Spectral behavior of preconditioned non-Hermitian multilevel block Toeplitz matrices with matrix-valued symbol

This note is devoted to preconditioning strategies for non-Hermitian multilevel block Toeplitz linear systems associated with a multivariate Lebesgue integrable matrix-valued symbol. In particular, we consider special preconditioned matrices, where the preconditioner has a band multilevel block Toeplitz structure, and we complement known results on the localization of the spectrum with global distribution results for the eigenvalues of the preconditioned matrices. In this respect, our main result is as follows. Let Ik:=(-π,π)k, let Ms be the linear space of complex s×s matrices, and let f,g:Ik→Ms be functions whose components fij,gij:Ik→C,i, j=1,⋯,s, belong to L∞. Consider the matrices Tn-1(g)Tn(f), where n:=(n1,⋯,nk) varies in Nk and Tn(f),Tn(g) are the multilevel block Toeplitz matrices of size n1⋯nks generated by f,g. Then {Tn-1(g)Tn(f)}nNk ∼λg- 1f, i.e. the family of matrices {Tn-1(g)Tn(f)}n Nk has a global (asymptotic) spectral distribution described by the function g-1f, provided g possesses certain properties (which ensure in particular the invertibility of Tn(g) for all n) and the following topological conditions are met: the essential range of g-1f, defined as the union of the essential ranges of the eigenvalue functions λj(g-1f),j=1,⋯,s, does not disconnect the complex plane and has empty interior. This result generalizes the one obtained by Donatelli, Neytcheva, Serra-Capizzano in a previous work, concerning the non-preconditioned case g=1. The last part of this note is devoted to numerical experiments, which confirm the theoretical analysis and suggest the choice of optimal GMRES preconditioning techniques to be used for the considered linear systems.