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From owner-reliable_computing [at] interval [dot] louisiana.edu Sun Jan  2 17:08:15 2005
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Date: Sun, 2 Jan 2005 16:06:52 -0700 (MST)
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From: Go05 <go05 [at] dali [dot] ace.ual.es>
Date: Tue, 28 Dec 2004 10:06:18 +0100 (CET)
Subject: Workshop in Spain on Global Optimization

Dear colleagues,

Happy new year on behalf of the organising committee of the

INTERNATIONAL WORKSHOP ON GLOBAL OPTIMIZATION GO05, Almeria, Spain

Before you start making new plans for 2005, let us remind you of the workshop.
We have been arranging facilities in the pitoresque village of San Jose,
which makes it possible to keep the fee low and being in a beautiful and 
calm environment.

Timing: 18th-22th September 2005
Participants arrive on Saturday 17 and leave Thursday 22, afternoon.

Global schedule:
Deadline for the submission of abstracts May 15th, 2005
Notification of acceptance June 15th, 2005
Deadline for early registration (reduced fee) June 30th, 2005
Deadline for last minute registration (450) July 28th, 2005
On-site reception September 17th, 2005
Start of conference September 18th, 2005

Publication:
Two special issues of the Journal of Global Optimization will publish 
papers emerging from the talks after the regular refereeing procedure.

Organising committee:
Emilio Carrizosa, Sevilla
Tibor Csendes, Szeged
Inmaculada Garcia, Almeria
Eligius Hendrix, Wageningen
Panos Pardalos, Florida

Local organisers:
Miguel Cobo
Leocadio Casado
Pilar Ortigosa
Boglarka Toth
Consolacion Gil
Raul Banos
Juana Redondo

Keep updated via http://dali.ace.ual.es/~go05/

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From owner-reliable_computing [at] interval [dot] louisiana.edu Mon Jan  3 08:36:51 2005
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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: SciCADE
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------------- Begin Forwarded Message -------------

From: krj [at] cs [dot] toronto.edu

... there would be two sessions on
Validated methods for differential equations at the upcoming SciCADE 
meeting in Nagoya, Japan, May 23-27, 2005. 
The URL for the conference is 

 	http://www.math.human.nagoya-u.ac.jp/scicade05/

...

The focus
of the conference is not validated computing, but there will be two
sessions on validated methods for differential equations.  However, it
is a very good conference for anyone interested in numerical methods
for differential equations.

Best regards,
Ken
------------- End Forwarded Message -------------



From owner-reliable_computing [at] interval [dot] louisiana.edu Mon Jan  3 13:10:31 2005
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Date: Mon, 03 Jan 2005 20:08:59 +0100
From: Arnold Neumaier <Arnold.Neumaier [at] univie [dot] ac.at>
Organization: University of Vienna
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Subject: quadratic equation with interval coefficients
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Is there a complete discussion in the literature of the solution
of a single quadratic equation in one variable,
with interval coefficients? Perhaps even an implementation
that computes a correctly rounded enclosure for the solution set?

I worked out a solution myself, but this problem is so natural
that it was probably treated before.


Arnold Neumaier



From owner-reliable_computing [at] interval [dot] louisiana.edu Mon Jan  3 13:46:13 2005
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Date: Mon, 03 Jan 2005 13:46:52 -0600
To: Arnold Neumaier <Arnold.Neumaier [at] univie [dot] ac.at>,
        interval <reliable_computing [at] interval [dot] louisiana.edu>
From: "R. Baker Kearfott" <rbk [at] louisiana [dot] edu>
Subject: Re: quadratic equation with interval coefficients
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Arnold,

I looked at it some years ago, when I was programming the elementary
functions for GlobSol's constraint propagation.  I even wrote a routine
for it.  However, I never published it, and it is not contained
in the distribution version of GlobSol.  Furthermore, I don't think
I proved that it was the tightest possible machine enclosure for the
solution set.  Thus, if I'm the only one who did it, it's "fair game"
for you to write it up :-)

Best regards,

Baker

At 08:08 PM 1/3/2005 +0100, Arnold Neumaier wrote:
>Is there a complete discussion in the literature of the solution
>of a single quadratic equation in one variable,
>with interval coefficients? Perhaps even an implementation
>that computes a correctly rounded enclosure for the solution set?
>
>I worked out a solution myself, but this problem is so natural
>that it was probably treated before.
>
>
>Arnold Neumaier
>
>
>

---------------------------------------------------------------
R. Baker Kearfott,    rbk [at] louisiana [dot] edu   (337) 482-5346 (fax)
(337) 482-5270 (work)                     (337) 993-1827 (home)
URL: http://interval.louisiana.edu/kearfott.html
Department of Mathematics, University of Louisiana at Lafayette
(Room 217 Maxim D. Doucet Hall, 1403 Johnston Street)
Box 4-1010, Lafayette, LA 70504-1010, USA
---------------------------------------------------------------


From owner-reliable_computing [at] interval [dot] louisiana.edu Mon Jan  3 14:27:22 2005
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Date: Mon, 3 Jan 2005 13:22:50 -0700 (MST)
From: Vladik Kreinovich <vladik [at] cs [dot] utep.edu>
Reply-To: Vladik Kreinovich <vladik [at] cs [dot] utep.edu>
Subject: Re: quadratic equation with interval coefficients
To: Arnold.Neumaier [at] univie [dot] ac.at, reliable_computing [at] interval [dot] louisiana.edu,
        rbk [at] louisiana [dot] edu
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It is described in the paper E. R. Hansen and G. W. Walster, Sharp bounds on 
interval polynomial roots, Reliable Computing, 2002, Vol. 8, No. 2, pp. 
115-122. 

This is how it works for the example that was sent to the list:

From vladik Mon Dec 27 22:52:10 2004
Date: Mon, 27 Dec 2004 22:52:10 -0700 (MST)
From: Vladik Kreinovich <vladik [at] cs [dot] utep.edu>
Subject: Re: requing solution
To: mailtomallik [at] yahoo [dot] com
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Dear Collegue, 

If I understand you correctly, you want to find the range of all the values x 
for which ax^2+bx+c=0 for some a in [1,2], b in [2,3], and c in [3,4]. 

For any given x, this condition is equivalent to the fact that 0 belongs to the 
range of ax^2+bx+c when a in [1,2], b in [2,3], and c in [3,4]. Since the 
dependence of ax^2+bx+c on a, b, anc is monotonic, this range can be explicitly 
computed.

For x>=0, the range is [x^2+2x+3,2x^2+3x+4]. 

For each of these quadratic bounds, we can explicitly find the roots and thus, 
describe the ranges where this bound is negative and positive. Roots of an 
interval polynomial correspond to the case when the lower bound is non-positive 
and the upper bound is non-negative. For these particular bounds, both are 
always positive for x>=0, so there are no positive roots of the interval 
polynomial. 

For x<0, the range is [x^2+3x+2,2x^2+2x+4]. Same thing. The first function has 
roots at x=-2 and x=-1, so it is non-positive for -2<= x<= -1. The second is 
always non-negative. So, the set of possible roots of the original interval 
polynomial are [-2,-1]. 


Vladik




> X-Sender: rbk5287 [at] pop [dot] louisiana.edu
> Mime-Version: 1.0
> Date: Mon, 03 Jan 2005 13:46:52 -0600
> To: Arnold Neumaier <Arnold.Neumaier [at] univie [dot] ac.at>, interval 
<reliable_computing [at] interval [dot] louisiana.edu>
> From: "R. Baker Kearfott" <rbk [at] louisiana [dot] edu>
> Subject: Re: quadratic equation with interval coefficients
> 
> Arnold,
> 
> I looked at it some years ago, when I was programming the elementary
> functions for GlobSol's constraint propagation.  I even wrote a routine
> for it.  However, I never published it, and it is not contained
> in the distribution version of GlobSol.  Furthermore, I don't think
> I proved that it was the tightest possible machine enclosure for the
> solution set.  Thus, if I'm the only one who did it, it's "fair game"
> for you to write it up :-)
> 
> Best regards,
> 
> Baker
> 
> At 08:08 PM 1/3/2005 +0100, Arnold Neumaier wrote:
> >Is there a complete discussion in the literature of the solution
> >of a single quadratic equation in one variable,
> >with interval coefficients? Perhaps even an implementation
> >that computes a correctly rounded enclosure for the solution set?
> >
> >I worked out a solution myself, but this problem is so natural
> >that it was probably treated before.
> >
> >
> >Arnold Neumaier
> >
> >
> >
> 
> ---------------------------------------------------------------
> R. Baker Kearfott,    rbk [at] louisiana [dot] edu   (337) 482-5346 (fax)
> (337) 482-5270 (work)                     (337) 993-1827 (home)
> URL: http://interval.louisiana.edu/kearfott.html
> Department of Mathematics, University of Louisiana at Lafayette
> (Room 217 Maxim D. Doucet Hall, 1403 Johnston Street)
> Box 4-1010, Lafayette, LA 70504-1010, USA
> ---------------------------------------------------------------


From owner-reliable_computing [at] interval [dot] louisiana.edu Mon Jan  3 15:40:53 2005
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Date: Mon, 03 Jan 2005 22:38:53 +0100
From: Arnold Neumaier <Arnold.Neumaier [at] univie [dot] ac.at>
Organization: University of Vienna
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Subject: Re: quadratic equation with interval coefficients
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Vladik Kreinovich wrote:
> It is described in the paper E. R. Hansen and G. W. Walster, Sharp bounds on 
> interval polynomial roots, Reliable Computing, 2002, Vol. 8, No. 2, pp. 
> 115-122. 



Thanks. I hope they didn't patent it!


Arnold


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Date: Mon, 3 Jan 2005 20:08:12 -0700 (MST)
From: Vladik Kreinovich <vladik [at] cs [dot] utep.edu>
Reply-To: Vladik Kreinovich <vladik [at] cs [dot] utep.edu>
Subject: Re: quadratic equation with interval coefficients
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FYI. Vladik

------------- Begin Forwarded Message -------------
From: "G. William Walster" <Bill.Walster [at] sun [dot] com>

...

the algorithm is also contained in Chapter 8 of
the second edition of "Global Optimization Using Interval Analysis."

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



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Date: Mon, 03 Jan 2005 21:35:59 -0600
Subject: Re: quadratic equation with interval coefficients
From: George Corliss <George.Corliss [at] marquette [dot] edu>
To: Arnold Neumaier <Arnold.Neumaier [at] univie [dot] ac.at>,
        Vladik Kreinovich <vladik [at] cs [dot] utep.edu>
CC: <reliable_computing [at] interval [dot] louisiana.edu>, <rbk [at] louisiana [dot] edu>
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Arnold and all,

>> It is described in the paper E. R. Hansen and G. W. Walster, Sharp bounds on
>> interval polynomial roots, Reliable Computing, 2002, Vol. 8, No. 2, pp.
>> 115-122. 
> 
> Thanks. I hope they didn't patent it!

They have applied for one.  Patent application number: 20030055857
Method and apparatus for computing roots of a polynomial equation with
interval coefficients

You may view the application at
http://appft1.uspto.gov/netacgi/nph-Parser?Sect1=PTO2&Sect2=HITOFF&u=%2Fneta
html%2FPTO%2Fsearch-adv.html&r=16&p=1&f=G&l=50&d=PG01&S1=walster.IN.&OS=IN/w
alster&RS=IN/walster


Of the applications Bill has submitted, 8 patents have been issued.  See
http://patft.uspto.gov/netacgi/nph-Parser?Sect1=PTO2&Sect2=HITOFF&u=%2Fnetah
tml%2Fsearch-adv.htm&r=0&p=1&f=S&l=50&Query=IN%2Fwalster&d=ptxt

Dr. George F. Corliss
Electrical and Computer Engineering
Marquette University
PO Box 1881 
1515 W. Wisconsin Ave.
Milwaukee WI 53201-1881 USA
414-288-6599; Fax: 288-5579; Dept. 288-6280
George.Corliss [at] Marquette [dot] edu



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From: "Andrzej Pownuk" <Andrzej.Pownuk [at] polsl [dot] pl>
To: <reliable_computing [at] interval [dot] louisiana.edu>
Subject: RE: quadratic equation with interval coefficients
Date: Tue, 4 Jan 2005 08:11:27 +0100
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Dear George,

According to my knowledge,
these patents are not valid (at least) in Europe.

Art. 52 (EUROPEAN PATENT CONVENTION)
says that:

(a) discoveries, scientific theories and mathematical methods;
(b) aesthetic creations;
(c) schemes, rules and methods for performing mental acts, playing games or
doing business, and programs for computers;
(d) presentations of information.

cannot be patentable
http://www.european-patent-office.org/legal/epc/e/ar52.html

This law is still valid.
http://www.gnu.org/thankpoland.html

Then at this moment
these patents are problems
for people living in the USA.

I am not a lawyer.
Correct me if I am wrong.

Regards,

Andrzej Pownuk

---------------------------------------------
Ph.D., research associate at:
Chair of Theoretical Mechanics
Faculty of Civil Engineering
Silesian University of Technology
ul. Krzywoustego 7
44-100  Gliwice, Poland

Tel/fax: 0048 32 2371542
Mobile:  0048 606 550147

URL:    http://zeus.polsl.gliwice.pl/~pownuk
E-mail: Andrzej.Pownuk [at] polsl [dot] pl
---------------------------------------------


> Arnold and all,
>
> >> It is described in the paper E. R. Hansen and G. W. Walster, Sharp
> bounds on
> >> interval polynomial roots, Reliable Computing, 2002, Vol. 8, No. 2, pp.
> >> 115-122.
> >
> > Thanks. I hope they didn't patent it!
>
> They have applied for one.  Patent application number: 20030055857
> Method and apparatus for computing roots of a polynomial equation with
> interval coefficients
>
> You may view the application at
> http://appft1.uspto.gov/netacgi/nph-
> Parser?Sect1=PTO2&Sect2=HITOFF&u=%2Fneta
> html%2FPTO%2Fsearch-
> adv.html&r=16&p=1&f=G&l=50&d=PG01&S1=walster.IN.&OS=IN/w
> alster&RS=IN/walster
>
>
> Of the applications Bill has submitted, 8 patents have been issued.  See
> http://patft.uspto.gov/netacgi/nph-
> Parser?Sect1=PTO2&Sect2=HITOFF&u=%2Fnetah
> tml%2Fsearch-adv.htm&r=0&p=1&f=S&l=50&Query=IN%2Fwalster&d=ptxt
>
> Dr. George F. Corliss
> Electrical and Computer Engineering
> Marquette University
> PO Box 1881
> 1515 W. Wisconsin Ave.
> Milwaukee WI 53201-1881 USA
> 414-288-6599; Fax: 288-5579; Dept. 288-6280
> George.Corliss [at] Marquette [dot] edu
>




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Subject: Re: quadratic equation with interval coefficients
From: Stefan Ratschan <stefan.ratschan@mpi-sb.mpg.de>
To: reliable_computing [at] interval [dot] louisiana.edu
In-Reply-To: <200501032022.j03KMoN09441 [at] cs [dot] utep.edu>
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On Mon, 2005-01-03 at 21:22, Vladik Kreinovich wrote:
> It is described in the paper E. R. Hansen and G. W. Walster, Sharp bounds on 
> interval polynomial roots, Reliable Computing, 2002, Vol. 8, No. 2, pp. 
> 115-122. 

To me it seems that this method is a reinvention of what already appears in Section 7 of

@Article{Hong:94b,
  author = 	 "Hoon Hong and Volker Stahl",
  title = 	 "Safe Starting Regions by Fixed Points and Tightening",
  journal =	 "Computing",
  year =	 1994,
  volume =	 53,
  pages =	 "323--335",
}

... this might also be relevant for the patent application ...

Stefan Ratschan



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Date: Tue, 04 Jan 2005 08:33:01 -0600
To: "r.   corless" <rcorless [at] uwo [dot] ca>,
        George Corliss <George.Corliss [at] marquette [dot] edu>
From: "R. Baker Kearfott" <rbk [at] louisiana [dot] edu>
Subject: Re: quadratic equation with interval coefficients
Cc: Arnold Neumaier <Arnold.Neumaier [at] univie [dot] ac.at>,
        Vladik Kreinovich <vladik [at] cs [dot] utep.edu>,
        reliable_computing [at] interval [dot] louisiana.edu
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Hmmm.... let me play devil's advocate:
It brings up an interesting question of standards for
original work for publication.  Presumably computing the
roots of interval polynomials was previously
published as a research publication.  An algorithm or procedure
might possibly be derivable by any of us;  nonetheless, a
clear, readable written record of it could still be valuable
to all, and writing it down in a clear and understandable
way may take some effort.  So, although it may not be surprising
and although any of a number of us might have been able to derive it
in a fairly straightforward way, might the written record still be
valuable to the community as a whole?

Best regards,

Baker

At 09:12 AM 1/4/2005 -0500, r.   corless wrote:
>
>Interesting.  I was speaking to an intellectual-property lawyer last week
>(he will marry my sister-in-law in July, this is just family, I am not
>applying for patents myself :-) and he pointed out to me that the
>"hard" question for patents is not "originality" but "obviousness".  A
>patent will not be granted for something that is "obvious" to someone
>"skilled in the art".
>
>I understand that phrase "skilled in the art" to mean an ordinarily
>competent interval practitioner in this case, and not to someone elite
>(and it is clear that there are elite practitioners on this mailing list,
>to be sure).
>
>So, here's the question for the community: Is the solution of
>interval polynomials "obvious", given competent knowledge of interval
>arithmetic?  
>
>Would you set it on an exam?
>
>-r
>
>
>
>On Mon, 3 Jan 2005, George Corliss wrote:
>
>> Arnold and all,
>> 
>> >> It is described in the paper E. R. Hansen and G. W. Walster, Sharp
bounds on
>> >> interval polynomial roots, Reliable Computing, 2002, Vol. 8, No. 2, pp.
>> >> 115-122. 
>> > 
>> > Thanks. I hope they didn't patent it!
>> 
>> They have applied for one.  Patent application number: 20030055857
>> Method and apparatus for computing roots of a polynomial equation with
>> interval coefficients
>> 
>> You may view the application at
>> http://appft1.uspto.gov/netacgi/nph-Parser?Sect1=PTO2&Sect2=HITOFF&u=%2Fneta
>> html%2FPTO%2Fsearch-adv.html&r=16&p=1&f=G&l=50&d=PG01&S1=walster.IN.&OS=IN/w
>> alster&RS=IN/walster
>> 
>> 
>> Of the applications Bill has submitted, 8 patents have been issued.  See
>> http://patft.uspto.gov/netacgi/nph-Parser?Sect1=PTO2&Sect2=HITOFF&u=%2Fnetah
>> tml%2Fsearch-adv.htm&r=0&p=1&f=S&l=50&Query=IN%2Fwalster&d=ptxt
>> 
>> Dr. George F. Corliss
>> Electrical and Computer Engineering
>> Marquette University
>> PO Box 1881 
>> 1515 W. Wisconsin Ave.
>> Milwaukee WI 53201-1881 USA
>> 414-288-6599; Fax: 288-5579; Dept. 288-6280
>> George.Corliss [at] Marquette [dot] edu
>> 
>> 
>> 
>
>

---------------------------------------------------------------
R. Baker Kearfott,    rbk [at] louisiana [dot] edu   (337) 482-5346 (fax)
(337) 482-5270 (work)                     (337) 993-1827 (home)
URL: http://interval.louisiana.edu/kearfott.html
Department of Mathematics, University of Louisiana at Lafayette
(Room 217 Maxim D. Doucet Hall, 1403 Johnston Street)
Box 4-1010, Lafayette, LA 70504-1010, USA
---------------------------------------------------------------


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Date: Mon, 03 Jan 2005 11:35:29 -0800
From: James Demmel <demmel [at] eecs [dot] berkeley.edu>
Subject: Re: quadratic equation with interval coefficients
To: Arnold Neumaier <Arnold.Neumaier [at] univie [dot] ac.at>
Cc: interval <reliable_computing [at] interval [dot] louisiana.edu>
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Yves Nievergelt, ynievergelt [at] ewu [dot] edu, has a paper on a related result.
Jim Demmel

Arnold Neumaier wrote:

> Is there a complete discussion in the literature of the solution
> of a single quadratic equation in one variable,
> with interval coefficients? Perhaps even an implementation
> that computes a correctly rounded enclosure for the solution set?
>
> I worked out a solution myself, but this problem is so natural
> that it was probably treated before.
>
> Arnold Neumaier


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From: James Demmel <demmel [at] eecs [dot] berkeley.edu>
Subject: Re: quadratic equation with interval coefficients
To: Arnold Neumaier <Arnold.Neumaier [at] univie [dot] ac.at>
Cc: interval <reliable_computing [at] interval [dot] louisiana.edu>
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Yves Nievergelt, ynievergelt [at] ewu [dot] edu, has a paper on a related result.
Jim Demmel

Arnold Neumaier wrote:

> Is there a complete discussion in the literature of the solution
> of a single quadratic equation in one variable,
> with interval coefficients? Perhaps even an implementation
> that computes a correctly rounded enclosure for the solution set?
>
> I worked out a solution myself, but this problem is so natural
> that it was probably treated before.
>
> Arnold Neumaier


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Date: Tue, 04 Jan 2005 09:15:25 -0600
To: reliable_computing [at] interval [dot] louisiana.edu
From: "R. Baker Kearfott" <rbk [at] louisiana [dot] edu>
Subject: Re: quadratic equation with interval coefficients
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Hmmm.... let me play devil's advocate:
It brings up an interesting question of standards for
original work for publication.  Presumably computing the
roots of interval polynomials was previously
published as a research publication.  An algorithm or procedure
might possibly be derivable by any of us;  nonetheless, a
clear, readable written record of it could still be valuable
to all, and writing it down in a clear and understandable
way may take some effort.  So, although it may not be surprising
and although any of a number of us might have been able to derive it
in a fairly straightforward way, might the written record still be
valuable to the community as a whole?

Best regards,

Baker

>At 09:12 AM 1/4/2005 -0500, r.   corless wrote:
>>
>>Interesting.  I was speaking to an intellectual-property lawyer last week
>>(he will marry my sister-in-law in July, this is just family, I am not
>>applying for patents myself :-) and he pointed out to me that the
>>"hard" question for patents is not "originality" but "obviousness".  A
>>patent will not be granted for something that is "obvious" to someone
>>"skilled in the art".
>>
>>I understand that phrase "skilled in the art" to mean an ordinarily
>>competent interval practitioner in this case, and not to someone elite
>>(and it is clear that there are elite practitioners on this mailing list,
>>to be sure).
>>
>>So, here's the question for the community: Is the solution of
>>interval polynomials "obvious", given competent knowledge of interval
>>arithmetic?  
>>
>>Would you set it on an exam?
>>
>>-r
>>
>>
>>
>>On Mon, 3 Jan 2005, George Corliss wrote:
>>
>>> Arnold and all,
>>> 
>>> >> It is described in the paper E. R. Hansen and G. W. Walster, Sharp
bounds on
>>> >> interval polynomial roots, Reliable Computing, 2002, Vol. 8, No. 2, pp.
>>> >> 115-122. 
>>> > 
>>> > Thanks. I hope they didn't patent it!
>>> 
>>> They have applied for one.  Patent application number: 20030055857
>>> Method and apparatus for computing roots of a polynomial equation with
>>> interval coefficients
>>> 
>>> You may view the application at
>>> http://appft1.uspto.gov/netacgi/nph-Parser?Sect1=PTO2&Sect2=HITOFF&u=%2Fneta
>>> html%2FPTO%2Fsearch-adv.html&r=16&p=1&f=G&l=50&d=PG01&S1=walster.IN.&OS=IN/w
>>> alster&RS=IN/walster
>>> 
>>> 
>>> Of the applications Bill has submitted, 8 patents have been issued.  See
>>> http://patft.uspto.gov/netacgi/nph-Parser?Sect1=PTO2&Sect2=HITOFF&u=%2Fnetah
>>> tml%2Fsearch-adv.htm&r=0&p=1&f=S&l=50&Query=IN%2Fwalster&d=ptxt
>>> 
>>> Dr. George F. Corliss
>>> Electrical and Computer Engineering
>>> Marquette University
>>> PO Box 1881 
>>> 1515 W. Wisconsin Ave.
>>> Milwaukee WI 53201-1881 USA
>>> 414-288-6599; Fax: 288-5579; Dept. 288-6280
>>> George.Corliss [at] Marquette [dot] edu
>>> 
>>> 
>>> 
>>
>>
>

---------------------------------------------------------------
R. Baker Kearfott,    rbk [at] louisiana [dot] edu   (337) 482-5346 (fax)
(337) 482-5270 (work)                     (337) 993-1827 (home)
URL: http://interval.louisiana.edu/kearfott.html
Department of Mathematics, University of Louisiana at Lafayette
(Room 217 Maxim D. Doucet Hall, 1403 Johnston Street)
Box 4-1010, Lafayette, LA 70504-1010, USA
---------------------------------------------------------------


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Date: Mon, 03 Jan 2005 11:35:29 -0800
From: James Demmel <demmel [at] eecs [dot] berkeley.edu>
Subject: Re: quadratic equation with interval coefficients
To: Arnold Neumaier <Arnold.Neumaier [at] univie [dot] ac.at>
Cc: interval <reliable_computing [at] interval [dot] louisiana.edu>
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Yves Nievergelt, ynievergelt [at] ewu [dot] edu, has a paper on a related result.
Jim Demmel

Arnold Neumaier wrote:

> Is there a complete discussion in the literature of the solution
> of a single quadratic equation in one variable,
> with interval coefficients? Perhaps even an implementation
> that computes a correctly rounded enclosure for the solution set?
>
> I worked out a solution myself, but this problem is so natural
> that it was probably treated before.
>
> Arnold Neumaier


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   Linear Algebra and its Applications
  Volume 396,  Pages 1-392 (1 February 2005)
 
 
 1. Editorial board
Pages ii-iii
 
 2. On the solution of Steinâ^À^Ùs equation and Fisherâ^À^Ùs information
matrix of an ARMAX process
Pages 1-34
André Klein and Peter Spreij 
 3. Nilpotent linear transformations and the solvability of
power-associative nilalgebras
Pages 35-53
Ivan Correa, Irvin Roy Hentzel, Pedro Pablo Julca and Luiz Antonio
Peresi 
 4. A structure-preserving doubling algorithm for continuous-time
algebraic Riccati equations
Pages 55-80
E.K.-W. Chu, H.-Y. Fan and W.-W. Lin 
 5. On generalized H -matrices
Pages 81-90
Ting-Zhu Huang, Shu-Qian Shen and Hou-Biao Li 
 6. Linear/additive preservers of rank 2 on spaces of alternate matrices
over fields
Pages 91-102
Xian Zhang 
 7. On the sensitivity of Lanczos recursions to the spectrum
Pages 103-125
Vladimir Druskin, Liliana Borcea and Leonid Knizhnerman 
 8. Geometry of skew-Hermitian matrices
Pages 127-157
Li-Ping Huang and Zhe-Xian Wan 
 9. Linear maps preserving Drazin inverses of matrices over fields
Pages 159-173
Changjiang Bu 
 10. Some functions reversing the order of positive operators
Pages 175-187
Josip PeÄ^ÍariÄ^Ç and Jadranka MiÄ^ÇiÄ^Ç 
 11. Symmetric triality relations and structurable algebras
Pages 189-222
Susumu Okubo 
 12. On the comparison of some realizability criteria for the real
nonnegative inverse eigenvalue problem
Pages 223-241
Ricardo Soto, Alberto Borobia and Julio Moro 
 13. The exponent and circumdiameter of primitive digraphs
Pages 243-258
L.F. Dame, D.D. Olesky and P. van den Driessche 
 14. A Perron Theorem for positive componentwise bilinear maps
Pages 259-272
Joseph E. Carroll, Timothy Lauck and Roland H. Lamberson 
 15. On sensitivity of eigenvalues and eigendecompositions of matrices
Pages 273-301
R. Alam and S. Bora 
 16. Combinatorial designs with two singular values II. Partial geometric
designs
Pages 303-316
E.R. van Dam and E. Spence 
 17. A pathway to matrix-variate gamma and normal densities
Pages 317-328
A.M. Mathai 
 18. Convex invertible sets and matrix sign function
Pages 329-352
Izchak Lewkowicz, Leiba Rodman and Elad J. Yarkoni 
 19. Convergence of logarithmic trace inequalities via generalized
Lieâ^À^ÓTrotter formulae
Pages 353-372
Takayuki Furuta 
 20. The geometric mean decomposition
Pages 373-384
Yi Jiang, William W. Hager and Jian Li 
 21. The structure of alternating-Hamiltonian matrices
Pages 385-390
William C. Waterhouse 
 22. Author index
Pages 391-392
 
Subscribers to LAA or ScenceDirect may access every paper of LAA
published since volume 1 (1968) and also papers now in press,
see www.sciencedirect.com . 





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Date: Wed, 05 Jan 2005 12:51:37 +0100
From: Arnold Neumaier <Arnold.Neumaier [at] univie [dot] ac.at>
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George Corliss wrote:
> 
>>>It is described in the paper E. R. Hansen and G. W. Walster, Sharp bounds on
>>>interval polynomial roots, Reliable Computing, 2002, Vol. 8, No. 2, pp.
>>>115-122. 
>>
>>Thanks. I hope they didn't patent it!
> 
> They have applied for one.  Patent application number: 20030055857
> Method and apparatus for computing roots of a polynomial equation with
> interval coefficients
> 
> You may view the application at
> http://appft1.uspto.gov/netacgi/nph-Parser?Sect1=PTO2&Sect2=HITOFF&u=%2Fneta
> html%2FPTO%2Fsearch-adv.html&r=16&p=1&f=G&l=50&d=PG01&S1=walster.IN.&OS=IN/w
> alster&RS=IN/walster


Ultimately their patents will just mean that, for fear of possible
law suits, nobody apart from SUN will be using the results of their
research, but instead use innocent replacements. Using the latter
may even be an advantage, as in the present case.


The algorithm they try to patent sometimes gives a gross
overestimation of the zero set. For example, in BACKGROUND
[0018],[0019],[0064],[0065], one gets [-inf,inf] if A=B=0 notin C,
although the solution set is empty.
It seems that it is implicitly assumed that A is nonzero.

Their formulas also lead to overflow when B.sup^2 > realmax,
with very poor results if A=C=B/2 and B is huge.


In order to prevent that there will be a patent on similar
elementary tasks, I post a solution for the constraint propagation
on univariate quadratic expressions, with or without interval
coefficents. dvi,ps,and pdf versions can be downloaded from
the address given below; the latex original is attached here
to document that the paper was created today.



Arnold Neumaier


=======================================================================

                         A. Neumaier

Constraint propagation for univariate quadratic constraints

                 Manuscript (January 5, 2005)

      http://www.mat.univie.ac.at/~neum/papers.html#cpquad

                           Abstract

We present formulas for rigorous constraint propagation of
quadratic equality or inequality constraints involving a single
nonlinear variable. Since the analysis is very elementary,
probably everything in here was known for a long time.
The present approach, based on directed rounding only, provide
efficient alternatives to the procedures discussed by
     E. R. Hansen and G. W. Walster,
     Sharp bounds on interval polynomial roots,
     Reliable Computing 8 (2002), 115--122.
(who only treat the solution of a quadratic equation with interval
coefficients, and treat incorrectly the case where the coefficient
of the quadratic term contains numbers of both signs),
which employ interval arithmetic.

In view of pending patent applications by these authors,
who by these activities threaten to curb the freedom
of research on interval methods, the following is explicitly stated:

Various modifications to the methods described will be readily apparent
to those skilled in the art, and the general principles defined herein
may be applied to such modifications without departing from
the spirit and scope of the present methods. Thus, the present
methods are not intended to be limited to the formulas shown, but is
to be accorded the widest scope consistent with the principles and
features disclosed herein.


--------------090408020206050502070202
Content-Type: application/x-tex;
 name="cpquad.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline;
 filename="cpquad.tex"

\documentclass[12pt]{article}

\def\a {{\bf a}}
\def\b {{\bf b}}
\def\c {{\bf c}}
\def\x {{\bf x}}
\def\ol{\overline}
\def\ul{\underline}

\parindent0pt
\parskip 2ex plus 1pt minus 1pt


\begin{document}


\begin{center}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\LARGE \bf Constraint propagation for } \\

{\LARGE \bf univariate quadratic constraints} \\

\vspace{1cm}

\centerline{\sl {\large \bf Arnold Neumaier}}

\vspace{0.5cm}

\centerline{\sl Institut f\"ur Mathematik, Universit\"at Wien}
\centerline{\sl Strudlhofgasse 4, A-1090 Wien, Austria}
\centerline{\sl email: Arnold.Neumaier [at] univie [dot] ac.at}
\centerline{\sl WWW: http://www.mat.univie.ac.at/$\sim$neum/}


\end{center}

\vspace{0.5cm}

\hfill January 5, 2004


\vspace{0.5cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf Abstract.} 

We present formulas for rigorous constraint propagation of
quadratic equality or inequality constraints involving a single
nonlinear variable. Since the analysis is very elementary, 
probably everything in here has been known for a long time.
The present approach, based on directed rounding only, provides 
efficient alternatives to the procedures discussed by 
{\sc Hansen \& Walster} \cite{HanW} (who only treat the solution
of a quadratic equation with interval coefficients, and treat 
incorrectly the case where the coefficient of the quadratic term 
contains numbers of both signs), which employ interval arithmetic.

In view of pending patent applications by these authors, 
who by these activities threaten to curb the freedom 
of research on interval methods, the following is explicitly stated:

Various modifications to the methods described will be readily apparent 
to those skilled in the art, and the general principles defined herein 
may be applied to such modifications without departing from
the spirit and scope of the present methods. Thus, the present
methods are not intended to be limited to the formulas shown, but are
to be accorded the widest scope consistent with the principles and
features disclosed herein.
 
Notation is as in my book {\sc Neumaier} \cite{Neu.int}.
 

\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Bounds for quadratic expressions} 

To find a rigorous upper bound on  
\[
u=\sup\,\{ax^2+bx \mid x \in \x\},
\]
we note that
\[
u=\max\,\{\ul x(a\ul x+b),\ol x(a\ol x+b)\},
\]
except in case that $ax^2+bx$ attains its global maximum in the
interior of $\x$.
This is the case iff $a<0$ and $t=-b/(2a)$ is in the interior of $\x$,
in which case $u=b^2/(-4a)$, attained at $t$. 

If $\ul x\ge0$, we get a rigorous upper bound in finite precision
arithmetic by computing with upward rounding as follows 
({\tt xl} = $\ul x$, {\tt xu} = $\ol x$):
\begin{verbatim}
  roundup;
  u=max(xl*(a*xl+b),xu*(a*xu+b));
  s=b/2; t=s/(-a);
  if t>xl, r=(-2*a)*xu;
    if r>b, u=max(u,s*t); end;
  end;
\end{verbatim}
With some extra analysis, it could be determined in most cases which
of the three cases is the worst case; however, if the unconstrained 
maximum of the quadratic is very close to a bound (or to both bounds),
two (or three) of the cases might apply due to uncertainty caused
by rounding errors.
 
\bigskip
Finding a rigorous enclosure for the interval  
\[
\c=\sup\,\{ax^2+bx \mid x \in \x,~ a \in \a,~ b\in \b\}
\]
can be reduced to the above for $\ul x \ge 0$, using
\[
\ol c = \sup\, \{\ol ax^2+\ol bx \mid x \in \x\},~~~
\ul c = -\sup\, \{-\ul ax^2-\ul bx \mid x \in \x\}.
\]
The case $\ol x\le0$ can be reduced to this by changing the sign of $x$,
and the general case by splitting $\x$ at zero if necessary.

\bigskip
Essentially the same analysis holds for rigorous upper bounds on
\[
u=\sup\Big\{~\sum_{i=1}^{n} a_i x^i ~\Big|~ x \in \x~\Big\}
\]
and for rigorous enclosures of   
\[
\c=\sup\Big\{~\sum_{i=1}^{n} a_i x^i ~\Big|~ x \in \x,~ a\in\a~\Big\},
\]
except that finding the interior extrema is more involved.
It can be done with closed formulas for $n\le 5$ (though already 
$n=4$ is quite cumbersome and not recommended), and in general 
(recommended for $n>3$) using a root enclosure algorithm for the 
derivative, such as that in {\sc Neumaier} \cite{Neu.roots}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Solving quadratic constraints} 

To find the set
\[
X=\{x \ge 0 \mid ax^2+2bx \ge c\},
\]
we proceed as follows. If $a=0$, the constraint is in fact linear,
and we have 
\[
X= \cases{
   \emptyset         & if $c>0$, $b\le 0$,\cr
   [0.5c/b,\infty]      & if $c>0$, $b>0$, \cr
   [0,0.5c/b]           & if $c\le 0$, $b<0$, \cr
   [0,\infty]        & if $c\le 0$, $b\ge 0$,
         }
\]
which can be nested such that only two compares are needed in any
particular case. For a rigorous enclosure in finite precision 
arithmetic, rounding must be downwards in the second case, 
and upwards in the third case.

If $a \ne 0$, the behavior is governed by the zeros
of the quadratic equation $ax^2+2bx-c=0$, given by
\[
t_1=\frac{-b-\sqrt{\Delta}}{a}=\frac{c}{b-\sqrt{\Delta}},~~~~~~~~~
t_2=\frac{-b+\sqrt{\Delta}}{a}=\frac{c}{b+\sqrt{\Delta}},
\]
where $\Delta:=b^2+ac$. If $\Delta\ge 0$, the zeros are real and 
the nonnegative zeros determine 
\[
X=\cases{
[0,\infty]\setminus~]t_1,t_2[ & if $a>0$,\cr
[0,\infty]\cap\,    [t_2,t_1] & if $a<0$.
        }
\]
Depending on the signs of $a$, $b$ and $c$ we find
\[
X=\cases{
  \emptyset                   & if $a< 0$, $b\le 0$, $c> 0$,\cr
  [0,-(c/z)]                  & if $a< 0$, $b\le 0$, $c\le 0$,\cr
  [0,z/(-a)]                  & if $a< 0$, $b\ge 0$, $c\le 0$,\cr
  [-((-c)/z),z/(-a)]          & if $a< 0$, $b\ge 0$, $c> 0$,\cr
  [0,-(c/z)]\cup[z/a,\infty]  & if $a> 0$, $b\le 0$, $c\le 0$,\cr
  [z/a,\infty]                & if $a> 0$, $b\le 0$, $c> 0$,\cr  
  [-((-c)/z),\infty]          & if $a> 0$, $b\ge 0$, $c> 0$,\cr
  [0,\infty]                  & if $a> 0$, $b\ge 0$, $c\le 0$,
       }
\]
where 
\[
z=|b|+\sqrt{\Delta}. 
\]
These formulas are numerically stable, and can be nested such that 
only three compares are needed in any particular case. 
(There are avoidable overflow problems for huge $|b|$, which can be 
cured by using for huge $|b|$ instead of $\sqrt{b^2+ac}$ the 
formula $|b|\sqrt{1+ac/b^2}$.)

Rigorous results in the presence of rounding errors are obtained if
lower bounds are rounded downwards, and upper bounds are rounded 
upwards. With the bracketing as given, this happens if in 
cases 2,5 and 6 all computations (including those of 
$\Delta=\sqrt{b^2+ac}$ and $z=|b|+\sqrt{\Delta}$) 
are done with rounding downwards, and in the other cases with rounding 
upwards. (However, this does {\bf not} hold for the version guarded 
against overflow, where further care is needed for the directed 
rounding of $\sqrt{\Delta}=|b|\sqrt{1+ac/b^2}$.)

If (the exact) $\Delta$ is negative, there is no real solution,
and $X$ is empty if $c>0$ and $[0,\infty]$ otherwise. 
The case when the sign of $\Delta$ cannot be determined due to rounding
errors needs special consideration. In the first and last case, 
the conclusion holds independent of the sign of $\Delta$, so that the 
latter need only be computed for cases 2--7. In the cases 2, 3, 6, and 7
we have $ac\ge 0$, so that $\Delta\ge 0$ automatically. 
This leaves cases 4 and 5. Now it is easily checked that with the 
recommended rounding and, in place of cases 4 and 5,
\[
X=\cases{
  \emptyset           & if $a< 0$, $b\ge 0$, $c> 0$, $\Delta< 0$,\cr
  [-((-c)/z),z/(-a)]  & if $a< 0$, $b\ge 0$, $c> 0$, $\Delta\ge 0$,\cr
  [0,-(c/z)]\cup[z/a,\infty]  
                      & if $a> 0$, $b\le 0$, $c\le 0$, $\Delta\ge 0$,\cr
  [0,\infty]          & if $a> 0$, $b\le 0$, $c\le 0$, $\Delta< 0$,
       }
\]
a rigorous enclosure is computed in all cases.

\bigskip
Finding the set
\[
X'=\{x \ge 0 \mid ax^2+2bx \in \c \mbox{ for some } a\in\a, b\in\b\}
\]
can be reduced to the previous task since
\[
X'=\{x \ge 0 \mid \ul a x^2+2\ul b x \le \ol c\} \cap
\{x \ge 0 \mid \ul a x^2+2\ul b x \le \ol c\}.
\]
The sets
\[
X''=\{x \in \x_0 \mid ax^2+2bx \ge c\}
\]
and
\[
X'''=\{x\in\x_0 \mid ax^2+2bx \in\c \mbox{ for some } a\in\a, b\in\b\}
\]
can be obtained by intersecting the result of the above tasks with 
$\x_0$ if $\ul x_0\ge0$, by negating $x$, $\x_0$, and $\b$ if 
$\ol x_0\le 0$, and by splitting $\x_0$ at zero if $0$ is in the 
interior of $\x_0$. By modifying the code appropriately, one can 
also avoid computing roots which can be seen to lie outside $\x_0$.

\bigskip
With minor changes, these formulas also apply for strict inequalities 
and interior enclosures. Also, it is clear that polynomial inequalities
and inclusions of interval polynomials can be solved by a 
straightforward adaptation of the above arguments.


\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}

\bibitem{HanW} E. R. Hansen and G. W. Walster, 
Sharp bounds on interval polynomial roots, 
Reliable Computing 8 (2002), 115--122. 

\bibitem{Neu.int} A. Neumaier, 
Interval Methods for Systems of Equations,
Cambridge Univ. Press, Cambridge 1990.

\bibitem{Neu.roots} A. Neumaier, 
Enclosing clusters of zeros of polynomials, 
J. Comput. Appl. Math. 156 (2003), 389--401. 


\end{thebibliography}
\end {document}




--------------090408020206050502070202--


From owner-reliable_computing [at] interval [dot] louisiana.edu Wed Jan  5 06:03:45 2005
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Date: Wed, 05 Jan 2005 13:03:18 +0100
From: Arnold Neumaier <Arnold.Neumaier [at] univie [dot] ac.at>
Organization: University of Vienna
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CC: George Corliss <George.Corliss [at] marquette [dot] edu>,
        Vladik Kreinovich <vladik [at] cs [dot] utep.edu>,
        reliable_computing [at] interval [dot] louisiana.edu, rbk [at] louisiana [dot] edu
Subject: Re: quadratic equation with interval coefficients
References: <Pine.SOL.4.21.0501040902130.29296-100000 [at] panther [dot] uwo.ca>
In-Reply-To: <Pine.SOL.4.21.0501040902130.29296-100000 [at] panther [dot] uwo.ca>
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r. corless wrote:
> Interesting.  I was speaking to an intellectual-property lawyer last week
> (he will marry my sister-in-law in July, this is just family, I am not
> applying for patents myself :-) and he pointed out to me that the
> "hard" question for patents is not "originality" but "obviousness".  A
> patent will not be granted for something that is "obvious" to someone
> "skilled in the art".
> 
> I understand that phrase "skilled in the art" to mean an ordinarily
> competent interval practitioner in this case, and not to someone elite
> (and it is clear that there are elite practitioners on this mailing list,
> to be sure).
> 
> So, here's the question for the community: Is the solution of
> interval polynomials "obvious", given competent knowledge of interval
> arithmetic?  
> 
> Would you set it on an exam?

The only case worked out in detail by Hansen and Walster is that of
quadratics. With some simple hints, a complete solution for
quadratics can be found by diligent students in a few hours,
at the same level of quality as Hansen and Walster's, or even better.

Indeed, they write on p.119 of their paper, ''The simpler algorithm
has probably also been often used; but the authors are unaware of any
publication describing it.'' Thus they admit that the contents is
''readily apparent to practitioners skilled in the art''
(a quote from their patent application; they use it in an opposite
context), and hence not patentable.

What do the law experts say?


Everything is in fact very elementary.
The only difficulty is that there are many cases to consider,
more than an average student would have the guts to consider.
So the students should be told that they need to use monotonicity
arguments and have to consider 16 cases, depending on the signs of
a,b,c,x. See my other mail for a neatly polished solution, with all
bells and whistles (which probably could be expected only by very
good students).

It takes much less work than the procedure of Hansen and Walster
who always compute up to 8 interval enclosures of roots.


Arnold Neumaier



From owner-reliable_computing [at] interval [dot] louisiana.edu Wed Jan  5 06:28:17 2005
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Subject: Re: quadratic equation with interval coefficients
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Arnold Neumaier wrote:

> The present approach, based on directed rounding only, provide
> efficient alternatives to the procedures discussed by
>     E. R. Hansen and G. W. Walster,
>     Sharp bounds on interval polynomial roots,
>     Reliable Computing 8 (2002), 115--122.
> (who only treat the solution of a quadratic equation with interval
> coefficients, and treat incorrectly the case where the coefficient
> of the quadratic term contains numbers of both signs),
> which employ interval arithmetic.

Actually their treatment is correct and my understanding of their
algorithm was at first mistaken; I had forgotten to take this
comment out of the abstract. Sorry.


Arnold Neumaier



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        Arnold Neumaier <Arnold.Neumaier [at] univie [dot] ac.at>
Subject: Re: Costs of floating-point rounding control and predecessor/successor
        computation
References: <CMM.0.92.0.1104415973.beebe [at] psi [dot] math.utah.edu>
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Nelson H. F. Beebe wrote:
> While I cannot comment on the efficiency of Matlab's rounding mode
> access in the Intlab package, I can supply fresh evidence that it
> should have minimal cost.
> 
> On IA-32, rounding mode access is available via the floating-point
> control word, which takes only one instruction to fetch or set.

While I can't comment on the efficiency of rounding mode
access in interpreted languages, I would like to point out that
rounding mode changes usually cause a complete pipeline flush
in modern processors, i.e. the processor waits until all previous
operations have completed.  On a Pentium 4, this is much more
visible than on a Pentium III due to much longer pipelines on
the P4.

The code

   for (k = 1; k <= 100000000; ++k)
     sum += BASE;

takes 0.96 sec on my Pentium III 850 Mhz, compared to 2.3 sec
for 1/100 the work in Nelson Beebe's interpreted language on
his 600 MHz CPU.  (The overhead of the interpreted language
is a factor of 200 in this particular case.)

Adding another addition (here: subtraction)

   for (k = 1; k <= 100000000; ++k) {
     sum += BASE;
     sum -= 0.5;
   }

now takes 1.32 sec on my machine, clearly showing the pipeline
effects: We're doing twice the number of floating-point operations,
but we need a meagre 30% additional time.

Adding a setting of the rounding mode instead of the second addition,
the loop now takes 2.87 sec on my machine, thus the differences
are much larger than those visible in the interpreted language.

My C library's nextafter() function is about 10 times slower
on subnormals than on normals (using glibc 2.3.3).

Jens Maurer


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forwarding; apologies for multiple copies
------------- Begin Forwarded Message -------------

From: Nacim RAMDANI <ramdani@univ-paris12.fr>

Dear colleagues
This mail is an invitation to participate to an invited session I plan to 
organize during the ECC-CDC 2005 conference 
http://www.esi2.us.es/~cdcecc05/, dealing with theory and application of 
interval computations.

Interval analysis has been used since more than a decade now, in both 
applied mathematics and engineering. On the one hand, it is a reliable tool 
for uncertainty propagation and thus is used for computing rigorous error 
bounds when evaluating functions or solving differential equations. On the 
other hand, allied with constraint propagation, interval analysis becomes a 
powerful tool for solving, in a guaranteed way, engineering problems such 
as global optimization, bounded-error estimation or robust control.
The invited session aims at bringing together researchers from the interval 
computations community in order to carry out presentations and discussions 
of recent advances in the field.

If you are interested to participate to this invited session, and I hope 
you will, please let me know as soon as possible.
Please also note that the deadline for submission is March 1st, therefore 
if you intend to participate to this session, please send the full paper in 
the correct format, to ramdani@univ-paris12.fr, no later than february 28th.

Looking forward reading from you soon.
Best Regards
Dr Nacim RAMDANI


I wish you a happy new year 2005.



  Je vous présente mes meilleurs voeux pour la nouvelle année 2005.

  Nacim RAMDANI
  Maître de Conférences

  phone :    +33 (0)1 45 17 18 36
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  email :   ramdani@univ-paris12.fr

  CERTES  - UPRES-EA 3481
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To: martineceberio [at] yahoo [dot] com
From: VIRTUALCONCEPT_PUBLICITY/LIPSI/ESTIA/CCIBPB [at] bayonne [dot] cci.fr

Dear Colleague,
due to Christmas time, a lot of authors wanting to submit an abstract to 
Virtual Concept 2005 have asked us a possibility for extending deadlines.

Deadlines for abstract submissions have been extended to the 4th of 
February. Other dates are still the same.

You can submit a 200 words abstract on our website for:
- an article: http://www.virtualconcept.estia.fr/call4paper.php?logout=1
- an invited session: 
http://www.virtualconcept.estia.fr/call4invited.php?logout=1
- a tutorial: 
http://www.virtualconcept.estia.fr/call4tutorial.php?logout=1
 
It will be an honor for us to welcome you in Virtual Concept 2005 that 
will foresee to gather more than 500 people interested by the use of 
Virtual Reality and Simulation in Industry.
 
Happy Christmas and happy new year.

Publicity and Scientific Committee Chairs of Virtual Concept 2005
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From: "Svetoslav Markov" <smarkov [at] bio [dot] bas.bg>
To: Arnold Neumaier <Arnold.Neumaier [at] univie [dot] ac.at>
Date: Fri, 07 Jan 2005 13:52:13 +0200
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Subject: Re: quadratic equation with interval coefficients
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Dear Arnold,

the solution  of a single quadratic equation in one variable
with interval coefficients has been discussed in detail in
Section 4 of the paper:

  N. S. Dimitrova,  S.  M.  Markov.  Ueber die
intervall-arithmetische Berechnung des Wertebereichs einer Funktion
mit Anwendungen, Freiburger Intervall-Berichte, Univ.  Freiburg, 81/4
(1981), 1--22.

Svetoslav Markov


On 3 Jan 2005 at 20:08, Arnold Neumaier wrote:

> Is there a complete discussion in the literature of the solution
> of a single quadratic equation in one variable,
> with interval coefficients? Perhaps even an implementation
> that computes a correctly rounded enclosure for the solution set?
> 
> I worked out a solution myself, but this problem is so natural
> that it was probably treated before.
> 
> 
> Arnold Neumaier
> 

 

From owner-reliable_computing [at] interval [dot] louisiana.edu Fri Jan  7 09:39:21 2005
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Date: Fri, 7 Jan 2005 08:38:59 -0700 (MST)
From: Vladik Kreinovich <vladik [at] cs [dot] utep.edu>
Reply-To: Vladik Kreinovich <vladik [at] cs [dot] utep.edu>
Subject: IFSA'05: deadline extended to January 17, 2005
To: reliable_computing [at] interval [dot] louisiana.edu, interval [at] cs [dot] utep.edu
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We originally planned two interval-related sessions: Weldon Lodwick is planning 
to organize a session on fuzzy optimization and Chenyi Hu was planning to 
organize a session on intervals and fuzzy in general. It looks like we did not 
have anough people for the second session, so everyone interested in welcome to 
submit to Weldon Lodwick's session. 

Please note that the deadline for conference submissions has been extended to 
January 17, 2005. 

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

>>> "IFSA2005 Secretariat" <ifsa2005 [at] em [dot] tsinghua.edu.cn> ss).

...

Thank you very much in advance!

Best regards,

IFSA2005/Beijing Secretariat
School of Economics and Management 
Tsinghua University, Beijing 100084, China
Phone/Fax: 86-10-62789925
Email: ifsa2005 [at] em [dot] tsinghua.edu.cn 
Webpage : http://ifsa2005.em.tsinghua.edu.cn/ 

>**************************************************************************
>CALL FOR PAPERS
>IFSA2005 World Congress
>July 28-31, 2005, Beijing China 
>http://ifsa2005.em.tsinghua.edu.cn 
>
>The 11th World Congress of International Fuzzy Systems Association
>(IFSA 2005) will be held in Beijing China, July 28-31, 2005. As a
>major bi-annual event of IFSA, the Congress aims at bringing together
>scholars and practitioners from academia and industries to present
the
>latest development in theories and applications of fuzzy logic and
>soft computing. The scientific program will include keynote/plenary
>talks and technical parallel sessions that address important issues
of
>interest in the fields. The Congress will serve as a platform not
only
>for knowledge sharing, but also for stimulating new ideas in
>broadening and deepening theoretical and applied explorations of
fuzzy
>logic and soft computing. Notably, IFSA2005 will take place in the
>year of IFSA's 20th anniversary, which may well be an event of
memory
>and celebration in the course of its evolution. 
>
>This IFSA World Congress is to be the first-time-ever in Mainland
>China. China is a dynamic nation with a rapid economic growth and
huge
>market. The conference site is Beijing, which has been the capital of
>China for about five hundred years, and is now one of the largest
>international cities in the world. Nowadays, as the cultural,
>educational and Hi-tech center of the nation, Beijing possesses a
>large number of world-class conference facilities, communication
>infrastructures and hotels, and has successfully hosted many
important
>international conferences. In addition, Beijing is an ideal place for
>sightseeing. Its rich cultures and historical attractions such as the
>Great Wall, Forbidden City, Summer Palace, and Temple of Heaven will
>provide participants with unique experiences for social activities. 
>
>Topics
>We solicit papers on theoretical issues and their applications
related
>to fuzzy logic and soft computing. Suggested topics include but are
>not limited to: 
>* Mathematical Foundations of Fuzzy Set Theory 
>* Fuzzy Logic and Approximate Reasoning 
>* Neural Networks, Genetic Algorithms and Soft Computing 
>* Fuzzy Control, Robots and Intelligent Techniques 
>* Expert Systems and Computational Intelligence 
>* Knowledge Discovery and Data Mining 
>* Rough Sets and Evidence Theory 
>* Uncertainty in Decision Sciences and Optimization 
>* Signal/Image Processing and Pattern Recognition 
>* Fuzzy Databases and Information Retrieval 
>* Hybrid Systems 
>
>Honorary Chair:     Lotfi A. Zadeh, USA
>General Chair:      Yingming Liu, China
>Organizing Chair:   Guoqing Chen, China
>Program Chair:      Mingsheng Ying, China
>
>Advisory Committee
>Zeungnam Bien, Korea     Witold Pedrycz, Canada
>George J. Klir, USA      Philippe Smets, Belgium
>Laszlo T. Koczy, Hungary Michio Sugeno, Japan
>Yingming Liu, China      Lotfi A. Zadeh, USA
>Ebrahim Mamdani, UK      Chunjun Zhao, China
>Zdzislaw Pawlak, Poland  Hans-Jurgen Zimmermann, Germany
>
>Local Organization Committee
>
>Yixiang Chen, China
>Dexue Zhang, China
>Wenxiu Zhang, China
>Chongyou Zheng, China
>
>Program Committee (Not Completed)
>
>M. Berthold (Germany)        Maokang Luo (China)
>James C. Bezdek (USA)        Luis Magdalena (Spain)
>Taner Bilgic (Turkey)        Trevor Martin (UK)
>Piero Bonissone (USA)        Kyung Chan Min (Korea)
>Bernadette Bouchon-Meunier (France)  Masao Mukaidono (Japan)
>Kai-Yuan Cai (China)         Vesa Niskanen (Finland)
>Christer Carlsson (Finland)  Vilem Novak (Czech Republic)
>Oscar Castillo (Mexico)      Fred Petry (USA)
>Guoqing Chen (China)         Nguyen Hoang Phuong (Vietnam)
>Didier Dubois (France)       Henri Prade (France)
>Takeshi Furuhashi (Japan)    Arthur Ramer (Australia)
>Giangiacomo Gerla (Italy)    Frank Chung-Hoon Rhee (Korea)
>Lluis Godo (Spain)           Da Ruan (Belgium)
>Fernando Gomide (Brazil)     Elie Sanchez (France)
>Kaoru Hirota (Japan)         Sandra Sandri(Brazil)
>Janusz Kacprzyk (Poland)     Thomas Sudkamp (USA)
>Okiay Kaynak (Turkey)        I. Burhan Turksen (Canada)
>Jim Keller (USA)             M. Amparo Vila Miranda (Spain)
>Etienne E. Kerre (Belgium)   Guojun Wang (China)
>Erich P. Klement (Austria)   Li-Xin Wang (Hong Kong China)
>Donald Kraft (USA)           Paul P. Wang (USA)
>Rudolf Kruse (Germany)       Congxin Wu (China)
>Jonathan Lee (Taiwan China)  Ronald R. Yager (USA)
>Hongxing Li (China)          John Yen (USA)
>C.T.Lin (Taiwan China)       Mingsheng Ying (China)
>Zhiqiang Liu  (Hong Kong China)
>
>Important Dates 
>
>Deadline for Submission:    January 17, 2005 (extended)  
>Notification of Acceptance: February 28, 2005
>Final Version due:          April 20, 2005 
>Conference:                 July 28-31, 2005 
>
>Organized by
>Fuzzy Mathematics and Fuzzy Systems Association of China 
>  (The IFSA China Chapter) 
>Tsinghua University 
>Sichuan University 
>Supported by
>National Natural Science Foundation of China 
>The Systems Engineering Society of China 
>China Automation Association 
>China Computer Federation 
>Chinese Mathematical Society 
>
>Detailed information about the paper submission, proceedings,
>registration, accommodation, journal special issues, student paper
>award, conference program, etc., will be available at the conference
>web site (http://ifsa2005.em.tsinghua.edu.cn) and in forthcoming
>CFPs. You may also contact us at: 
>    Professor Guoqing Chen
>    IFSA2005 Organizing Committee and Secretariat
>    School of Economics and Management
>    Tsinghua University, Beijing 100084, China 
>    Tel./Fax: 86-10-62789925
>    Email: ifsa2005 [at] em [dot] tsinghua.edu.cn 
>
>------------- End Forwarded Message -------------



From owner-reliable_computing [at] interval [dot] louisiana.edu Mon Jan 10 10:56:31 2005
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Date: Mon, 10 Jan 2005 17:55:57 +0100
From: Arnold Neumaier <Arnold.Neumaier [at] univie [dot] ac.at>
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To: Svetoslav Markov <smarkov [at] bio [dot] bas.bg>
CC: reliable_computing [at] interval [dot] louisiana.edu
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Svetoslav Markov wrote:

> the solution  of a single quadratic equation in one variable
> with interval coefficients has been discussed in detail in
> Section 4 of the paper:
> 
>   N. S. Dimitrova,  S.  M.  Markov.  Ueber die
> intervall-arithmetische Berechnung des Wertebereichs einer Funktion
> mit Anwendungen, Freiburger Intervall-Berichte, Univ.  Freiburg, 81/4
> (1981), 1--22.

A scanned copy of the paper (with thanks to Prof. Markov) is at
http://www.mat.univie.ac.at/~neum/contrib/intquad.pdf

Section 4 contains a complete discussion of the determination of
{ x | x^2 + p x + q = 0 for some p in \p, q in \q }
for intervals \p, \q. This is slightly less general than
what Hansen and Walster (and I) treat, where x^2 also has an
interval coefficient. Note that simply dividing by this
coefficient would result in overestimation of the solution set.


Arnold Neumaier



From owner-reliable_computing [at] interval [dot] louisiana.edu Wed Jan 12 08:52:31 2005
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Date: Wed, 12 Jan 2005 08:51:54 -0600
To: reliable_computing [at] interval [dot] louisiana.edu
From: "R. Baker Kearfott" <rbk [at] louisiana [dot] edu>
Subject: Freiburger Intervallberichte: Volunteer?
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Colleagues,

I see steady reference to articles in the "Freiburger Intervallberichte,"
and I personally have a need from time to time to look up articles
therein.  Also, my understanding is that, since Freiburger Intervallberichte
was a preprint series, copyright issues are not involved.

I see significant benefit in having electronic (i.e. scanned) copies
of the Freiburger Intervallberichte on the web (such as on 
interval.louisiana.edu).  However, someone will actually have to 
scan the series (ideally into PDF), a significant but not prohibitive
task.  We thus need a volunteer (or a volunteering of resources) to
scan the series.

Does anyone wish to volunteer?  Also, does anyone have comments concerning
this proposed endeavor?

Sincerely,

R. Baker Kearfott

---------------------------------------------------------------
R. Baker Kearfott,    rbk [at] louisiana [dot] edu   (337) 482-5346 (fax)
(337) 482-5270 (work)                     (337) 993-1827 (home)
URL: http://interval.louisiana.edu/kearfott.html
Department of Mathematics, University of Louisiana at Lafayette
(Room 217 Maxim D. Doucet Hall, 1403 Johnston Street)
Box 4-1010, Lafayette, LA 70504-1010, USA
---------------------------------------------------------------



From owner-reliable_computing [at] interval [dot] louisiana.edu Wed Jan 12 08:59:46 2005
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From: "Patrizia Lamberti" <plamberti [at] unisa [dot] it>
To: <reliable_computing [at] interval [dot] louisiana.edu>
Subject: polynomial variability
Date: Wed, 12 Jan 2005 15:59:09 +0100
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------=_NextPart_000_00AE_01C4F8BF.A6A0E220
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I am a PhD Student in Information and Electric Engineering at the =
University of Salerno, Italy.

=20

I am investigating the possibility of determining the range of minimum =
variability of a polynomial by means of interval analysis.

=20

I prefer to describe the problem by using a simple example.

=20

Let us consider the 5th order polynomial =
f(x)=3Dx^5+60*x^4+30*x^3-10*x^2+6, which has 2 minima and 3 maxima in =
the range I=3D[-1.0,0.7]. Let us inspect the behaviour of f over I by =
considering a translating WIDE interval =
X(k)=3D[x0(k)-delta,x0(k)+delta], included in I, with =
x0=3D{-0.7,-0.6,.,0.3,0.4} and delta=3D0.3. Let us calculate the Taylor =
expansion of f(x) centred in x0(k), k=3D1,2,., and evaluate its interval =
extension F_TM(X(k)).

I have found that, among the N values F_TM(X(k)) calculated, the one of =
minimum radius is obtained for the X(k) wherein the real function f(x) =
is subjected to the minimum variation (namely, in that X(k) the united =
extension of f(X(k)) assumes the minimum radius).

=20

Note that:

1)      the F_TM(X(k)) are calculated over intervals X(k) of fixed, and =
big, amplitude 2*delta;

2)      I don't have looked for the true range of f over I or X(k), but =
I it was enough for me to look for the range of minimum (even if =
unknown) radius of f(x).

=20

Now, I have some questions:

1)      is this result of general validity, namely for any polynomial =
and/or function in general? Does it exist some theorem helpful to =
demonstrate it in general?

2)      has this problem already afforded in literature?

=20

Please, do not hesitate to ask for further details.

=20

Best regards

=20

Patrizia Lamberti

=20

------=_NextPart_000_00AE_01C4F8BF.A6A0E220
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
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<STYLE></STYLE>
</HEAD>
<BODY bgColor=3D#ffffff>
<DIV><FONT face=3DArial size=3D2>
<DIV><FONT face=3DArial size=3D2>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: =
justify"><FONT=20
size=3D3><FONT face=3D"Times New Roman"><SPAN lang=3DEN-GB=20
style=3D"mso-ansi-language: EN-GB">I am a PhD Student in Information and =
Electric=20
Engineering at the </SPAN><?xml:namespace prefix =3D st1 ns =3D=20
"urn:schemas-microsoft-com:office:smarttags" =
/><st1:place><st1:PlaceType><SPAN=20
lang=3DEN-GB=20
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lang=3DEN-GB style=3D"mso-ansi-language: EN-GB"> of =
</SPAN><st1:PlaceName><SPAN=20
lang=3DEN-GB=20
style=3D"mso-ansi-language: =
EN-GB">Salerno</SPAN></st1:PlaceName></st1:place><SPAN=20
lang=3DEN-GB style=3D"mso-ansi-language: EN-GB">,=20
</SPAN><st1:country-region><st1:place><SPAN lang=3DEN-GB=20
style=3D"mso-ansi-language: =
EN-GB">Italy</SPAN></st1:place></st1:country-region><SPAN=20
lang=3DEN-GB style=3D"mso-ansi-language: EN-GB">.<?xml:namespace prefix =
=3D o ns =3D=20
"urn:schemas-microsoft-com:office:office" =
/><o:p></o:p></SPAN></FONT></FONT></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: =
justify"><SPAN=20
lang=3DEN-GB style=3D"mso-ansi-language: EN-GB"><o:p><FONT face=3D"Times =
New Roman"=20
size=3D3>&nbsp;</FONT></o:p></SPAN></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: =
justify"><SPAN=20
lang=3DEN-GB style=3D"mso-ansi-language: EN-GB"><FONT size=3D3><FONT=20
face=3D"Times New Roman">I am investigating the possibility of =
determining the=20
range of minimum variability of a polynomial by means of interval=20
analysis.<o:p></o:p></FONT></FONT></SPAN></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: =
justify"><SPAN=20
lang=3DEN-GB style=3D"mso-ansi-language: EN-GB"><o:p><FONT face=3D"Times =
New Roman"=20
size=3D3>&nbsp;</FONT></o:p></SPAN></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: =
justify"><SPAN=20
lang=3DEN-GB style=3D"mso-ansi-language: EN-GB"><FONT size=3D3><FONT=20
face=3D"Times New Roman">I prefer to describe the problem by using a =
simple=20
example.<o:p></o:p></FONT></FONT></SPAN></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: =
justify"><SPAN=20
lang=3DEN-GB style=3D"mso-ansi-language: EN-GB"><o:p><FONT face=3D"Times =
New Roman"=20
size=3D3>&nbsp;</FONT></o:p></SPAN></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: =
justify"><FONT=20
size=3D3><FONT face=3D"Times New Roman"><SPAN lang=3DEN-GB=20
style=3D"mso-ansi-language: EN-GB">Let us consider the 5<SUP>th</SUP> =
order=20
polynomial f(x)=3Dx^5+60*x^4+30*x^3-10*x^2+6, which has 2 minima and 3 =
maxima in=20
the range I=3D[-1.0,0.7]. Let us inspect the behaviour of f over I by =
considering=20
a translating WIDE interval X(k)=3D[x0(k)-delta,x0(k)+delta], included =
in I, with=20
x0=3D{-0.7,-0.6,=85,0.3,0.4} and delta=3D0.3. Let us calculate the=20
</SPAN><st1:City><st1:place><SPAN lang=3DEN-GB=20
style=3D"mso-ansi-language: =
EN-GB">Taylor</SPAN></st1:place></st1:City><SPAN=20
lang=3DEN-GB style=3D"mso-ansi-language: EN-GB"> expansion of f(x) =
centred in x0(k),=20
k=3D1,2,=85, and evaluate its interval extension=20
F_TM(X(k)).<o:p></o:p></SPAN></FONT></FONT></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: =
justify"><SPAN=20
lang=3DEN-GB style=3D"mso-ansi-language: EN-GB"><FONT size=3D3><FONT=20
face=3D"Times New Roman">I have found that, among the N values =
F_TM(X(k))=20
calculated, the one of minimum radius is obtained for the X(k) wherein =
the real=20
function f(x) is subjected to the minimum variation (namely, in that =
X(k) the=20
united extension of f(X(k)) assumes the minimum=20
radius).<o:p></o:p></FONT></FONT></SPAN></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: =
justify"><SPAN=20
lang=3DEN-GB style=3D"mso-ansi-language: EN-GB"><o:p><FONT face=3D"Times =
New Roman"=20
size=3D3>&nbsp;</FONT></o:p></SPAN></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: =
justify"><SPAN=20
lang=3DEN-GB style=3D"mso-ansi-language: EN-GB"><FONT size=3D3><FONT=20
face=3D"Times New Roman">Note that:<o:p></o:p></FONT></FONT></SPAN></P>
<P class=3DMsoNormal=20
style=3D"MARGIN: 0cm 0cm 0pt 36pt; TEXT-INDENT: -18pt; TEXT-ALIGN: =
justify; mso-list: l1 level1 lfo1; tab-stops: list 36.0pt"><FONT=20
face=3D"Times New Roman"><SPAN lang=3DEN-GB style=3D"mso-ansi-language: =
EN-GB"><SPAN=20
style=3D"mso-list: Ignore"><FONT size=3D3>1)</FONT><SPAN=20
style=3D"FONT: 7pt 'Times New Roman'">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=20
</SPAN></SPAN></SPAN><SPAN lang=3DEN-GB style=3D"mso-ansi-language: =
EN-GB"><FONT=20
size=3D3>the F_TM(X(k)) are calculated over intervals X(k) of fixed, and =
big,=20
amplitude 2*delta;<o:p></o:p></FONT></SPAN></FONT></P>
<P class=3DMsoNormal=20
style=3D"MARGIN: 0cm 0cm 0pt 36pt; TEXT-INDENT: -18pt; TEXT-ALIGN: =
justify; mso-list: l1 level1 lfo1; tab-stops: list 36.0pt"><FONT=20
face=3D"Times New Roman"><SPAN lang=3DEN-GB style=3D"mso-ansi-language: =
EN-GB"><SPAN=20
style=3D"mso-list: Ignore"><FONT size=3D3>2)</FONT><SPAN=20
style=3D"FONT: 7pt 'Times New Roman'">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=20
</SPAN></SPAN></SPAN><SPAN lang=3DEN-GB style=3D"mso-ansi-language: =
EN-GB"><FONT=20
size=3D3>I don=92t have looked for the true range of f over I or X(k), =
but I it was=20
enough for me to look for the range of minimum (even if unknown) radius =
of=20
f(x).<o:p></o:p></FONT></SPAN></FONT></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: =
justify"><SPAN=20
lang=3DEN-GB style=3D"mso-ansi-language: EN-GB"><o:p><FONT face=3D"Times =
New Roman"=20
size=3D3>&nbsp;</FONT></o:p></SPAN></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: =
justify"><SPAN=20
lang=3DEN-GB style=3D"mso-ansi-language: EN-GB"><FONT size=3D3><FONT=20
face=3D"Times New Roman">Now, I have some=20
questions:<o:p></o:p></FONT></FONT></SPAN></P>
<P class=3DMsoNormal=20
style=3D"MARGIN: 0cm 0cm 0pt 36pt; TEXT-INDENT: -18pt; TEXT-ALIGN: =
justify; mso-list: l0 level1 lfo2; tab-stops: list 36.0pt"><FONT=20
face=3D"Times New Roman"><SPAN lang=3DEN-GB style=3D"mso-ansi-language: =
EN-GB"><SPAN=20
style=3D"mso-list: Ignore"><FONT size=3D3>1)</FONT><SPAN=20
style=3D"FONT: 7pt 'Times New Roman'">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=20
</SPAN></SPAN></SPAN><SPAN lang=3DEN-GB style=3D"mso-ansi-language: =
EN-GB"><FONT=20
size=3D3>is this result of general validity, namely for any polynomial =
and/or=20
function in general? Does it exist some theorem helpful to demonstrate =
it in=20
general?<o:p></o:p></FONT></SPAN></FONT></P>
<P class=3DMsoNormal=20
style=3D"MARGIN: 0cm 0cm 0pt 36pt; TEXT-INDENT: -18pt; TEXT-ALIGN: =
justify; mso-list: l0 level1 lfo2; tab-stops: list 36.0pt"><FONT=20
face=3D"Times New Roman"><SPAN lang=3DEN-GB style=3D"mso-ansi-language: =
EN-GB"><SPAN=20
style=3D"mso-list: Ignore"><FONT size=3D3>2)</FONT><SPAN=20
style=3D"FONT: 7pt 'Times New Roman'">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=20
</SPAN></SPAN></SPAN><SPAN lang=3DEN-GB style=3D"mso-ansi-language: =
EN-GB"><FONT=20
size=3D3>has this problem already afforded in=20
literature?<o:p></o:p></FONT></SPAN></FONT></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt 18pt; TEXT-ALIGN: =
justify"><SPAN=20
lang=3DEN-GB style=3D"mso-ansi-language: EN-GB"><o:p><FONT face=3D"Times =
New Roman"=20
size=3D3>&nbsp;</FONT></o:p></SPAN></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: =
justify"><SPAN=20
lang=3DEN-GB style=3D"mso-ansi-language: EN-GB"><FONT size=3D3><FONT=20
face=3D"Times New Roman">Please, do not hesitate to ask for further=20
details.<o:p></o:p></FONT></FONT></SPAN></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: =
justify"><SPAN=20
lang=3DEN-GB style=3D"mso-ansi-language: EN-GB"><o:p><FONT face=3D"Times =
New Roman"=20
size=3D3>&nbsp;</FONT></o:p></SPAN></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: =
justify"><SPAN=20
lang=3DEN-GB style=3D"mso-ansi-language: EN-GB"><FONT size=3D3><FONT=20
face=3D"Times New Roman">Best =
regards<o:p></o:p></FONT></FONT></SPAN></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: =
justify"><SPAN=20
lang=3DEN-GB style=3D"mso-ansi-language: EN-GB"><o:p><FONT face=3D"Times =
New Roman"=20
size=3D3>&nbsp;</FONT></o:p></SPAN></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: =
justify"><SPAN=20
lang=3DEN-GB style=3D"mso-ansi-language: EN-GB"><FONT size=3D3><FONT=20
face=3D"Times New Roman">Patrizia =
Lamberti<o:p></o:p></FONT></FONT></SPAN></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: =
justify"><SPAN=20
lang=3DEN-GB style=3D"mso-ansi-language: EN-GB"><o:p><FONT face=3D"Times =
New Roman"=20
size=3D3>&nbsp;</FONT></o:p></SPAN></P></FONT></DIV></FONT></DIV></BODY><=
/HTML>

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From owner-reliable_computing [at] interval [dot] louisiana.edu Wed Jan 12 16:46:45 2005
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Message-ID: <001c01c4f8ed$75f45160$6500a8c0@megan>
From: "Jiawang Nie" <njw [at] Math [dot] Berkeley.EDU>
To: "Patrizia Lamberti" <plamberti [at] unisa [dot] it>,
        <reliable_computing [at] interval [dot] louisiana.edu>
Cc: "James Demmel" <demmel [at] cs [dot] berkeley.edu>
References: <00b401c4f8b7$4de76350$7fa4cdc1@plportatile>
Subject: Re: polynomial variability
Date: Wed, 12 Jan 2005 13:27:02 -0800
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Dear  Patrizia:

      Your question about polynomial variability is very interesting, and r=
elated to our work.=20

      The global minimum or maximum of a univariate polynomial on an interv=
al [a,b] can be found very efficiently by software SOSTOOLS or Gloptipoly (=
 search in google for the references and websits). So the true range of f(x=
)  over your X(k) can be found.=20

      The interval extensions F_TM(X(k)) evaluated via taylor expansion ( f=
irst order ?) of  f(x) reflect the true range of f(x) over X(k) when delta =
is tiny. If delta is not sufficiently small, the relationship may not be cl=
ear.

=20=20=20=20=20=20=20=20
       A more general problem about sensitivity analysis of solutions to  p=
olynomial systems is discussed in one of our papers.  The following paper m=
ay be useful to you:
             http://front.math.ucdavis.edu/math.OC/0411122
Given a multivariate system of polynomials p(x,mu), where the mu's can  be =
thought of as parameters determining the coefficient of p(), we give an alg=
orithm  for computing the minimal ellipsoid contining all the real solution=
s x of p(x,mu)=3D0.  For example, if q() is a single polynomial in the scal=
ar x with coefficients in=20
intervals with given centers and radii, and you want bounds on the real zer=
os,then this can be formulated as finding  the minimum ellipsoid containing=
 the  roots x of a system like

      sum_{i=3D0 to n}   mu_i * x^i  =3D 0
      (mu_i - center_i)^2 + mu'_i^2 =3D radius^2   for  i =3D 0 to n

The paper uses techniques from semidefinite programming.


Regards,
Jiawang Nie and Jim Demmel

  ----- Original Message -----=20
  From: Patrizia Lamberti=20
  To: reliable_computing [at] interval [dot] louisiana.edu=20
  Sent: Wednesday, January 12, 2005 6:59 AM
  Subject: polynomial variability


  I am a PhD Student in Information and Electric Engineering at the Univers=
ity of Salerno, Italy.

=20=20=20

  I am investigating the possibility of determining the range of minimum va=
riability of a polynomial by means of interval analysis.

=20=20=20

  I prefer to describe the problem by using a simple example.

=20=20=20

  Let us consider the 5th order polynomial f(x)=3Dx^5+60*x^4+30*x^3-10*x^2+=
6, which has 2 minima and 3 maxima in the range I=3D[-1.0,0.7]. Let us insp=
ect the behaviour of f over I by considering a translating WIDE interval X(=
k)=3D[x0(k)-delta,x0(k)+delta], included in I, with x0=3D{-0.7,-0.6,.,0.3,0=
.4} and delta=3D0.3. Let us calculate the Taylor expansion of f(x) centred =
in x0(k), k=3D1,2,., and evaluate its interval extension F_TM(X(k)).

  I have found that, among the N values F_TM(X(k)) calculated, the one of m=
inimum radius is obtained for the X(k) wherein the real function f(x) is su=
bjected to the minimum variation (namely, in that X(k) the united extension=
 of f(X(k)) assumes the minimum radius).

=20=20=20

  Note that:

  1)      the F_TM(X(k)) are calculated over intervals X(k) of fixed, and b=
ig, amplitude 2*delta;

  2)      I don't have looked for the true range of f over I or X(k), but I=
 it was enough for me to look for the range of minimum (even if unknown) ra=
dius of f(x).

=20=20=20

  Now, I have some questions:

  1)      is this result of general validity, namely for any polynomial and=
/or function in general? Does it exist some theorem helpful to demonstrate =
it in general?

  2)      has this problem already afforded in literature?

=20=20=20

  Please, do not hesitate to ask for further details.

=20=20=20

  Best regards

=20=20=20

  Patrizia Lamberti

=20=20=20

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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<HTML DEFANGED_xmlns:st1 =3D "urn:schemas-microsoft-com:office:smarttags" D=
EFANGED_xmlns:o =3D=20
"urn:schemas-microsoft-com:office:office"><HEAD>
<META http-equiv=3DContent-Type content=3D"text/html; charset=3Diso-8859-1">
<META content=3D"MSHTML 6.00.2900.2523" name=3DGENERATOR><DEFANGED_STYLE_0 =
<>
<DEFANGED_style_0 </STYLE>
</HEAD>
<BODY bgColor=3D#ffffff>
<DEFANGED_DIV><FONT face=3DArial size=3D2>
<DEFANGED_DIV><FONT face=3DArial size=3D2><FONT face=3DArial size=3D2></FON=
T></FONT>&nbsp;</DEFANGED_DIV>
<DEFANGED_DIV><FONT face=3DArial size=3D2>Dear&nbsp; Patrizia:</FONT></DEFA=
NGED_DIV>
<DEFANGED_DIV><FONT face=3DArial size=3D2></FONT>&nbsp;</DEFANGED_DIV>
<DEFANGED_DIV><FONT face=3DArial size=3D2>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Yo=
ur question about=20
polynomial variability is very interesting, and related to our work.=20
</FONT></DEFANGED_DIV>
<DEFANGED_DIV><FONT face=3DArial size=3D2></FONT>&nbsp;</DEFANGED_DIV>
<DEFANGED_DIV><FONT face=3DArial size=3D2>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Th=
e global minimum=20
or maximum of a univariate polynomial on an interval [a,b] can be found ver=
y=20
efficiently by software SOSTOOLS or Gloptipoly ( search in google for the=
=20
references and websits). So the true range of f(x)&nbsp; over your X(k) can=
 be=20
found. </FONT></DEFANGED_DIV>
<DEFANGED_DIV><FONT face=3DArial size=3D2></FONT>&nbsp;</DEFANGED_DIV>
<DEFANGED_DIV><FONT face=3DArial size=3D2>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Th=
e interval=20
extensions <FONT face=3D"Times New Roman" size=3D3>F_TM(X(k)) evaluated via=
 taylor=20
expansion ( first order ?) of&nbsp; f(x) reflect the true range of f(x) ove=
r=20
X(k) when delta is tiny. If delta is not sufficiently small, the relationsh=
ip=20
may not be clear.</FONT></FONT></DEFANGED_DIV>
<DEFANGED_DIV><FONT face=3DArial size=3D2></FONT>&nbsp;</DEFANGED_DIV>
<DEFANGED_DIV><FONT face=3DArial size=3D2>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;=20
</FONT></DEFANGED_DIV>
<DEFANGED_DIV><FONT face=3DArial size=3D2>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;A more=20
general problem about&nbsp;sensitivity analysis of solutions=20
to&nbsp;&nbsp;polynomial systems is discussed in one of our papers.<FONT=20
face=3D"Times New Roman" size=3D3>&nbsp; The </FONT></FONT><FONT face=3DAri=
al=20
size=3D2><FONT face=3D"Times New Roman" size=3D3>following paper may be use=
ful to=20
you:<BR>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
=20
&nbsp;</FONT><A href=3D""><FONT face=3D"Times New Roman"=20
size=3D3>http://front.math.ucdavis.edu/math.OC/0411122</FONT></A><BR><FONT=
=20
face=3D"Times New Roman" size=3D3>Given a multivariate system of polynomial=
s=20
p(x,mu), where the mu's can&nbsp; be thought of as parameters determining t=
he=20
coefficient of p(), we give an algorithm&nbsp; for computing the minimal=20
ellipsoid contining all the real solutions x of p(x,mu)=3D0.&nbsp; For exam=
ple, if=20
q() is a single polynomial in the scalar x with coefficients in <BR>interva=
ls=20
with given centers and radii, and you want bounds on the real zeros,then th=
is=20
can be formulated as finding&nbsp; the minimum ellipsoid containing the&nbs=
p;=20
roots x of a system like<BR><BR>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; sum_{i=3D0 t=
o=20
n}&nbsp;&nbsp; mu_i * x^i&nbsp; =3D 0<BR>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (mu=
_i -=20
center_i)^2 + mu'_i^2 =3D radius^2&nbsp;&nbsp; for&nbsp; i =3D 0 to n<BR><B=
R>The=20
paper uses techniques from semidefinite=20
programming.<BR></FONT></FONT></DEFANGED_DIV><FONT face=3DArial size=3D2><F=
ONT=20
face=3D"Times New Roman" size=3D3>
<DEFANGED_DIV><BR>Regards,<BR>Jiawang Nie and Jim=20
Demmel</FONT><BR></DEFANGED_DIV></FONT></FONT></DEFANGED_DIV>
<BLOCKQUOTE dir=3D"ltr"=20
DEFANGED_style=3D"PADDING-RIGHT: 0px; PADDING-LEFT: 5px; MARGIN-LEFT: 5px; =
BORDER-LEFT: #000000 2px solid; MARGIN-RIGHT: 0px">
  <DEFANGED_DIV style=3D"FONT: 10pt arial">----- Original Message ----- </D=
EFANGED_DIV>
  <DEFANGED_DIV=20
  style=3D"BACKGROUND: #e4e4e4; FONT: 10pt arial; font-color: black"><B>Fro=
m:</B>=20
  <A title=3Dplamberti [at] unisa [dot] it href=3D"mailto:plamberti [at] unisa [dot] it">Patrizia=
=20
  Lamberti</A> </DEFANGED_DIV>
  <DEFANGED_DIV style=3D"FONT: 10pt arial"><B>To:</B> <A=20
  title=3Dreliable_computing [at] interval [dot] louisiana.edu=20
  href=3D"mailto:reliable_computing [at] interval [dot] louisiana.edu">reliable_comput=
ing [at] interval [dot] louisiana.edu</A>=20
  </DEFANGED_DIV>
  <DEFANGED_DIV style=3D"FONT: 10pt arial"><B>Sent:</B> Wednesday, January =
12, 2005 6:59=20
  AM</DEFANGED_DIV>
  <DEFANGED_DIV style=3D"FONT: 10pt arial"><B>Subject:</B> polynomial varia=
bility</DEFANGED_DIV>
  <DEFANGED_DIV><BR></DEFANGED_DIV><DEFANGED_DIV><FONT face=3DArial size=3D=
2><DEFANGED_DIV><FONT=20
  face=3DArial size=3D2>
  <P class=3D"MsoNormal"=20
  DEFANGED_style=3D"MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><FONT size=3D=
3><FONT=20
  face=3D"Times New Roman"><DEFANGED_SPAN lang=3DEN-GB=20
  style=3D"mso-ansi-language: EN-GB">I am a PhD Student in Information and=
=20
  Electric Engineering at the=20
  </DEFANGED_SPAN><DEFANGED_ST1:PLACE><DEFANGED_ST1:PLACETYPE><DEFANGED_SPA=
N=20
  lang=3DEN-GB=20
  style=3D"mso-ansi-language: EN-GB">University</DEFANGED_SPAN></DEFANGED_S=
T1:PLACETYPE><DEFANGED_SPAN=20
  lang=3DEN-GB style=3D"mso-ansi-language: EN-GB"> of=20
  </DEFANGED_SPAN><DEFANGED_ST1:PLACENAME><DEFANGED_SPAN lang=3DEN-GB=20
  style=3D"mso-ansi-language: EN-GB">Salerno</DEFANGED_SPAN></DEFANGED_ST1:=
PLACENAME></DEFANGED_ST1:PLACE><DEFANGED_SPAN=20
  lang=3DEN-GB style=3D"mso-ansi-language: EN-GB">,=20
  </DEFANGED_SPAN><DEFANGED_ST1:COUNTRY-REGION><DEFANGED_ST1:PLACE><DEFANGE=
D_SPAN=20
  lang=3DEN-GB=20
  style=3D"mso-ansi-language: EN-GB">Italy</DEFANGED_SPAN></DEFANGED_ST1:PL=
ACE></DEFANGED_ST1:COUNTRY-REGION><DEFANGED_SPAN=20
  lang=3DEN-GB=20
  style=3D"mso-ansi-language: EN-GB">.<o:p></o:p></DEFANGED_SPAN></FONT></F=
ONT></P>
  <P class=3D"MsoNormal"=20
  DEFANGED_style=3D"MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><DEFANGED_SPA=
N=20
  lang=3DEN-GB style=3D"mso-ansi-language: EN-GB"><o:p><FONT face=3D"Times =
New Roman"=20
  size=3D3>&nbsp;</FONT></o:p></DEFANGED_SPAN></P>
  <P class=3D"MsoNormal"=20
  DEFANGED_style=3D"MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><DEFANGED_SPA=
N=20
  lang=3DEN-GB style=3D"mso-ansi-language: EN-GB"><FONT size=3D3><FONT=20
  face=3D"Times New Roman">I am investigating the possibility of determinin=
g the=20
  range of minimum variability of a polynomial by means of interval=20
  analysis.<o:p></o:p></FONT></FONT></DEFANGED_SPAN></P>
  <P class=3D"MsoNormal"=20
  DEFANGED_style=3D"MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><DEFANGED_SPA=
N=20
  lang=3DEN-GB style=3D"mso-ansi-language: EN-GB"><o:p><FONT face=3D"Times =
New Roman"=20
  size=3D3>&nbsp;</FONT></o:p></DEFANGED_SPAN></P>
  <P class=3D"MsoNormal"=20
  DEFANGED_style=3D"MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><DEFANGED_SPA=
N=20
  lang=3DEN-GB style=3D"mso-ansi-language: EN-GB"><FONT size=3D3><FONT=20
  face=3D"Times New Roman">I prefer to describe the problem by using a simp=
le=20
  example.<o:p></o:p></FONT></FONT></DEFANGED_SPAN></P>
  <P class=3D"MsoNormal"=20
  DEFANGED_style=3D"MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><DEFANGED_SPA=
N=20
  lang=3DEN-GB style=3D"mso-ansi-language: EN-GB"><o:p><FONT face=3D"Times =
New Roman"=20
  size=3D3>&nbsp;</FONT></o:p></DEFANGED_SPAN></P>
  <P class=3D"MsoNormal"=20
  DEFANGED_style=3D"MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><FONT size=3D=
3><FONT=20
  face=3D"Times New Roman"><DEFANGED_SPAN lang=3DEN-GB=20
  style=3D"mso-ansi-language: EN-GB">Let us consider the 5<SUP>th</SUP> ord=
er=20
  polynomial f(x)=3Dx^5+60*x^4+30*x^3-10*x^2+6, which has 2 minima and 3 ma=
xima in=20
  the range I=3D[-1.0,0.7]. Let us inspect the behaviour of f over I by=20
  considering a translating WIDE interval X(k)=3D[x0(k)-delta,x0(k)+delta],=
=20
  included in I, with x0=3D{-0.7,-0.6,=85,0.3,0.4} and delta=3D0.3. Let us =
calculate=20
  the </DEFANGED_SPAN><DEFANGED_ST1:CITY><DEFANGED_ST1:PLACE><DEFANGED_SPAN=
=20
  lang=3DEN-GB=20
  style=3D"mso-ansi-language: EN-GB">Taylor</DEFANGED_SPAN></DEFANGED_ST1:P=
LACE></DEFANGED_ST1:CITY><DEFANGED_SPAN=20
  lang=3DEN-GB style=3D"mso-ansi-language: EN-GB"> expansion of f(x) centre=
d in=20
  x0(k), k=3D1,2,=85, and evaluate its interval extension=20
  F_TM(X(k)).<o:p></o:p></DEFANGED_SPAN></FONT></FONT></P>
  <P class=3D"MsoNormal"=20
  DEFANGED_style=3D"MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><DEFANGED_SPA=
N=20
  lang=3DEN-GB style=3D"mso-ansi-language: EN-GB"><FONT size=3D3><FONT=20
  face=3D"Times New Roman">I have found that, among the N values F_TM(X(k))=
=20
  calculated, the one of minimum radius is obtained for the X(k) wherein th=
e=20
  rea