Re: [Scilab-users] Is cond([]) 0 or 1 ? (bug 15579)

[Samuel Gougeon]

Le 27/05/2018 à 23:23, philippe a écrit :

Le 27/05/2018 à 19:51, Samuel Gougeon a écrit :

[...]

Please let us know more about your proposal for 1 instead of 0.

I can't say it now, the best value also depends on the compatibility of
this value with others assigned to matrix functions with argument [].
I would like to check it before, but with the example you mention 0 looks
a better choice.


As far as i understand, rcond(X) should return the inverse of cond(X,1).

In Scilab, rcond([]) returns []. This looks pretty inconsistent with any 
Scilab cond([],p) result,

and does not help much for this longly empty cond([]) discussion.

In Octave, rcond([]) returns %inf, that's consistent with cond([])==0.

However, still, a guy whose det() is 1 and norm() is 0 is really a 
special guy.


If we have no evidence/reference on the fact norm([]) is 1 rather than 0,
i guess that we will change Scilab's cond([]) to 0.
May the Scilab creator having initially set 1 read us and correct us if 
it's needed.


Cheers
Samuel

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Re: [Scilab-users] Is cond([]) 0 or 1 ? (bug 15579)

[Stéphane Mottelet]

Le 28/05/2018 à 10:58, Rafael Guerra a écrit :


No, it is not clear but then, I am not an Emeritus Professor of 
Mathematics.


Sorry but I have no other references on this empty subject.

Regards,

Rafael

*From:*users [mailto:users-boun...@lists.scilab.org] *On Behalf Of 
*Samuel Gougeon

*Sent:* Monday, May 28, 2018 12:09 AM
*To:* Users mailing list for Scilab <users@lists.scilab.org>
*Subject:* Re: [Scilab-users] Is cond([]) 0 or 1 ? (bug 15579)

Le 27/05/2018 à 23:49, Rafael Guerra a écrit :

You guys may have missed the paper sent before on the algebra of
empty matrices by Carl de Boor (Emeritus Professor in Mathematics
and Computer Science), so I am sending it again:


https://pdfs.semanticscholar.org/0f3b/c36f19d5c6a761c19fbc3c4ebde2f31b0a10.pdf

<https://antispam.utc.fr/proxy/2/c3RlcGhhbmUubW90dGVsZXRAdXRjLmZy/pdfs.semanticscholar.org/0f3b/c36f19d5c6a761c19fbc3c4ebde2f31b0a10.pdf>

He argues that the condition number of the square empty matrix []
= 0 and its det([]) = 1


No no, i did not miss it. His argument is that norm([]) is 0:

/Any norm of an empty matrix is zero, as the supremum of the empty set 
of nonnegative numbers. This implies that the condition number of the 
square empty matrix [] is 0./


This does not prevent asking to Philippe why he proposes 1, noticeably 
if he also considers that norm([])==0, or not.


The De Boor argument "/as the supremum of the empty set of nonnegative 
numbers."/ is not clear to me./

/

Every property (even trivially false) is true for all elements of the 
empty set. Hence, any positive number can be taken as the supremum of 
the empty set. However, I don't see in which sense this could justify 
the discusses convention...


S.


/
/Is it clear for you? Would you have a second convergent independent 
reference?


Samuel



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--
Stéphane Mottelet
Ingénieur de recherche
EA 4297 Transformations Intégrées de la Matière Renouvelable
Département Génie des Procédés Industriels
Sorbonne Universités - Université de Technologie de Compiègne
CS 60319, 60203 Compiègne cedex
Tel : +33(0)344234688
http://www.utc.fr/~mottelet

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Re: [Scilab-users] Is cond([]) 0 or 1 ? (bug 15579)

[Rafael Guerra]

No, it is not clear but then, I am not an Emeritus Professor of Mathematics.
Sorry but I have no other references on this empty subject.

Regards,
Rafael

From: users [mailto:users-boun...@lists.scilab.org] On Behalf Of Samuel Gougeon
Sent: Monday, May 28, 2018 12:09 AM
To: Users mailing list for Scilab <users@lists.scilab.org>
Subject: Re: [Scilab-users] Is cond([]) 0 or 1 ? (bug 15579)

Le 27/05/2018 à 23:49, Rafael Guerra a écrit :

You guys may have missed the paper sent before on the algebra of empty matrices 
by Carl de Boor (Emeritus Professor in Mathematics and Computer Science), so I 
am sending it again:

https://pdfs.semanticscholar.org/0f3b/c36f19d5c6a761c19fbc3c4ebde2f31b0a10.pdf



He argues that the condition number of the square empty matrix [] = 0 and its 
det([]) = 1

No no, i did not miss it. His argument is that norm([]) is 0:

Any norm of an empty matrix is zero, as the supremum of the empty set of 
nonnegative numbers. This implies that the condition number of the square empty 
matrix [] is 0.

This does not prevent asking to Philippe why he proposes 1, noticeably if he 
also considers that norm([])==0, or not.

The De Boor argument "as the supremum of the empty set of nonnegative numbers." 
is not clear to me.

Is it clear for you? Would you have a second convergent independent reference?

Samuel
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Re: [Scilab-users] Is cond([]) 0 or 1 ? (bug 15579)

[Samuel Gougeon]

Le 27/05/2018 à 23:49, Rafael Guerra a écrit :


You guys may have missed the paper sent before on the algebra of empty 
matrices by Carl de Boor (Emeritus Professor in Mathematics and 
Computer Science), so I am sending it again:


https://pdfs.semanticscholar.org/0f3b/c36f19d5c6a761c19fbc3c4ebde2f31b0a10.pdf

He argues that the condition number of the square empty matrix [] = 0 
and its det([]) = 1




No no, i did not miss it. His argument is that norm([]) is 0:

/Any norm of an empty matrix is zero, as the supremum of the empty set 
of nonnegative numbers. This implies that the condition number of the 
square empty matrix [] is 0./


This does not prevent asking to Philippe why he proposes 1, noticeably 
if he also considers that norm([])==0, or not.


The De Boor argument "/as the supremum of the empty set of nonnegative 
numbers."/ is not clear to me./


/Is it clear for you? Would you have a second convergent independent 
reference?


Samuel

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Re: [Scilab-users] Is cond([]) 0 or 1 ? (bug 15579)

[Rafael Guerra]

You guys may have missed the paper sent before on the algebra of empty matrices 
by Carl de Boor (Emeritus Professor in Mathematics and Computer Science), so I 
am sending it again:

https://pdfs.semanticscholar.org/0f3b/c36f19d5c6a761c19fbc3c4ebde2f31b0a10.pdf



He argues that the condition number of the square empty matrix [] = 0 and its 
det([]) = 1



Regards,

Rafael



-Original Message-
From: users [mailto:users-boun...@lists.scilab.org] On Behalf Of philippe
Sent: Sunday, May 27, 2018 11:24 PM
To: users@lists.scilab.org
Subject: Re: [Scilab-users] Is cond([]) 0 or 1 ? (bug 15579)



Le 27/05/2018 à 19:51, Samuel Gougeon a écrit :

> [...]

>

> Please let us know more about your proposal for 1 instead of 0.



I can't say it now, the best value also depends on the compatibility of this 
value with others assigned to matrix functions with argument [].  I would like 
to check it before, but with the example you mention 0 looks a better choice.



Philippe








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Re: [Scilab-users] Is cond([]) 0 or 1 ? (bug 15579)

[philippe]

Le 27/05/2018 à 19:51, Samuel Gougeon a écrit :
> [...]
> 
> Please let us know more about your proposal for 1 instead of 0.

I can't say it now, the best value also depends on the compatibility of
this value with others assigned to matrix functions with argument [].  I
would like to check it before, but with the example you mention 0 looks
a better choice.

Philippe

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Re: [Scilab-users] Is cond([]) 0 or 1 ? (bug 15579)

[Samuel Gougeon]

Hello Philippe,

Le 24/05/2018 à 23:20, philippe a écrit :

Le 21/05/2018 à 15:46, Samuel Gougeon a écrit :

There are at least two way to do it :

   * either keep *cond([])* to *1* and set all *cond([], p)* to 1 instead
 of 0
   * or set *cond([])* to *0*.

I don't see any clear reason enforcing a choice rather than the other.
Do you?

from a mathematical point of view  the Condition number of a matrix A is
defined by

cond(A)=||A|| . ||A^(-1)||
its interest is that when solving A*x=y rounding errors on y (eps) are
amplified to be cond(A)*eps. The optimal value of cond(A) is 1 (for
identity matrix) so for me it looks natural that cond([])=1 .


I must confess that, unless claiming that *norm([])***is NOT 0,  i do 
not catch clearly the logical of your conclusion


 * /"The optimal value of cond(A) is 1 (for identity matrix) so for me
   it looks natural that cond([])=1"/

Indeed, cond([]) = norm([])*norm(inv([])) = 0*norm([]) = 0*0 = 0

By the way, according to the clear explanation you give about the 
meaning of the condition number, the value 0 is even more optimal: it 
says that small variations are not getting amplified, but killed.


Please let us know more about your proposal for 1 instead of 0.

Regards
Samuel

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Re: [Scilab-users] Is cond([]) 0 or 1 ? (bug 15579)

[philippe]

Le 21/05/2018 à 15:46, Samuel Gougeon a écrit :
> 
> There are at least two way to do it :
> 
>   * either keep *cond([])* to *1* and set all *cond([], p)* to 1 instead
> of 0
>   * or set *cond([])* to *0*.
> 
> I don't see any clear reason enforcing a choice rather than the other.
> Do you?

from a mathematical point of view  the Condition number of a matrix A is
defined by

cond(A)=||A|| . ||A^(-1)||
its interest is that when solving A*x=y rounding errors on y (eps) are
amplified to be cond(A)*eps. The optimal value of cond(A) is 1 (for
identity matrix) so for me it looks natural that cond([])=1 .

Philippe

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Re: [Scilab-users] Is cond([]) 0 or 1 ? (bug 15579)

[Rafael Guerra]

Hi Samuel,

Please read the following article on this interesting topic of the algebra of 
empty matrices:
https://pdfs.semanticscholar.org/0f3b/c36f19d5c6a761c19fbc3c4ebde2f31b0a10.pdf

It states that the condition number of the empty matrix should be 0.

Regards,
Rafael

From: users [mailto:users-boun...@lists.scilab.org] On Behalf Of Samuel Gougeon
Sent: Monday, May 21, 2018 3:46 PM
To: International users mailing list for Scilab. <users@lists.scilab.org>
Subject: [Scilab-users] Is cond([]) 0 or 1 ? (bug 15579)

Hello,

Another case with []:
The bug 15579<http://bugzilla.scilab.org/show_bug.cgi?id=15579> reports that 
cond([]) returns 1, while cond([],2) -- that computes the same 2-norm 
conditional number for any other matrix than [] -- returns 0.

Actually, cond([], p) returns 0 for any accepted p value.

At least since Scilab 4, and likely since the beginning of Scilab,  cond([]) is 
set to 1 as a specific case.
I am wondering why.

In my opinion, the mismatch between cond([]) and the equivalent cond([],2) must 
be fixed.
There are at least two way to do it :

  *   either keep cond([]) to 1 and set all cond([], p) to 1 instead of 0
  *   or set cond([]) to 0.

I don't see any clear reason enforcing a choice rather than the other.
Do you?

Let's note that

  *   Octave returns 0 in both cond([]) and cond([],p) cases
  *   Julia, Python, and R (with kappa()) yield an error
Regards
Samuel
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[Scilab-users] Is cond([]) 0 or 1 ? (bug 15579)

[Samuel Gougeon]

Hello,

Another case with []:
The bug 15579  reports 
that *cond([])* returns *1*, while *cond([],2)* -- that computes the 
same 2-norm conditional number for any other matrix than [] -- returns *0*.


Actually, *cond([], p)* returns 0 for any accepted p value.

At least since Scilab 4, and likely since the beginning of Scilab, 
*cond([])* is set to 1 as a specific case.

I am wondering why.

In my opinion, the mismatch between *cond([])* and the equivalent 
*cond([],2)* must be fixed.

There are at least two way to do it :

 * either keep *cond([])* to *1* and set all *cond([], p)* to 1 instead
   of 0
 * or set *cond([])* to *0*.

I don't see any clear reason enforcing a choice rather than the other.
Do you?

Let's note that

 * Octave returns 0 in both cond([]) and cond([],p) cases
 * Julia, Python, and R (with kappa()) yield an error

Regards
Samuel


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