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= aj-i, for aj-i∈ C, i, j≥ 1 , while E is a matrix representing a compact operator in ℓ2. The matrix A is finitely representable if ak= 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 (λ, v) such that Av= λv, with λ∈ C, v=(vj)j∈Z+, v≠ 0 , and ∑j=1∞|vj|2<∞. It is shown that the problem is reduced to a finite nonlinear eigenvalue problem of the kind WU(λ) β= 0 , where W is a constant matrix and U depends on λ 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(λ) = 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].

Computing eigenvalues of semi-infinite quasi-Toeplitz matrices

Iannazzo B.;Robol L.
2022

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= aj-i, for aj-i∈ C, i, j≥ 1 , while E is a matrix representing a compact operator in ℓ2. The matrix A is finitely representable if ak= 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 (λ, v) such that Av= λv, with λ∈ C, v=(vj)j∈Z+, v≠ 0 , and ∑j=1∞|vj|2<∞. It is shown that the problem is reduced to a finite nonlinear eigenvalue problem of the kind WU(λ) β= 0 , where W is a constant matrix and U depends on λ 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(λ) = 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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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11391/1535915
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