A unifying approach to abstract matrix algebra preconditioning

In earlier papers Tyrtyshnikov [42] and the first author [14] considered th e analysis of clustering properties of the spectra of specific Toeplitz pre conditioned matrices obtained by means of the best known matrix algebras. Here we generalize this technique to a generic Banach algebra of matrices b y devising general preconditioners related to "convergent" approximation pr ocesses [36]. Finally, as case study, we focus our attention on the Tau preconditioning b y showing how and why the best matrix algebra preconditioners for symmetric Toeplitz systems can be constructed in this class. Mathematics Subject Cla ssification (1991): 65F10, 65F35.