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), is an element of 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 (2, v) such that Av = lambda v, with lambda is an element of C, v = (v(j))(j is an element of z+), v not equal 0, and Sigma(infinity)(j=1) vertical bar v(j)vertical bar(2) < infinity. It is shown that the problem is reduced to a finite nonlinear eigenvalue problem of the kind WU(lambda)beta = 0, where W is a constant matrix and U depends on lambda and can be given in terms of either a Van-dermonde 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].