Preconditioned HSS method for large multilevel block Toeplitz linear systems via the notion of matrix‐valued symbol

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.