Re: Re: smooth eigenvalues and eigenvectors as a function of frequency

• To: mathgroup at smc.vnet.net
• Subject: [mg60365] Re: [mg60321] Re: smooth eigenvalues and eigenvectors as a function of frequency
• From: Maria Cristina Dias Tavares <cristina at dsce.fee.unicamp.br>
• Date: Wed, 14 Sep 2005 03:27:32 -0400 (EDT)
• References: <dg05rf\$a1u\$1@smc.vnet.net> <200509131006.GAA09672@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Hi

Once I had this problem in another environment and I used Newton
Raphson to identify the eigenvalues. The important thing to avoid
switchover was to correctly give the seed. I have used as seed the
previous calculated point and that worked.

For instance, when dealing with non-transposed transmission line,
the former seed should be the eigenvector for ideally transposed line.
The following seeds should be calculated as I described above.

Regards,

Eckhard Hennig wrote:

>"Antonio Carlos Siqueira" <acsl at dee.ufrj.br> schrieb im Newsbeitrag
>news:dg05rf\$a1u\$1 at smc.vnet.net...
>
>
>>Dear MathGroup
>>
>>I am posting this message hoping that someone may have a better idea
>>than me and point me in some direction to the solution. I have to fit a
>>complex based function using a state-space approach and thus smooth
>>eigenvectors and eigenfunctions are needed. Using Eigensystem I
>>experienced some eigenvector/eigenvalues switchovers (from one
>>frequency step to the next).
>>I was wondering whether some sort of MapIndexed or some Sort can do the
>>trick to switch the eigenvector back. I know that if I can track the
>>direction of the eigenvalue.eigenvector dot product I might probably
>>identify the switchover.
>>
>>
>
>Hi Antonio,
>
>without having looked into all the details of your code (it's not complete
>anyway) I believe that your problem is one of identifying corresponding
>eigenvectors of two closely related eigenvalue problems: Let A(p) denote a
>square matrix whose entries are functions of a parameter p, and let deltap
>denote a (small) perturbation of p. Let the two eigenvalue problems be
>formulated as
>
>(1)     (A(p) - lambda.I) x = 0
>
>and
>
>(2)     (A(p+deltap) - lambda.I) x = 0
>
>Now assume that (lambda1_1, x1_1) is an eigenvalue/eigenvector pair of (1)
>and that you are interested in finding the corresponding pair (lambda2_1,
>x2_1) in the set of solutions {(lambda2_k, x2_k), k=1..n} of the perturbed
>equation (2). This can be achieved with the help of the modal assurance
>criterion (MAC), defined as
>
>  MAC(x1, x2) = abs(x1'.x2)^2/((x1'.x1)*(x2'.x2))
>
>where x' denotes the complex conjugate transpose of x.
>
>The MAC ranges from 0 to 1. The eigenpair of (2) corresponding to
>(lambda1_1, x1_1) is the one for which
>
>  MAC(x1_1, x2_k)
>
>is closest to 1.
>
>HTH,
>
>Eckhard
>
>
>

--
Profa Maria Cristina Tavares
Faculdade de Engenharia Elétrica e de Computação
UNICAMP / FEEC / DSCE
CP 6101 - CEP 13083-970
tel : (19) 3788 3738
fax : (19) 3289 1395
http://www.dsce.fee.unicamp.br/~cristina

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Hi<br>
<br>
&nbsp;&nbsp;&nbsp; Once I had this problem in another environment and I used Newton
Raphson to identify the eigenvalues. The important thing to avoid
switchover was to correctly give the seed. I have used as seed the
previous calculated point and that worked. <br>
<br>
&nbsp;&nbsp;&nbsp; For instance, when dealing with non-transposed transmission line,
the former seed should be the eigenvector for ideally transposed line.
The following seeds should be calculated as I described above. <br>
<br>
&nbsp;&nbsp;&nbsp; Regards,<br>
<br>
<br>
<br>
<br>
Eckhard Hennig wrote:<br>
<blockquote cite="mid200509131006.GAA09672 at smc.vnet.net" type="cite">
<pre wrap="">"Antonio Carlos Siqueira" <a class="moz-txt-link-rfc2396E" href="mailto:acsl at dee.ufrj.br">&lt;acsl at dee.ufrj.br&gt;</a> schrieb im Newsbeitrag
<a class="moz-txt-link-freetext" href="news:dg05rf\$a1u\$1 at smc.vnet.net">news:dg05rf\$a1u\$1 at smc.vnet.net</a>...
</pre>
<blockquote type="cite">
<pre wrap="">Dear MathGroup

I am posting this message hoping that someone may have a better idea
than me and point me in some direction to the solution. I have to fit a
complex based function using a state-space approach and thus smooth
eigenvectors and eigenfunctions are needed. Using Eigensystem I
experienced some eigenvector/eigenvalues switchovers (from one
frequency step to the next).
I was wondering whether some sort of MapIndexed or some Sort can do the
trick to switch the eigenvector back. I know that if I can track the
direction of the eigenvalue.eigenvector dot product I might probably
identify the switchover.
</pre>
</blockquote>
<pre wrap=""><!---->
Hi Antonio,

without having looked into all the details of your code (it's not complete
anyway) I believe that your problem is one of identifying corresponding
eigenvectors of two closely related eigenvalue problems: Let A(p) denote a
square matrix whose entries are functions of a parameter p, and let deltap
denote a (small) perturbation of p. Let the two eigenvalue problems be
formulated as

(1)     (A(p) - lambda.I) x = 0

and

(2)     (A(p+deltap) - lambda.I) x = 0

Now assume that (lambda1_1, x1_1) is an eigenvalue/eigenvector pair of (1)
and that you are interested in finding the corresponding pair (lambda2_1,
x2_1) in the set of solutions {(lambda2_k, x2_k), k=1..n} of the perturbed
equation (2). This can be achieved with the help of the modal assurance
criterion (MAC), defined as

MAC(x1, x2) = abs(x1'.x2)^2/((x1'.x1)*(x2'.x2))

where x' denotes the complex conjugate transpose of x.

The MAC ranges from 0 to 1. The eigenpair of (2) corresponding to
(lambda1_1, x1_1) is the one for which

MAC(x1_1, x2_k)

is closest to 1.

HTH,

Eckhard

</pre>
</blockquote>
<br>
<br>
<pre class="moz-signature" cols="72">--
Profa Maria Cristina Tavares
Faculdade de Engenharia El&eacute;trica e de Computa&ccedil;&atilde;o
UNICAMP / FEEC / DSCE
CP 6101 - CEP 13083-970
tel : (19) 3788 3738
fax : (19) 3289 1395