Spectral equivalence and matrix algebra preconditioners for multilevel Toeplitz systems: a negative result

Authors:
D. Noutsos;S. Serra Capizzano;P. Vassalos
Affiliations:
Department of Mathematics, University of Ioannina, Greece;Dipartimento CFM, Università Dell'Insubria - Sede Di Como, Italy;Department of Mathematics, University of Ioannina, Greece
Venue:
Contemporary mathematics
Year:
2001

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Abstract

In the last two decades a lot of matrix algebra optimal and suxper-linear preconditioners (those assuring a strong clustering at the unity) have been proposed for the solution of polynomially ill-conditioned Toeplitz linear systems. The corresponding generalizations to multilevel structures do not preserve optimality neither superlinearity (see e.g. [11]). Regarding the notion of superlinearity, it has been recently shown that this is simply impossible (see [15, 17, 18]). Here we propose some ideas and a proof technique for demonstrating that also the spectral equivalence and the essential spectral equivalence (up to a constant number of diverging eigenvalues) are impossible and therefore the search for optimal matrix algebra preconditioners in the multilevel setting cannot be successful.