Preconditioned HSS method for large multilevel block Toeplitz linear systems via the notion of matrix‐valued symbol
Marco Donatelli, Carlo Garoni, Mariarosa Mazza, Stefano Serra‐Capizzano, Debora Sesana · Numerical Linear Algebra with Applications · 2015
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 Tn(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 Tn(g), the resulting sequence of PHSS iteration matrices Mn 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 Mnwhen 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 Tn(g) for the matrix Tn(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.