Preconditioned Eigensolvers - an Oxymoron?

A short survey of some results on preconditioned iterative methods for symmetric eigenvalue problems is presented. The survey is in no way complete and reflects author''s personal interests and biases, with emphasis on author''s own contributions. The author tries to mention the majority of important theoretical results and ideas from the Soviet literature on the subject, adding references from the other side of the Iron Curtain usually just to preserve the integrity of the topic. An attempt is made to introduce a systematic classification of preconditioned eigensolvers, separating the choice of a preconditioner from the choice of an iterative method. A formal definition of a preconditioned eigensolver is given. Recent developments in the area are mainly ignored, in particular, on Davidson''s method. Domain decomposition methods for eigenproblems are included in the framework of preconditioned eigensolvers. The paper has more the style of a popular presentation, and can serve as an introduction into the area.

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