This talk will review the implicit restarting technique in Krylov projection
methods. Implicit restarting provides a fixed storage Krylov method and
has led to very effective algorithms and software for eigenvalue problems.
Considerable advances in the ability to deal with non-symmetric matrices have
resulted. With appropriate pre-conditioning, linear scaling can be achieved
when a fixed number of eigenvalues with specified properties are sought. A
typical example would be to find a few eigenvalues of largest real part in
order to conduct a linear stability analysis or a bifurcation analysis.
Implicit restarting is the basis for the eigenvalue software package ARPACK.
This package has been used extensively in many large scale applications,
including computational chemistry, molecular dynamics, structural analysis,
semi-conductor laser design, linear stability analysis in CFD and MHD.
Extensions of the methodology have led to effective regularization methods
for seismic inversion applications and also to a method for computing
balanced model reductions for large state space control systems.
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