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Re: Eigensystems

• To: mathgroup at smc.vnet.net
• Subject: [mg18906] Re: Eigensystems
• From: Eckhard Hennig <hennig at itwm.uni-kl.de>
• Date: Mon, 26 Jul 1999 14:27:47 -0400
• Organization: ITWM
• References: <7nef9i$238@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Tony Harker wrote in message <7nef9i$238 at smc.vnet.net>...
>
>  Can anybody recommend an efficient method for solving the modified
>eigensystem
>       A x = lambda B x
>numerically in Mathematica?


Dear Tony,

this question is raised every now and then in this newsgroup, and the
answers are usually on the line of "Try solving B^(-1)A - lambda x = 0. If B
is singular, then add some small perturbations first." To obtain a *useful*
;-) answer, you need to give some more details of the generalized eigenvalue
problem (GEP) you wish to solve. The answers to the following questions will
help to tell whether a particular numerical algorithm is appropriate (and
efficient) for your applications.

- how large is your GEP?
- is the GEP symmetric?
- is the GEP sparse or dense?
- is B singular (presumably, it is)?
- is rank(B) approximately equal to dim(B) or is rank(B) << dim(B)?
- what is the typical spectral radius of your GEPs (i.e. stiff or non-stiff
equations)?
- do you need to compute the complete spectrum or just one (or a few)
eigenvalue(s)?
- do you need to compute the (left and/or right) eigenvectors?
- for a particular GEP, can you specify good initial guesses for the
eigenpairs of interest?
- do you wish to solve parametric GEPs efficiently (i.e. track one or more
eigenvalues as some parameters are varied)?

Best regards,

  Eckhard

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Dipl.-Ing. Eckhard Hennig      mailto:hennig at itwm.uni-kl.de
Institut fuer Techno- und Wirtschaftsmathematik e.V. (ITWM)
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