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In this chapter we investigated the convergence behavior of the
reduction algorithms in detail. A general framework was provided,
which enables us to classify similarity transformations. Two easy to
check conditions should be placed on such a similarity transformation in
order to have the Arnoldi-Ritz values in the already reduced part of
the matrix. We showed that the tridiagonalization and the
reduction to semiseparable form satisfy the desired properties, and
therefore also have the predicted convergence behavior.
Moreover we indicated in this chapter that the reduction to
semiseparable form has an additional convergence behavior with respect
to the tridiagonalization. It was proven that during the reduction to
semiseparable form, some kind of nested subspace iteration is
performed. This convergence behavior clearly interacts with the
convergence towards the Arnoldi-Ritz values. The subspace iteration
starts converging as soon as the Arnoldi-Ritz values approximate the
dominant eigenvalues of the original matrix well enough.

Raf Vandebril
2004-05-03