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

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 symbolø.f; g/ describing the asymptotic eigenvalue distribution ofMn when n→∞and the matrix size diverges. By minimizing the infinity norm of the spectral radius of the symbolø.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.