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 Av= λv, 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 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].