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Mathematics of Computation

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How to prove that a preconditioner cannot be superlinear

Authors: S. Serra Capizzano and E. Tyrtyshnikov
Journal: Math. Comp. 72 (2003), 1305-1316
MSC (2000): Primary 15A12, 15A18, 65F10, 47B25
Published electronically: February 3, 2003
MathSciNet review: 1972737
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Abstract: In the general case of multilevel Toeplitz matrices, we recently proved that any multilevel circulant preconditioner is not superlinear (a cluster it may provide cannot be proper). The proof was based on the concept of quasi-equimodular matrices, although this concept does not apply, for example, to the sine-transform matrices. In this paper, with a new concept of partially equimodular matrices, we cover all trigonometric matrix algebras widely used in the literature. We propose a technique for proving the non-superlinearity of certain frequently used preconditioners for some representative sample multilevel matrices. At the same time, we show that these preconditioners are, in a certain sense, the best among the sublinear preconditioners (with only a general cluster) for multilevel Toeplitz matrices.

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Additional Information

S. Serra Capizzano
Affiliation: Dipartimento di Chimica, Fisica e Matematica, Università dell’Insubria - Sede di Como, Via Valleggio 11, 22100 Como, Italy

E. Tyrtyshnikov
Affiliation: Institute of Numerical Mathematics, Russian Academy of Sciences, Gubkina 8, Moscow 117333, Russia

Received by editor(s): May 5, 1998
Received by editor(s) in revised form: March 7, 2001
Published electronically: February 3, 2003
Additional Notes: The work of the second author was supported by the Russian Fund for Basic Research (under grant No. 97-01-00155) and Volkswagen-Stiftung.
Article copyright: © Copyright 2003 American Mathematical Society