We consider the nonlinear eigenvalue problem
T()x = 0 (1)
where T() is a family of sparse matrices.
Problems of this type arise in damped vibrations of structures, conservative gyroscopic
systems, lateral buckling problems, stability of delay-di erential systems,
uid-solid vibra-
tions, transparent boundary conditions, and the electronic behavior of quantum dot hetero-
structures, to name just a few.
Most of the examples mentioned above are large and sparse, and typically only a small
number of eigenvalues are of interest. Numerical methods have to exploit the sparseness
fully to be ecient in storage and computing time.
In this presentation we review iterative projection methods for sparse nonlinear eigen-
value problems which have proven to be very ecient. Here the eigenvalue problem is
projected to a subspace V of small dimension which yields approximate eigenpairs. If an
error tolerance is not met then the search space V is expanded in an iterative way with the
aim that some of the eigenvalues of the reduced matrix become good approximations of some
of the wanted eigenvalues of the given large matrix. Methods of this type are the nonlinear
Arnoldi method [1], the Jacobi{Davidson method [2, 3], and the rational Krylov method [4],
which are particularly ecient if the eigenvalues of (1) satisfy a minmax property [5].