In the past few years, Bogoya, Böttcher, Grudsky, and Maximenko obtained for the eigenvalues of a Toeplitz matrix Tn(f), under suitable assumptions on the generating function f, the precise asymptotic expansion 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). 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.
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