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*To*: "Kreinovich, Vladik" <vladik@xxxxxxxx>*Subject*: Re: [Reliable_computing] Question about strong regularity of (parametric) interval matrices*From*: Walter Mascarenhas <walter.mascarenhas@xxxxxxxxx>*Date*: Sun, 29 Oct 2017 18:07:46 -0200*Arc-authentication-results*: i=1; mx.google.com; dkim=pass header.i=@gmail.com header.s=20161025 header.b=FMpgIHeu; spf=pass (google.com: domain of walter.mascarenhas@xxxxxxxxx designates 209.85.223.169 as permitted sender) smtp.mailfrom=walter.mascarenhas@xxxxxxxxx*Arc-message-signature*: i=1; a=rsa-sha256; c=relaxed/relaxed; d=google.com; s=arc-20160816; h=cc:to:subject:message-id:date:from:references:in-reply-to :mime-version:dkim-signature:arc-authentication-results; bh=nyB1ErJ6G3dYGGU2WaFxJtuByaQ8SJ2SnOvariAOywo=; b=fqbRMMNNu1/fQMJ2NO4tGSnFBTEJPlvba1yRnt5ZdJ1XPEncJFkPcdg6h2f6qMZe5T O8BYBJGH+q0lEawj0YBAcwXpg8Sx+X4W6ii7ua7CX7OJD/Tyv1MuFP0EKYQIWh/EoWvM p+YSqaf2xHMHWZpDxFYnXbxwrgDlGwxl2Aq80qVhnuZobx/o0M2/zkc+hk2Tgn/diw+E uji1/FpHJWGb4Fq4jsYlww8hAqLNRgc8ugPyojczj7rdBVrYuSGiPcOakeW3uorKbDsz X3EN7MEjvTeejpFB8p0gLGdzFn2+e9XDJ3ElGnhnvtlDboI86UESl+eKlESndKw/bUCy rkAA==*Arc-seal*: i=1; a=rsa-sha256; t=1509307669; cv=none; d=google.com; s=arc-20160816; b=hy5O2XN1QBO4+ZY2GS+CYXDAJ06DbylgpWQSY5M/h/GHPBlLzBzSlDBSEwyY6KPWWQ O0MMZQqJlTNacjprtZOgbeUrg9iGH6yOyHBaRzmiDTIOLb+S81Pc61zha7AqRiIUz0DB QyD6ddjYCi2emLkW4lvPNPanwJpHh6OQ4MIo9fs78ApiTjjIlByGvsP5tdsDFzK64fJY Go2hkw6M2dGApzu/C7jqJ0519+FMSRHMqoxEnH0w4sZ/yc5UhidwziOxzSI7ANt89eUS inxpSlh3E0FCsNNVAsWPp2xowmFJA9V6UXZS80bLxsN2hKp3riG1hdb7jp3NXhwELQmx S6vw==*Cc*: skalna <skalna@xxxxxxxxxx>, "stds-1788@xxxxxxxx" <stds-1788@xxxxxxxx>, "reliable_computing@xxxxxxxxxxxxxxxxxxx" <reliable_computing@xxxxxxxxxxxxxxxxxxx>*Delivered-to*: mhonarc@xxxxxxxxxxxxxxxx*In-reply-to*: <bf6b135dd9b6472aaf28bd750b47773f@ITDSRVMBX011.utep.edu>*List-help*: <https://listserv.ieee.org/cgi-bin/wa?LIST=STDS-1788>, <mailto:LISTSERV@LISTSERV.IEEE.ORG?body=INFO%20STDS-1788>*List-owner*: <mailto:STDS-1788-request@LISTSERV.IEEE.ORG>*List-subscribe*: <mailto:STDS-1788-subscribe-request@LISTSERV.IEEE.ORG>*List-unsubscribe*: <mailto:STDS-1788-unsubscribe-request@LISTSERV.IEEE.ORG>*References*: <f76c74bb624124ea2fc65b23b6b4470d@agh.edu.pl> <bf6b135dd9b6472aaf28bd750b47773f@ITDSRVMBX011.utep.edu>*Sender*: stds-1788@xxxxxxxx

My few cents:

1) The definition "[A] is regular if and only all its elements are regular" seem so

clear and simple to me that there is no point in changing it.

2) I would use the terms "left regular" and "right regular" to express the

the conditions r \rho(|(mid(A))^{-1}rad([A])|)<1 and r \rho(|(rad([A])mid(A))^{-1}|)<1,

but of course this is all a matter of taste.

3) Indeed, definitions are somewhat arbitrary, but it would be nice if we could

reserve the simple terms, like "regular", to simple and relevant concepts.

4) One of my favorite definitions is the one of compact set in topology

("every covering by open subsets has a finite subcovering.") Perfect

definitions like this one, by Alexandrov and Urysohn 1929, make me

wonder whether definitions are indeed arbitrary. In fact, at same times,

a great definition is worth a thousand theorems, and it is definitely

unwise to mess with them.

Walter.

On Sun, Oct 29, 2017 at 3:59 PM, Kreinovich, Vladik <vladik@xxxxxxxx> wrote:

Dear Ivona,

Please note that you sent not to the interval mailing list but the interval standard mailing list, which is different, I am sending this reply to the acyiual interval community mailing list. The standard mailing list is dealing with the interval standard.

I have two answers to your question.

1) On the substance: the main objective is to check whether the interval matrix [A] is regular, i.e., if every matrix from this interval is regular. In general. This is known to be NP-hard, which means that unless P=NP, there is no hope of having a feasible algorithm that would always check correctly.

There are feasible algorithms for which, if the result is yes, the interval matrix is regular, if no, wit may be regular it may be not we do not know. It is definitely worthwhile to try to come up with new such algorithms.

One of these algorithms, as you mention correctly, is checking whether \rho(|(mid(A))^{-1}rad([A])|)<1.=, but I do not think that it is equivalent to checking whether the matrix [B]=mid(A))^{-1}[A] is regular, because checking this for an interval matrix [B] is, therefore, NP-hard as well.

Similarly, checking the regularity of the inverse product matrix [C]= }[A]mid(A)^{-1} is also NP-hard, so by itself does not lead to any new feasible algorithm, so I am not sure how this comment helps.

2) My opinion (which our professors in St. Petersburg university installed into us) is that definitions are somewhat arbitrary, one does not argue too much about the definitions.

I am not very familiar with the definition of strong regularity as \rho(|(mid(A))^{-1}rad([A])|)<1, you say that this is the definition in most papers, I have not seen any paper with such a definition, can you name one?

It indeed sounds somewhat arbitrary, if you want to propose a new definition fine, just use a different term. However, you did not provide any feasibly testable new definition, and if your definition is not feasibly testable, why not just say that an interval matrix is regular if all corresponding numerical matrices are regular?

______________________________

-----Original Message-----

From: stds-1788@xxxxxxxx [mailto:stds-1788@xxxxxxxx] On Behalf Of skalna

Sent: Sunday, October 29, 2017 11:30 AM

To: stds-1788@xxxxxxxx

Subject: Question about strong regularity of (parametric) interval matrices

Dear Members of Interval Community,

I have the following question. Most of papers on solving interval linear systems says that interval matrix is strongly regular if \rho(|(mid(A))^{-1}rad([A])|)<1.

or equivalently if

[B]=mid(A))^{-1}[A] is regular.

But we can also post-multiply the matrix, and it is probable that the matrix [B] will not be regular, but the matrix [C]=[A}A^{-1} will be regular.

So, shouldn't the definition of strong regularity of interval matrices be changed?

I am in fact more interested in parametric interval matrices, but parametric matrices are strictly connected with interval matrices.

Best regards,

Iwona Skalna

--

___ ____ __ __

| _ | _| | | | Iwona Skalna |

| | | | | | Department of Applied Computer Science |

|_|_|____|__|__| | ul. Gramatyka 10, 30-067 Kraków, Polska |

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**References**:**Question about strong regularity of (parametric) interval matrices***From:*skalna

**RE: Question about strong regularity of (parametric) interval matrices***From:*Kreinovich, Vladik

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