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Fast Toeplitz eigenvalue computations, joining interpolation-extrapolation matrix-less algorithms and simple-loop theory

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Abstract

Under appropriate technical assumptions, the simple-loop theory allows to derive various types of asymptotic expansions for the eigenvalues of Toeplitz matrices generated by a function f. Independently and under the milder hypothesis that f is even and monotone over [0,π], matrix-less algorithms have been developed for the fast eigenvalue computation of large Toeplitz matrices, within a linear complexity in the matrix order: behind the high efficiency of such algorithms there are the expansions predicted by the simple-loop theory, combined with the extrapolation idea. Here we focus our attention on a change of variable, followed by the asymptotic expansion of the new variable, and we adapt the matrix-less algorithm to the considered new setting. Numerical experiments show a higher precision (till machine precision) and the same linear computation cost, when compared with the matrix-less procedures already presented in the relevant literature. Among the advantages, we concisely mention the following: (a) when the coefficients of the simple-loop function are analytically known, the algorithm computes them perfectly; (b) while the proposed algorithm is better or at worst comparable to the previous ones for computing the inner eigenvalues, it is vastly better for the computation of the extreme eigenvalues; a mild deterioration in the quality of the numerical experiments is observed when dense Toeplitz matrices are considered, having generating function of low smoothness and not satisfying the simple-loop assumptions.

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Notes

  1. Note that the eigenvalues of Xn are real, because Tn(g) is symmetric positive definite and Xn is similar to the symmetric matrix \(T_{n}^{-\frac {1}{2}}(g)T_{n}(l)T_{n}^{-\frac {1}{2}}(g)\).

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Acknowledgements

We acknowledge the reviewers for their careful reading and insightful suggestions that improved our article. Part of the numerical experiments were calculated in the computer center Jürgen Tischer of the mathematics department at Universidad del Valle.

Funding

The first author was partially supported by Universidad del Valle. The third author is partially supported by INdAM.GNCS.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad del Valle, Calle 13 No.100-00, Cali, Colombia

    Manuel Bogoya

  2. Dipartimento di Scienza e Alta Tecnologia, University of Insubria, via Valleggio, 11, Como, 22100, Italy

    Manuel Bogoya & Stefano Serra-Capizzano

  3. Department of Information Technology, Division of Scientific Computing, Uppsala University, Box 337, SE-751 05, Uppsala, Sweden

    Sven-Erik Ekström & Stefano Serra-Capizzano

Authors
  1. Manuel Bogoya
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  2. Sven-Erik Ekström
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  3. Stefano Serra-Capizzano
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Correspondence to Manuel Bogoya.

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Bogoya, M., Ekström, SE. & Serra-Capizzano, S. Fast Toeplitz eigenvalue computations, joining interpolation-extrapolation matrix-less algorithms and simple-loop theory. Numer Algor 91, 1653–1676 (2022). https://doi.org/10.1007/s11075-022-01318-7

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  • DOI: https://doi.org/10.1007/s11075-022-01318-7

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