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Regularizing properties of a class of matrices including the optimal and the superoptimal preconditioners
Author(s) -
Cipolla Stefano,
Di Fiore Carmine,
Durastante Fabio,
Zellini Paolo
Publication year - 2019
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2225
Subject(s) - toeplitz matrix , mathematics , class (philosophy) , matrix (chemical analysis) , unitary matrix , unitary state , positive definite matrix , algebra over a field , pure mathematics , eigenvalues and eigenvectors , computer science , law , materials science , physics , quantum mechanics , artificial intelligence , political science , composite material
Summary In this work, given a positive definite matrix A , we introduce a class of matrices related to A , which is obtained by suitably combining projections of its powers onto algebras of matrices simultaneously diagonalized by a unitary transform. After a detailed theoretical study of some spectral properties of the matrices of this class, which suggests their use as regularizing preconditioners, we prove that such matrices can be cheaply computed when the matrix A has a Toeplitz structure. We provide numerical evidence of the advantages coming from the employment of the proposed preconditioners when used in regularizing procedures.