Computing eigenvalues of semi-infinite quasi-Toeplitz matrices.

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 ∈ C , i , j ≥ 1 , while E is a matrix representing a compact operator in ℓ 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 (λ , v) such that A v = λ v , with λ ∈ C , v = (v j) j ∈ Z + , v ≠ 0 , and ∑ j = 1 ∞ | v j | 2 < ∞ . It is shown that the problem is reduced to a finite nonlinear eigenvalue problem of the kind W U (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 W U (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]. [ABSTRACT FROM AUTHOR]