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Re: [Reliable_computing] Question about strong regularity of (parametric) interval matrices

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.


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  |
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