Computing eigenvalues of semi-infinite quasi-Toeplitz matrices
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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 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 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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