A matrix-less and parallel interpolation–extrapolation algorithm for computing the eigenvalues of preconditioned banded symmetric Toeplitz matrices
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In the past few years, Bogoya, Böttcher, Grudsky, and Maximenko obtained the precise asymptotic expansion for the eigenvalues of a Toeplitz matrix Tn(f), under suitable assumptions on the generating function f, as the matrix size n goes to infinity. On the basis of several numerical experiments, it was conjectured by Serra-Capizzano that a completely analogous expansion also holds for the eigenvalues of the preconditioned Toeplitz matrix Tn(u)− 1Tn(v), provided f = v/u is monotone and further conditions on u and v are satisfied. Based on this expansion, we here propose and analyze an interpolation–extrapolation algorithm for computing the eigenvalues of Tn(u)− 1Tn(v). The algorithm is suited for parallel implementation and it may be called “matrix-less” as it does not need to store the entries of the matrix. We illustrate the performance of the algorithm through numerical experiments and we also present its generalization to the case where f = v/u is non-monotone.
KeywordsPreconditioned Toeplitz matrices Eigenvalues Asymptotic eigenvalue expansion Polynomial interpolation Extrapolation
Mathematics Subject Classification (2010)15B05 65F15 65D05 65B05
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