The asymptotic spectrum of flipped multilevel Toeplitz matrices and of certain preconditionings

11/17/2020

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by   M. Mazza, et al.

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In this work, we perform a spectral analysis of flipped multilevel Toeplitz sequences, i.e., we study the asymptotic spectral behaviour of {Y_nT_n(f)}_n, where T_n(f) is a real, square multilevel Toeplitz matrix generated by a function f∈ L^1([-π,π]^d) and Y_n is the exchange matrix, which has 1s on the main anti-diagonal. In line with what we have shown for unilevel flipped Toeplitz matrix sequences, the asymptotic spectrum is determined by a 2× 2 matrix-valued function whose eigenvalues are ± |f|. Furthermore, we characterize the eigenvalue distribution of certain preconditioned flipped multilevel Toeplitz sequences with an analysis that covers both multilevel Toeplitz and circulant preconditioners. Finally, all our findings are illustrated by several numerical experiments.