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\title{Subspace Iteration Methods with Preconditioning for Eigenvalues of very
       Large Matrices}
\author{Henk van der Vorst~\thanks{Mathematical Institute,
        University of Utrecht, Budapestlaan 6, Utrecht, the
        Netherlands\protect\\
        {\tt e-mail: vorst@math.ruu.nl}}}
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\begin{abstract}
We will discuss iterative methods for the computation of
eigenvalues of $Ax=\lambda x$, for very large sparse matrices $A$.
In particular, we will pay
attention to the recently proposed class of Jacobi-Davidson methods, which
represent versatile approaches for the solution of large sparse eigenproblems.
In these methods correction equations have to be solved for a proper
update of the approximations for an eigenpair, and for the solution of these
correction equations we may employ the arsenal of techniques available
for iterative solution of linear sparse systems, including preconditioning.
An overview of variants of the Jacobi-Davidson method for standard linear
eigenproblems will be given.  Although these methods can be used for the
computation of any desired eigenpair, it turns out that they can be
employed in particular for interior eigenproblems, without the necessity
to solve accurately large linear systems, as is the case in the shift
and invert approach.\\[2mm]
This work reflects joint research with Gerard Sleijpen.
\end{abstract}

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