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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