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Computing eigenvalues of semi-infinite quasi-Toeplitz matrices

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Abstract

A quasi-Toeplitz (QT) matrix is a semi-infinite matrix of the form \(A=T(a)+E\) where T(a) is the Toeplitz matrix with entries \((T(a))_{i,j}=a_{j-i}\), for \(a_{j-i}\in \mathbb {C}\), \(i,j\ge 1\), while E is a matrix representing a compact operator in \(\ell ^2\). The matrix A is finitely representable if \(a_k=0\) for \(k<-m\) and for \(k>n\), given \(m,n>0\), and if E has a finite number of nonzero entries. The problem of numerically computing eigenpairs of a finitely representable QT matrix is investigated, i.e., pairs \((\lambda ,\varvec{v})\) such that \(A\varvec{v}=\lambda \varvec{v}\), with \(\lambda \in \mathbb {C}\), \(\varvec{v}=(v_j)_{j\in \mathbb {Z}^+}\), \(\varvec{v}\ne 0\), and \({\sum }_{j=1}^\infty |v_j|^2<\infty\). It is shown that the problem is reduced to a finite nonlinear eigenvalue problem of the kind \(WU(\lambda )\varvec{\beta }=0\), where W is a constant matrix and U depends on \(\lambda\) and can be given in terms of either a Vandermonde matrix or a companion matrix. Algorithms relying on Newton’s method applied to the equation det \(WU(\lambda )=0\) are analyzed. Numerical experiments show the effectiveness of this approach. The algorithms have been included in the CQT-Toolbox [Numer. Algorithms 81 (2019), no. 2, 741–769].

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Acknowledgements

The first author wishes to thank Matthew Colbrook and Mark Embree for helpful conversations and comments. The second author would like to thank Dimitri Breda for useful discussions.

Funding

This work has been partially supported by University of Pisa’s project PRA_2020_61, and by GNCS of INdAM.

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Correspondence to J. Meng.

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Dedicated to Claude Brezinski on the occasion of his 80th birthday.

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Bini, D.A., Iannazzo, B., Meini, B. et al. Computing eigenvalues of semi-infinite quasi-Toeplitz matrices. Numer Algor 92, 89–118 (2023). https://doi.org/10.1007/s11075-022-01381-0

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