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 ----------------------------------------------------------- Dipl.-Ing. Eckhard Hennig mailto:hennig at itwm.uni-kl.de Institut fuer Techno- und Wirtschaftsmathematik e.V. (ITWM) Erwin-Schroedinger-Strasse, 67663 Kaiserslautern, Germany Voice: +49-(0)631-205-3126 Fax: +49-(0)631-205-4139 http://www.itwm.uni-kl.de/as/employees/hennig.html ITWM - Makers of Analog Insydes for Mathematica http://www.itwm.uni-kl.de/as/products/ai -----------------------------------------------------------