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 --------------040509060604010109030607 <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"> <html> <head> <meta content="text/html;charset=ISO-8859-1" http-equiv="Content-Type"> <title></title> </head> <body bgcolor="#ffffff" text="#000000"> Hi<br> <br> 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> 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> 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"><acsl at dee.ufrj.br></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 Elétrica e de Computação UNICAMP / FEEC / DSCE CP 6101 - CEP 13083-970 tel : (19) 3788 3738 fax : (19) 3289 1395 <a class="moz-txt-link-freetext" href="http://www.dsce.fee.unicamp.br/~cristina";>http://www.dsce.fee.unicamp.br/~cristina</a></pre> </body> </html> --------------040509060604010109030607--

**References**:**Re: smooth eigenvalues and eigenvectors as a function of frequency***From:*"Eckhard Hennig" <aidev@n-o-s-p-a-m.kaninkolo.de>

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