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Preconditioned HSS method for large multilevel block Toeplitz linear systems via the notion of matrix‐valued symbol
Author(s) -
Donatelli Marco,
Garoni Carlo,
Mazza Mariarosa,
SerraCapizzano Stefano,
Sesana Debora
Publication year - 2016
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.2007
Subject(s) - toeplitz matrix , mathematics , hermitian matrix , circulant matrix , eigenvalues and eigenvectors , matrix (chemical analysis) , matrix norm , block matrix , preconditioner , combinatorics , block (permutation group theory) , discrete mathematics , linear system , pure mathematics , mathematical analysis , physics , materials science , quantum mechanics , composite material
Summary We perform a spectral analysis of the preconditioned Hermitian/skew‐Hermitian splitting (PHSS) method applied to multilevel block Toeplitz linear systems in which the coefficient matrix T n ( f ) is associated with a Lebesgue integrable matrix‐valued function f . When the preconditioner is chosen as a Hermitian positive definite multilevel block Toeplitz matrix T n ( g ), the resulting sequence of PHSS iteration matrices M n belongs to the generalized locally Toeplitz class. In this case, we are able to compute the symbol ϕ ( f , g ) describing the asymptotic eigenvalue distribution of M n when n → ∞ and the matrix size diverges. By minimizing the infinity norm of the spectral radius of the symbol ϕ ( f , g ), we are also able to identify effective PHSS preconditioners T n ( g ) for the matrix T n ( f ). A number of numerical experiments are presented and commented, showing that the theoretical results are confirmed and that the spectral analysis leads to efficient PHSS methods. Copyright © 2015 John Wiley & Sons, Ltd.