Preconditioned HSS method for large multilevel block Toeplitz linear systems via the notion of matrix‐valued symbol. Issue 1 (27th August 2015)

• Record Type: Journal Article
• Authors: Donatelli, Marco; Garoni, Carlo; Mazza, Mariarosa; Serra‐Capizzano, Stefano; Sesana, Debora
• Is Part Of: Numerical linear algebra with applications. Volume 23:Issue 1(2016:Jan.)
• Page Start: 83
• Page End: 119
• Publication Date: 2015-08-27
• Subjects: preconditioned HSS method -- Toeplitz matrix -- Toeplitz preconditioning -- eigenvalue distribution -- symbol; Algebras, Linear -- Periodicals; 512.5
• DOI: 10.1002/nla.2007
• Languages: English
• ISSNs: 1070-5325

Abstract

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.