          Sorry for your inconvienence. Please go through the following
> matter.
>
> ------------------------------------------------------------------------
> Recently, I have implemented an optimization algorithm in FORTRAN 95 to
> find global minimum based on your paper "On Computing the Range of Values"
> published in Computing, 28:225-237, 1982 with Asaithambi, N.S. and Shen
> Zuhe.
>          When I ran the algorithm on one of the test function in [1] it
> did not give me the correct global minimum.
>
> Test function in [1]
>
> f(x,y) = (x^2 + y^2)/200 - cos(x)*cos(y/sqrt(2)) + 1
> X=[-100,100], Y=[-100,100]
>
> Results obtained using the algorithm given in your paper
> Global minimum     = 0.145618691261803
> Global minimum box = [-3.1104241497814656,-3.1104241032153367]
>                      [-4.3556719087064267,-4.3556718155741691]
>
> But the results given in [1] are
> Global minimum         = 0
> Global minimum point   = (0,0)
>
>          When I went through each steps of the algorithm, I had come to
> know that the algorithm deletes the boxes, which contain the global
> minimum, which is "zero", due to steps 8 and step 13. Now in these steps
> the variable 'S' decides to keep/delete the reduced boxes (Z). The values
> of 'S' are based on the formulae given in your paper followed by equation
> (3.2).
>          I used new variable 'S1' in steps 8 and 13, which is based on the
> following conditions,
>
> S1 .neq. 0 if (Infimum (DiF(X)) <= 0 and Supremum (DiF(X)) >= 0 in any one
> direction)
> S1 .eq.  0 if (Infimum (DiF(X)) > 0 or Supremum (DiF(X)) < 0 in all the
> directions)
>
> (i.e. delete the box if the function is strictly monotonic in all the
> directions).
>
> The modified steps 8 and 13 are,
> Step 8:  if S1(1) is 0 and Z(1) does not intersect the boundary of Z(0),
>          discard Z(1) and go to step 12.
> Step 13: if S1(2) is 0 and Z(2) does not intersect the boundary of Z(0),
>          discard Z(2) and go to step 17.
>
> Results obtained after modification
> Global minimum     = 0.0E0
> Global minimum box = [0.0E0, 0.0E0] [0.0E0, 0.0E0]
>
>          I would highly appreciate if you kindly spend some of your
> precious time to go through the algorithm and tell me whether the changes
> I have made are correct or not.
>
> References:
> [1] C. Jansson, O. Knuppel. "A Global Minimization Method: The
> Multi-Dimensional Case." TUHH (Technische Universitat Hamburg-Harburg),
> Bericht 92.1, Januar, 1992, pp.47-48.
>
>
>
>


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To: reliable_computing [at] interval [dot] louisiana.edu, rmoore17 [at] columbus [dot] rr.com
Subject: Re: Fw: Difficulty in your paper
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I've just read through your letter, including the 'plea'; and I think I
interpret it a bit differently. As far as I can see, Mr. Kubal tries to
inform you (in a polite and somewhat involved manner) that he thinks he
found an error in a paper published by you, and also a remedy for the
error.

He writes that when he implemented your algorithm right off he got an
error, but that when he modified your algorithm in a couple of steps,
and implemented the modified algorithm, then he got the correct result.
At least, this is what I read into the text.

Joergen Backelin

From owner-reliable_computing [at] interval [dot] louisiana.edu Wed May 15 13:50:29 2002
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Date: Wed, 15 May 2002 13:50:21 -0500
To: "Ramon Moore" <rmoore17 [at] columbus [dot] rr.com>,
   "interval" <reliable_computing [at] interval [dot] louisiana.edu>
From: "R. Baker Kearfott" <rbk [at] louisiana [dot] edu>
Subject: Re: Fw: Difficulty in your paper
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--=====================_1021506621==_
Content-Type: text/plain; charset="us-ascii"

Friends,

For what it's worth, I tried this problem in GlobSol.  It found the
optimum (0, at (0,0)) without bisection; the output is attached :-)

I note that the gradient expressions are SUE's.

Best regards,

Baker

At 07:58 AM 5/15/2002 -0400, Ramon Moore wrote:
>Dear Colleagues,
>
>Is there a kind soul among you who will help the gentleman who sent me the
>plea below?
>I suspect he has a bug in his program, but I do not have the energy to help
>him find it.
>
>Ray (Ramon Moore)
>
>----- Original Message -----
>From: "N. S. Kubal _00423002_" <nskubal [at] ee [dot] iitb.ac.in>
>To: "Ramon Moore" <rmoore17 [at] columbus [dot] rr.com>
>Sent: Wednesday, May 15, 2002 3:44 AM
>Subject: Re: Difficulty in your paper
>
>
>> On Wed, 15 May 2002, Ramon Moore wrote:
>>
>> > Unfortunately I am unable to open the document you attached. Perhaps you
>can
>> > just explain your question in text in an email.
>> >
>> > R. Moore
>>
>> Dear Sir,
>>          Sorry for your inconvienence. Please go through the following
>> matter.
>>
>> ------------------------------------------------------------------------
>> Recently, I have implemented an optimization algorithm in FORTRAN 95 to
>> find global minimum based on your paper "On Computing the Range of Values"
>> published in Computing, 28:225-237, 1982 with Asaithambi, N.S. and Shen
>> Zuhe.
>>          When I ran the algorithm on one of the test function in [1] it
>> did not give me the correct global minimum.
>>
>> Test function in [1]
>>
>> f(x,y) = (x^2 + y^2)/200 - cos(x)*cos(y/sqrt(2)) + 1
>> X=[-100,100], Y=[-100,100]
>>
>> Results obtained using the algorithm given in your paper
>> Global minimum     = 0.145618691261803
>> Global minimum box = [-3.1104241497814656,-3.1104241032153367]
>>                      [-4.3556719087064267,-4.3556718155741691]
>>
>> But the results given in [1] are
>> Global minimum         = 0
>> Global minimum point   = (0,0)
>>
>>          When I went through each steps of the algorithm, I had come to
>> know that the algorithm deletes the boxes, which contain the global
>> minimum, which is "zero", due to steps 8 and step 13. Now in these steps
>> the variable 'S' decides to keep/delete the reduced boxes (Z). The values
>> of 'S' are based on the formulae given in your paper followed by equation
>> (3.2).
>>          I used new variable 'S1' in steps 8 and 13, which is based on the
>> following conditions,
>>
>> S1 .neq. 0 if (Infimum (DiF(X)) <= 0 and Supremum (DiF(X)) >= 0 in any one
>> direction)
>> S1 .eq.  0 if (Infimum (DiF(X)) > 0 or Supremum (DiF(X)) < 0 in all the
>> directions)
>>
>> (i.e. delete the box if the function is strictly monotonic in all the
>> directions).
>>
>> The modified steps 8 and 13 are,
>> Step 8:  if S1(1) is 0 and Z(1) does not intersect the boundary of Z(0),
>>          discard Z(1) and go to step 12.
>> Step 13: if S1(2) is 0 and Z(2) does not intersect the boundary of Z(0),
>>          discard Z(2) and go to step 17.
>>
>> Results obtained after modification
>> Global minimum     = 0.0E0
>> Global minimum box = [0.0E0, 0.0E0] [0.0E0, 0.0E0]
>>
>>          I would highly appreciate if you kindly spend some of your
>> precious time to go through the algorithm and tell me whether the changes
>> I have made are correct or not.
>>
>> References:
>> [1] C. Jansson, O. Knuppel. "A Global Minimization Method: The
>> Multi-Dimensional Case." TUHH (Technische Universitat Hamburg-Harburg),
>> Bericht 92.1, Januar, 1992, pp.47-48.
>>
>>
>>
>>
>
>

--=====================_1021506621==_
Content-Type: text/plain; charset="us-ascii"
Content-Disposition: attachment; filename="Moore.OT1"

 Output from FIND_GLOBAL_MIN on  05/15/2002  at  13:43:08.
 Version for the system is: October 10, 2000
 
 Codelist file name is: MooreG.CDL
 Box data file name is: Moore.DT1                     
 
 
 Initial box:
 [   -0.1000D+03,   0.1000D+03 ] [   -0.1000D+03,   0.1000D+03 ]
 
 BOUND_CONSTRAINT:
   F F   F F
 
 ---------------------------------------
 CONFIGURATION VALUES:
 
 EPS_DOMAIN:   0.1000D-07   MAXITR:   20000
 MAX_CPU_SECONDS:    0.1080D+05
 DO_INTERVAL_NEWTON: T  QUADRATIC: T  FULL_SPACE: F
 VERY_GOOD_INITIAL_GUESS: F
 USE_SUBSIT: F
 OUTPUT UNIT:           7  PRINT_LENGTH:           1
 PHI_MUST_CONVERGE: T
 EQ_CNS_MUST_CONVERGE: T
 INEQ_CNS_MUST_CONVERGE: T
 PHI_THICKNESS_FACTOR:   0.5000D+00
 EQ_CNS_THICKNESS_FACTOR:   0.5000D+00
 INEQ_CNS_THICKNESS_FACTOR:   0.5000D+00
 PHI_MUST_CONVERGE: T
 EQ_CNS_MUST_CONVERGE: T
 INEQ_CNS_MUST_CONVERGE: T
 PHI_CONVERGENCE_FACTOR:   0.1000D-13
 EQ_CNS_CONVERGENCE_FACTOR:   0.1000D-13
 INEQ_CNS_CONVERGENCE_FACTOR:   0.1000D-13
 CONTINUITY_ACROSS_BRANCHES: F
 SINGULAR_EXPANSION_FACTOR:   0.1000D+02
 HEURISTIC PARAMETER ALPHA:   0.5000D+00
 APPROX_OPT_BEFORE_BISECTION: T
 USE_TAYLOR_EQUALITY_CONSTRAINTS F
 USE_TAYLOR_INEQ_CONSTRAINTS F
 USE_TAYLOR_OBJECTIVE F
 USE_TAYLOR_EQ_CNS_GRD F
 USE_TAYLOR_GRAD F
 USE_TAYLOR_INEQ_CNS_GRD F
 USE_TAYLOR_REDUCED_INEWTON F
 COSY_POLYNOMIAL_ORDER           5
 LEAST_SQUARES_FUNCTIONS: F
 NONLINEAR_SYSTEM: F
 Default point optimizer was used. 
 
 THERE WERE NO BOXES IN THE LIST OF SMALL BOXES.
 
 LIST OF BOXES CONTAINING VERIFIED FEASIBLE POINTS:
 
 Box no.:           1
 Box coordinates:
 [   -0.4950D-09,   0.4950D-09 ] [   -0.1095-306,   0.1095-306 ]
PHI:
 [   -0.1110D-14,   0.2221D-14 ]
 B%LIUI(1,*):
 F F
 B%LIUI(2,*):
 F F
 B%SIDE(*):
 F F
 B%PEEL(*):
 F F
 Level:            0
 Box contains the following approximate root:
    0.0000D+00   0.0000D+00
 OBJECTIVE ENCLOSURE AT APPROXIMATE ROOT:
 [   -0.1332D-14,   0.2220D-14 ]
 Unknown =  T    Contains_root = T
 Changed coordinates:
 F F
 -------------------------------------------------
 ALGORITHM COMPLETED WITH LESS THAN THE MAXIMUM NUMBER,
       20000  OF BOXES.
 Number of evaluations of a row of the point
       expanded Jacobi matrix:           27
 No. dense interval residual evaluations -- gradient code list:           14
 Number of orig. system C-LP preconditioner rows:            2
 Total number of forward_substitutions:           48
 Number of Gauss--Seidel steps on the dense system:            2
 Number point dense residual evaluations, gradient codelist:            2
 No. point dense Hessian matrix evaluations -- gradient
       codelist:            1
 Total number of iterations of one of
       the point Newton methods:            1
 Total number of dense slope matrix evaluations:            8
 Total number second-order interval evaluations of the
       original function:            6
 Total number dense interval reduced gradient evaluations:            3
 Number of times the interval Newton method made a coordinate
       interval smaller:            2
 Number of times the approximate solver was called:            2
 Number Fritz-John matrix evaluations:            1
 BEST_ESTIMATE:    0.2220D-14
 Overall CPU time:    0.1001D-01
 CPU time in PEEL_BOUNDARY:    0.0000D+00
 CPU time in REDUCED_INTERVAL_NEWTON:    0.0000D+00

--=====================_1021506621==_
Content-Type: text/plain; charset="us-ascii"
Content-Disposition: attachment; filename="Moore.DT1"

1D-8
-100 100
-100 100
F F
F F


--=====================_1021506621==_
Content-Type: text/plain; charset="us-ascii"


---------------------------------------------------------------
R. Baker Kearfott,    rbk [at] louisiana [dot] edu   (337) 482-5346 (fax)
(337) 482-5270 (work)                     (337) 981-9744 (home)
URL: http://interval.louisiana.edu/kearfott.html
Department of Mathematics, University of Louisiana at Lafayette
Box 4-1010, Lafayette, LA 70504-1010, USA
---------------------------------------------------------------

--=====================_1021506621==_--


From owner-reliable_computing [at] interval [dot] louisiana.edu Wed May 15 14:00:54 2002
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Message-ID: <002d01c1fc42$b6513160$66ae1841 [at] columbus [dot] rr.com>
From: "Ramon Moore" <rmoore17 [at] columbus [dot] rr.com>
To: "N. S. Kubal _00423002_" <nskubal [at] ee [dot] iitb.ac.in>
Cc: "interval" <reliable_computing [at] interval [dot] louisiana.edu>
Subject: Fw: Fw: Difficulty in your paper
Date: Wed, 15 May 2002 14:59:53 -0400
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Dear N. S. Kubal,

I am forwarding to you a response to your question from Prof. Kearfott, whom
I thank for his help.
As you can see, the algorithm suggested in the paper of 1982 has been
supplanted by others by now.
In any case, I am sorry to say that I have no energy at this time for
digging back into the details
of that 20 year-old paper. Perhaps your suggested changes are necessary. I
do not know.
I no longer have a working program implementing the algorithm of the 1982
paper,
and I do not wish to write another one at this point in time.
I am hoping that others may help with a clear answer, if you are really
determined to know.
Best wishes,
Ramon Moore

----- Original Message -----
From: "R. Baker Kearfott" <rbk [at] louisiana [dot] edu>
To: "Ramon Moore" <rmoore17 [at] columbus [dot] rr.com>; "interval"
<reliable_computing [at] interval [dot] louisiana.edu>
Sent: Wednesday, May 15, 2002 2:50 PM
Subject: Re: Fw: Difficulty in your paper


> Friends,
>
> For what it's worth, I tried this problem in GlobSol.  It found the
> optimum (0, at (0,0)) without bisection; the output is attached :-)
>
> I note that the gradient expressions are SUE's.
>
> Best regards,
>
> Baker
>
> At 07:58 AM 5/15/2002 -0400, Ramon Moore wrote:
> >Dear Colleagues,
> >
> >Is there a kind soul among you who will help the gentleman who sent me
the
> >plea below?
> >I suspect he has a bug in his program, but I do not have the energy to
help
> >him find it.
> >
> >Ray (Ramon Moore)
> >
> >----- Original Message -----
> >From: "N. S. Kubal _00423002_" <nskubal [at] ee [dot] iitb.ac.in>
> >To: "Ramon Moore" <rmoore17 [at] columbus [dot] rr.com>
> >Sent: Wednesday, May 15, 2002 3:44 AM
> >Subject: Re: Difficulty in your paper
> >
> >
> >> On Wed, 15 May 2002, Ramon Moore wrote:
> >>
> >> > Unfortunately I am unable to open the document you attached. Perhaps
you
> >can
> >> > just explain your question in text in an email.
> >> >
> >> > R. Moore
> >>
> >> Dear Sir,
> >>          Sorry for your inconvienence. Please go through the following
> >> matter.
> >>
>
>> ------------------------------------------------------------------------
> >> Recently, I have implemented an optimization algorithm in FORTRAN 95 to
> >> find global minimum based on your paper "On Computing the Range of
Values"
> >> published in Computing, 28:225-237, 1982 with Asaithambi, N.S. and Shen
> >> Zuhe.
> >>          When I ran the algorithm on one of the test function in [1] it
> >> did not give me the correct global minimum.
> >>
> >> Test function in [1]
> >>
> >> f(x,y) = (x^2 + y^2)/200 - cos(x)*cos(y/sqrt(2)) + 1
> >> X=[-100,100], Y=[-100,100]
> >>
> >> Results obtained using the algorithm given in your paper
> >> Global minimum     = 0.145618691261803
> >> Global minimum box = [-3.1104241497814656,-3.1104241032153367]
> >>                      [-4.3556719087064267,-4.3556718155741691]
> >>
> >> But the results given in [1] are
> >> Global minimum         = 0
> >> Global minimum point   = (0,0)
> >>
> >>          When I went through each steps of the algorithm, I had come to
> >> know that the algorithm deletes the boxes, which contain the global
> >> minimum, which is "zero", due to steps 8 and step 13. Now in these
steps
> >> the variable 'S' decides to keep/delete the reduced boxes (Z). The
values
> >> of 'S' are based on the formulae given in your paper followed by
equation
> >> (3.2).
> >>          I used new variable 'S1' in steps 8 and 13, which is based on
the
> >> following conditions,
> >>
> >> S1 .neq. 0 if (Infimum (DiF(X)) <= 0 and Supremum (DiF(X)) >= 0 in any
one
> >> direction)
> >> S1 .eq.  0 if (Infimum (DiF(X)) > 0 or Supremum (DiF(X)) < 0 in all the
> >> directions)
> >>
> >> (i.e. delete the box if the function is strictly monotonic in all the
> >> directions).
> >>
> >> The modified steps 8 and 13 are,
> >> Step 8:  if S1(1) is 0 and Z(1) does not intersect the boundary of
Z(0),
> >>          discard Z(1) and go to step 12.
> >> Step 13: if S1(2) is 0 and Z(2) does not intersect the boundary of
Z(0),
> >>          discard Z(2) and go to step 17.
> >>
> >> Results obtained after modification
> >> Global minimum     = 0.0E0
> >> Global minimum box = [0.0E0, 0.0E0] [0.0E0, 0.0E0]
> >>
> >>          I would highly appreciate if you kindly spend some of your
> >> precious time to go through the algorithm and tell me whether the
changes
> >> I have made are correct or not.
> >>
> >> References:
> >> [1] C. Jansson, O. Knuppel. "A Global Minimization Method: The
> >> Multi-Dimensional Case." TUHH (Technische Universitat Hamburg-Harburg),
> >> Bericht 92.1, Januar, 1992, pp.47-48.
> >>
> >>
> >>
> >>
> >
> >
>


----------------------------------------------------------------------------
----


>
> ---------------------------------------------------------------
> R. Baker Kearfott,    rbk [at] louisiana [dot] edu   (337) 482-5346 (fax)
> (337) 482-5270 (work)                     (337) 981-9744 (home)
> URL: http://interval.louisiana.edu/kearfott.html
> Department of Mathematics, University of Louisiana at Lafayette
> Box 4-1010, Lafayette, LA 70504-1010, USA
> ---------------------------------------------------------------
>

------=_NextPart_000_002A_01C1FC21.2BA5EFE0
Content-Type: application/octet-stream;
	name="Moore.OT1"
Content-Transfer-Encoding: quoted-printable
Content-Disposition: attachment;
	filename="Moore.OT1"

 Output from FIND_GLOBAL_MIN on  05/15/2002  at  13:43:08.
 Version for the system is: October 10, 2000
=20
 Codelist file name is: MooreG.CDL
 Box data file name is: Moore.DT1                    =20
=20
=20
 Initial box:
 [   -0.1000D+03,   0.1000D+03 ] [   -0.1000D+03,   0.1000D+03 ]
=20
 BOUND_CONSTRAINT:
   F F   F F
=20
 ---------------------------------------
 CONFIGURATION VALUES:
=20
 EPS_DOMAIN:   0.1000D-07   MAXITR:   20000
 MAX_CPU_SECONDS:    0.1080D+05
 DO_INTERVAL_NEWTON: T  QUADRATIC: T  FULL_SPACE: F
 VERY_GOOD_INITIAL_GUESS: F
 USE_SUBSIT: F
 OUTPUT UNIT:           7  PRINT_LENGTH:           1
 PHI_MUST_CONVERGE: T
 EQ_CNS_MUST_CONVERGE: T
 INEQ_CNS_MUST_CONVERGE: T
 PHI_THICKNESS_FACTOR:   0.5000D+00
 EQ_CNS_THICKNESS_FACTOR:   0.5000D+00
 INEQ_CNS_THICKNESS_FACTOR:   0.5000D+00
 PHI_MUST_CONVERGE: T
 EQ_CNS_MUST_CONVERGE: T
 INEQ_CNS_MUST_CONVERGE: T
 PHI_CONVERGENCE_FACTOR:   0.1000D-13
 EQ_CNS_CONVERGENCE_FACTOR:   0.1000D-13
 INEQ_CNS_CONVERGENCE_FACTOR:   0.1000D-13
 CONTINUITY_ACROSS_BRANCHES: F
 SINGULAR_EXPANSION_FACTOR:   0.1000D+02
 HEURISTIC PARAMETER ALPHA:   0.5000D+00
 APPROX_OPT_BEFORE_BISECTION: T
 USE_TAYLOR_EQUALITY_CONSTRAINTS F
 USE_TAYLOR_INEQ_CONSTRAINTS F
 USE_TAYLOR_OBJECTIVE F
 USE_TAYLOR_EQ_CNS_GRD F
 USE_TAYLOR_GRAD F
 USE_TAYLOR_INEQ_CNS_GRD F
 USE_TAYLOR_REDUCED_INEWTON F
 COSY_POLYNOMIAL_ORDER           5
 LEAST_SQUARES_FUNCTIONS: F
 NONLINEAR_SYSTEM: F
 Default point optimizer was used.=20
=20
 THERE WERE NO BOXES IN THE LIST OF SMALL BOXES.
=20
 LIST OF BOXES CONTAINING VERIFIED FEASIBLE POINTS:
=20
 Box no.:           1
 Box coordinates:
 [   -0.4950D-09,   0.4950D-09 ] [   -0.1095-306,   0.1095-306 ]
PHI:
 [   -0.1110D-14,   0.2221D-14 ]
 B%LIUI(1,*):
 F F
 B%LIUI(2,*):
 F F
 B%SIDE(*):
 F F
 B%PEEL(*):
 F F
 Level:            0
 Box contains the following approximate root:
    0.0000D+00   0.0000D+00
 OBJECTIVE ENCLOSURE AT APPROXIMATE ROOT:
 [   -0.1332D-14,   0.2220D-14 ]
 Unknown =3D  T    Contains_root =3D T
 Changed coordinates:
 F F
 -------------------------------------------------
 ALGORITHM COMPLETED WITH LESS THAN THE MAXIMUM NUMBER,
       20000  OF BOXES.
 Number of evaluations of a row of the point
       expanded Jacobi matrix:           27
 No. dense interval residual evaluations -- gradient code list:          =
 14
 Number of orig. system C-LP preconditioner rows:            2
 Total number of forward_substitutions:           48
 Number of Gauss--Seidel steps on the dense system:            2
 Number point dense residual evaluations, gradient codelist:            =
2
 No. point dense Hessian matrix evaluations -- gradient
       codelist:            1
 Total number of iterations of one of
       the point Newton methods:            1
 Total number of dense slope matrix evaluations:            8
 Total number second-order interval evaluations of the
       original function:            6
 Total number dense interval reduced gradient evaluations:            3
 Number of times the interval Newton method made a coordinate
       interval smaller:            2
 Number of times the approximate solver was called:            2
 Number Fritz-John matrix evaluations:            1
 BEST_ESTIMATE:    0.2220D-14
 Overall CPU time:    0.1001D-01
 CPU time in PEEL_BOUNDARY:    0.0000D+00
 CPU time in REDUCED_INTERVAL_NEWTON:    0.0000D+00

------=_NextPart_000_002A_01C1FC21.2BA5EFE0
Content-Type: application/octet-stream;
	name="Moore.DT1"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment;
	filename="Moore.DT1"

1D-8
-100 100
-100 100
F F
F F


------=_NextPart_000_002A_01C1FC21.2BA5EFE0--


From owner-reliable_computing [at] interval [dot] louisiana.edu Wed May 15 15:51:55 2002
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Date: Wed, 15 May 2002 22:51:33 +0200
From: Arnold Neumaier <Arnold.Neumaier [at] univie [dot] ac.at>
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Subject: Taylor forms - use and limits
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Following last months discussion on the mailing list about the 
Taylor models of Martin Berz, I wrote a survey article
detailing on 37 pages my current knowledge about Taylor arithmetic 
in its various forms. Below you can find the abstract and the 
conclusions. It can be downloaded from the following address:

          Arnold Neumaier, Taylor forms - use and limits
      http://www.mat.univie.ac.at/~neum/papers.html#taylor



Arnold Neumaier



Abstract
========


This review is a response to recent discussions on the reliable 
computing mailing list, and to continuing uncertainties about the
properties and merits of Taylor forms, multivariate higher degree 
generalizations of centered forms. They were invented around 
1980 by Lanford, documented in detail in 1984 
by Eckmann, Koch and Wittwer, and independently studied and 
popularized since 1996 by Berz, Makino and Hoefkens.
A highlight is their application to the verified integration of 
asteroid dynamics in the solar system in 2001, although the details
given are not sufficient to check the validity of their claims.

Apart from summarizing what Taylor forms are and do,
this review puts them into the perspective of more traditional methods,
in particular centered forms, discusses the major applications, 
and analyzes some of their elementary properties.
Particular emphasis is given to overestimation properties and the 
wrapping effect. A deliberate attempt has been made to offer value 
statements with appropriate justifications; but all opinions given 
are my own and might be controversial.



Conclusions
===========

1. The various versions of Taylor arithmetic constitute a significant 
enhancement of the toolkit of interval analysis techniques. Indeed,
interval coefficient versions were used by mathematical physicists to 
prove estimates important for computer-assisted proofs, 
and floating-point coefficient versions were used by Berz and his group
to verify solutions of celestial mechanics problems that so far
defied interval techniques.

Berz and his group also used Taylor models for applications to 
multivariate integration over a box, differential algebraic equations, 
and other problems that have not been discussed before in the interval 
literature. 


2. Claims that Taylor arithmetic approximates ranges to higher than 
second order are unfounded, at least for the currently implemented 
versions that do not attempt to compute the exact range of the 
quadratic term.

What is true is that in problems where in real arithmetic substantial 
cancellation of leading digits occurs, Taylor arithmetic may drastically
reduce overestimation over intervals narrow enough such that the 
remainder intervals remain sufficiently small. These are sometimes
quite small intervals and sometimes quite wide ones, depending on the 
problem. If dependent intervals are present but without the cancellation
structure, gains from a Taylor arithmetic are slight, and the much 
higher cost and storage does not justify their use.


3. Claims that Taylor arithmetic eliminates the wrapping effect are
unfounded. In the asteroid integrations by Berz et al., the wrapping 
effect is reduced by means of a technique called shrink wrapping 
which is poorly documented, but without which rounding errors are 
quickly magnified to huge intervals, ruining the gain from 
cancellation. 

What is true is that Taylor arithmetic may drastically reduce the 
part of the wrapping effect due to nonlinearities, 
since the nonlinear dependence on initial conditions is taken 
into account to a large extent, while methods like Lohner's AWA 
incur here additional wrapping from the need to enclose bended boxes 
into parallelepipeds. 
However, the wrapping incurred by the error term is not addressed 
at all by Taylor arithmetic, and shrink wrapping seems to be a good 
remedy only for problems whose Jacobians
have no eigenvalues with significantly differing real parts, or where 
drastic phase space volume reductions compensate the wrapping. 
This is apparently the case for the asteroid problem and 
(for some time) for the Lorenz attractor, but it excludes already 
simple 2-dimensional Hamiltonian dynamical systems with a hyperbolic 
fixed point, which are very benign from an analytic point of view.


4. In the COSY implementation of Taylor models used by Berz et al. 
for the claimed verified integration of asteroid dynamics in the 
solar system,
rounding error control is completely undocumented, so that it is 
difficult to assess whether the implementation is mathematically 
rigorous. Moreover, the (almost lacking) details given about shrink 
wrapping are not sufficient to check the validity of their claims.

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From: "Ramon Moore" <rmoore17 [at] columbus [dot] rr.com>
To: "N. S. Kubal _00423002_" <nskubal [at] ee [dot] iitb.ac.in>
Cc: "interval" <reliable_computing [at] interval [dot] louisiana.edu>
Subject: question
Date: Wed, 15 May 2002 16:54:26 -0400
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Dear N. S. Kubal,

I have tried to understand the difference between what you propose:

I used new variable 'S1' in steps 8 and 13, which is based on
the following conditions,
    S1 .neq. 0 if (Infimum (DiF(X)) <=3D 0 and Supremum (DiF(X)) >=3D 0 =
in any one direction)
    S1 .eq.  0 if (Infimum (DiF(X)) > 0 or Supremum (DiF(X)) < 0 in all =
the directions)
    (i.e. delete the box if the function is strictly monotonic in all =
the directions).

    The modified steps 8 and 13 are,
    Step 8:  if S1(1) is 0 and Z(1) does not intersect the boundary of =
Z(0),
    discard Z(1) and go to step 12.
    Step 13: if S1(2) is 0 and Z(2) does not intersect the boundary of =
Z(0),
    discard Z(2) and go to step 17.

and what was in the 1982 paper at steps 8 and 13

    Step 8: if S(1) is empty [meaning that the function is strictly =
monotonic in all directions] and Z(1) does not         intersect the =
boundary of Z(0),
    discard Z(1) and go to step 12.
    . . .
    Step 13: etc.

I do not see any difference whatever. I do not understand how you got =
the wrong answer other than by having a bug in the implementation.

Ramon Moore

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<BODY bgColor=3D#ffffff>
<DIV><FONT face=3DArial>Dear N. S. Kubal,</FONT></DIV>
<DIV><FONT face=3DArial></FONT>&nbsp;</DIV>
<DIV><FONT face=3DArial>I have tried to understand the difference =
between what you=20
propose:</FONT></DIV>
<DIV><FONT face=3DArial></FONT>&nbsp;</DIV>
<DIV>I used new variable 'S1' in steps 8 and 13, which is based =
on<BR>the=20
following conditions,<BR>&nbsp;&nbsp;&nbsp; S1 .neq. 0 if (Infimum =
(DiF(X))=20
&lt;=3D 0 and Supremum (DiF(X)) &gt;=3D 0 in any one=20
direction)<BR>&nbsp;&nbsp;&nbsp; S1 .eq.&nbsp; 0 if (Infimum (DiF(X)) =
&gt; 0 or=20
Supremum (DiF(X)) &lt; 0 in all the directions)<BR>&nbsp;&nbsp;&nbsp; =
(i.e.=20
delete the box if the function is strictly monotonic in all the=20
directions).<BR><BR><FONT face=3DArial>&nbsp;&nbsp;&nbsp; </FONT>The =
modified=20
steps 8 and 13 are,<BR>&nbsp;&nbsp;&nbsp; Step 8:&nbsp; if S1(1) is 0 =
and Z(1)=20
does not intersect the boundary of Z(0),<BR>&nbsp;&nbsp;&nbsp; discard =
Z(1) and=20
go to step 12.<BR>&nbsp;&nbsp;&nbsp; Step 13: if S1(2) is 0 and Z(2) =
does not=20
intersect the boundary of Z(0),<BR>&nbsp;&nbsp;&nbsp; discard Z(2) and =
go to=20
step 17.</DIV>
<DIV>&nbsp;</DIV>
<DIV><FONT face=3DArial>and what was in the 1982 paper at steps 8 and=20
13</FONT></DIV>
<DIV><FONT face=3DArial></FONT>&nbsp;</DIV>
<DIV><FONT face=3DArial>&nbsp;&nbsp;&nbsp; Step 8: if S(1) is empty =
[meaning that=20
the function is strictly monotonic in all directions] and Z(1) does not=20
&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; intersect the boundary of=20
Z(0),</FONT></DIV>
<DIV><FONT face=3DArial>&nbsp;&nbsp;&nbsp; discard Z(1) and go to step=20
12.</FONT></DIV>
<DIV><FONT face=3DArial>&nbsp;&nbsp;&nbsp; . . .</FONT></DIV>
<DIV><FONT face=3DArial>&nbsp;&nbsp;&nbsp; Step 13: etc.</FONT></DIV>
<DIV><FONT face=3DArial></FONT>&nbsp;</DIV>
<DIV><FONT face=3DArial>I do not see any difference whatever. I do not =
understand=20
how you got the wrong answer other than by having a bug in the=20
implementation.</FONT></DIV>
<DIV><FONT face=3DArial></FONT>&nbsp;</DIV>
<DIV><FONT face=3DArial>Ramon Moore</FONT></DIV></BODY></HTML>

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From owner-reliable_computing [at] interval [dot] louisiana.edu Fri May 17 09:46:35 2002
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From: "Ramon Moore" <rmoore17 [at] columbus [dot] rr.com>
To: "N. S. Kubal _00423002_" <nskubal [at] ee [dot] iitb.ac.in>
Cc: "interval" <reliable_computing [at] interval [dot] louisiana.edu>
References: <Pine.LNX.4.33.0205171912060.22708-102000 [at] bharati [dot] ee.iitb.ac.in>
Subject: Re: question
Date: Fri, 17 May 2002 10:45:23 -0400
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Dear Nandkishor,

I apologize for taking so long to read carefully the problem you raised.
Yes, you are correct. The algorithm given in the 1982 paper throws the baby
out
with the bath water in the example you showed. In fact, on page 2 of your
attachment (pdf version), at step 7 in Iteration 2, we already have the
exact solution
as b_1= 0 and Z(1)=([0,0],[0,0]). Clearly we do NOT want to delete that.
Your way of fixing the problem is fine, testing for strict monotonicity, as
you have
pointed out in your example. In the 1982 paper, on page 232, after (3.2) is:
"where S is the set of integers i such that D_i F(X) properly contains zero
and . . ."
but the following definition of (u_i,v_i) should have been, as you point
out,
using strict inequalities to correspond with what was intended in the
definition of S.

Thank you very much for pointing this out.
Best Wishes:-

Ramon Moore


----- Original Message -----
From: "N. S. Kubal _00423002_" <nskubal [at] ee [dot] iitb.ac.in>
To: "Ramon Moore" <rmoore17 [at] columbus [dot] rr.com>
Sent: Friday, May 17, 2002 9:39 AM
Subject: Re: question


>
>
> Dear Sir,
> I am bothering you again with the same doubt I have raised in one of my
> mails to you. By strictly monotonic I understand   D_F(X)<0 or D_F(X)>0.
> But in your paper 'S' is empty when sup(D_F(X))<=0 or inf(D_F(X))>=0. When
> 'S' is defined as given in the paper it deletes the solution box.  This is
> the reason why I felt there is a need of new variable in place of 'S'. I
> attached a document (.ps as well as .pdf) which gives results of each
> steps of the algorithm as given in the paper for a particular function and
> domain.
>
> With Regards
>
> Nandkishor Kubal
>
>
>
>
>
> On Thu, 16 May 2002, Ramon Moore wrote:
>
> > Dear N. S. Kubal,
> >
> > I have tried to understand the difference between what you propose:
> >
> > I used new variable 'S1' in steps 8 and 13, which is based on
> > the following conditions,
> >     S1 .neq. 0 if (Infimum (DiF(X)) <= 0 and Supremum (DiF(X)) >= 0 in
any one direction)
> >     S1 .eq.  0 if (Infimum (DiF(X)) > 0 or Supremum (DiF(X)) < 0 in all
the directions)
> >     (i.e. delete the box if the function is strictly monotonic in all
the directions).
> >
> >     The modified steps 8 and 13 are,
> >     Step 8:  if S1(1) is 0 and Z(1) does not intersect the boundary of
Z(0),
> >     discard Z(1) and go to step 12.
> >     Step 13: if S1(2) is 0 and Z(2) does not intersect the boundary of
Z(0),
> >     discard Z(2) and go to step 17.
> >
> > and what was in the 1982 paper at steps 8 and 13
> >
> >     Step 8: if S(1) is empty [meaning that the function is strictly
monotonic in all directions] and Z(1) does not         intersect the
boundary of Z(0),
> >     discard Z(1) and go to step 12.
> >     . . .
> >     Step 13: etc.
> >
> > I do not see any difference whatever. I do not understand how you got
the wrong answer other than by having a bug in the implementation.
> >
> > Ramon Moore
> >
>
> --
>
****************************************************************************
*
> contact me on
>
> nskubal [at] ee [dot] iitb.ac.in
> nanduskubal [at] yahoo [dot] com
>
> Department                              Residence
>
> NANDKISHOR S. KUBAL                     ROOM NO 287
> SYSTEM AND CONTROL ENGG.                HOSTEL NO 1
> ELECTRICAL DEPT.
> IIT,BOMBAY                              PHONE NO
> POWAI,MUMBAI.
>                                         (internal) 5601
>                                         (external) 5721017 / 5720093
> PHONE NO
> 5722545 EXT 5741
>


From owner-reliable_computing [at] interval [dot] louisiana.edu Fri May 17 11:23:57 2002
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Date: Fri, 17 May 2002 09:23:45 -0700 (PDT)
From: William Walster <Bill.Walster [at] Eng [dot] Sun.COM>
Subject: Re: Fw: Difficulty in your paper
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Dear Ray and Nandkishor,

The following URL contains a small test program that our interval team
prepared some time ago.  It gets the correct answer to Griewank's problem.

http://wwws.sun.com/software/Developer-products/fortran/interval/general/Minimum
.html#griewank2.f90

While it is important to find and fix problems, whatever their source, I
think it is important to point out that there is no doubt about the 
correctness of the interval approach to solving this and other nonlinear
problems.

It is *this* fact for which I will always be grateful to you, Ray.

Best regards,

Bill


	G. William (Bill) Walster, Ph.D. 
	Interval Technology Engineering Manager
	Sun Microsystems, Inc.
	16 Network Circle, MS UMPK16-304
	Menlo Park, CA 94025
	(650) 786-9004 Direct
	(650) 786-9551 Fax

	bill.walster [at] eng [dot] sun.com



From owner-reliable_computing [at] interval [dot] louisiana.edu Sun May 19 17:42:09 2002
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Date: Sun, 19 May 2002 16:41:38 -0600 (MDT)
From: Vladik Kreinovich <vladik [at] cs [dot] utep.edu>
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From: Arnold Neumaier <Arnold.Neumaier [at] univie [dot] ac.at>
Date: Fri, 10 May 2002 12:26:58 +0200
Subject: The Mathematics Genealogy Project

I came across a very interesting WWW-site:

The Mathematics Genealogy Project
http://hcoonce.math.mankato.msus.edu/index.html

The site contains names, advisors, and titles of dissertations
of mathematicians with a Ph.D., throughout the history

Currently (May 6th, 2002) there are records of 55249 mathematicians.
While fairly complete on US mathematicians, the directory
is still quite incomplete for the rest of the world.
The authors invite submissions updating incomplete or missing
information.

Please contribute and make the directory more complete and
hence more useful.

Arnold Neumaier



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From: "Ramon Moore" <rmoore17 [at] columbus [dot] rr.com>
To: "Vladik Kreinovich" <vladik [at] cs [dot] utep.edu>
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Fascinating! Thanks. Just for fun I looked up a few people I know or knew,
and there they surely were!
I looked up Charles Noble, who was an 85 year old emeritus professor (and
former chairman) of the mathematics dept at Berkeley when I was a student
there. I lived as a guest in his house for more than a year during 1949-50.
I recall many fascinating stories he told about his days as a student at
Go"ttingen at the end of the 19th century. Indeed, if you look him up in the
Math Geneology webpages, you will find he got his Ph.D. from, David Hilbert
in 1901. If you look up Hilbert, you will find that he was advisor to 71
Ph.D.students. When you look up Henri Poincare', however, you will find no
students. I was bit surprised by that, because Charles Noble told stories
about hikes together with both Hilbert and Poincare', when Henri was
visiting at Go"ttingen. I do not know of any Ph.D. advisees of Poincare',
but I know he was around students, at least at Go"ttingen.

Anyway, thanks Arnold and Vladik for pointing out this fun website.

Ramon Moore



----- Original Message -----
From: "Vladik Kreinovich" <vladik [at] cs [dot] utep.edu>
To: <reliable_computing [at] interval [dot] louisiana.edu>
Sent: Sunday, May 19, 2002 6:41 PM
Subject: from NA Digest


> From: Arnold Neumaier <Arnold.Neumaier [at] univie [dot] ac.at>
> Date: Fri, 10 May 2002 12:26:58 +0200
> Subject: The Mathematics Genealogy Project
>
> I came across a very interesting WWW-site:
>
> The Mathematics Genealogy Project
> http://hcoonce.math.mankato.msus.edu/index.html
>
> The site contains names, advisors, and titles of dissertations
> of mathematicians with a Ph.D., throughout the history
>
> Currently (May 6th, 2002) there are records of 55249 mathematicians.
> While fairly complete on US mathematicians, the directory
> is still quite incomplete for the rest of the world.
> The authors invite submissions updating incomplete or missing
> information.
>
> Please contribute and make the directory more complete and
> hence more useful.
>
> Arnold Neumaier
>
>
>


From owner-reliable_computing [at] interval [dot] louisiana.edu Mon May 20 13:30:39 2002
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Subject: Reliable Computing, Volume 8, issue 4, 2002
Date: Mon, 20 May 2002 22:15:27 +0400
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                     4-2002
                Reliable Computing
             Volume 8, issue 4, 2002

             Mathematical Research

Sparse Systems in Fixed Point Form
Ramon E. Moore
249-265

On Existence and Uniqueness Verification for  Non-Smooth Functions
R. Baker Kearfott
267-282

Interval Estimations of Solution Sets to  Real-Valued Systems of Linear
or Non-Linear Equations
Miguel A. Sainz, Ernest Gardenyes, Lambert Jorba
283-305

         Short Communication

Interval Arithmetic on Multimedia Architectures
Juergen Wolff von Gudenberg
307-312

         Letter to the Editor

Grand Challenges and Scientific Standards in Interval Analysis
Arnold Neumaier
313-320



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&nbsp;&nbsp;=20
Volume 8, issue 4, 2002</FONT></DIV>
<DIV>&nbsp;</DIV>
<DIV><FONT face=3D"Arial Cyr"=20
size=3D2>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;=20
Mathematical Research</FONT></DIV>
<DIV>&nbsp;</DIV>
<DIV><FONT face=3D"Arial Cyr" size=3D2>Sparse Systems in Fixed Point =
Form<BR>Ramon=20
E. Moore<BR>249-265</FONT></DIV>
<DIV>&nbsp;</DIV>
<DIV><FONT face=3D"Arial Cyr" size=3D2>On Existence and Uniqueness =
Verification=20
for&nbsp; Non-Smooth Functions<BR>R. Baker =
Kearfott<BR>267-282</FONT></DIV>
<DIV>&nbsp;</DIV>
<DIV><FONT face=3D"Arial Cyr" size=3D2>Interval Estimations of Solution =
Sets=20
to&nbsp; Real-Valued Systems of Linear<BR>or Non-Linear =
Equations<BR>Miguel A.=20
Sainz, Ernest Gardenyes, Lambert Jorba<BR>283-305</FONT></DIV>
<DIV>&nbsp;</DIV>
<DIV><FONT face=3D"Arial Cyr"=20
size=3D2>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Short=20
Communication</FONT></DIV>
<DIV>&nbsp;</DIV>
<DIV><FONT face=3D"Arial Cyr" size=3D2>Interval Arithmetic on Multimedia =

Architectures<BR>Juergen Wolff von Gudenberg<BR>307-312</FONT></DIV>
<DIV>&nbsp;</DIV>
<DIV><FONT face=3D"Arial Cyr"=20
size=3D2>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Letter to the=20
Editor</FONT></DIV>
<DIV>&nbsp;</DIV>
<DIV><FONT face=3D"Arial Cyr" size=3D2>Grand Challenges and Scientific =
Standards in=20
Interval Analysis<BR>Arnold Neumaier<BR>313-320</FONT></DIV>
<DIV>&nbsp;</DIV>
<DIV>&nbsp;</DIV></BODY></HTML>

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