Computing eigenvalues of semi-infinite quasi-Toeplitz matrices

A quasi-Toeplitz (QT) matrix is a semi-infinite of the form \(A=T(a)+E\) where T(a) Toeplitz with entries \((T(a))_{i,j}=a_{j-i}\), for \(a_{j-i}\in \mathbb {C}\), \(i,j\ge 1\), while E representing compact operator in \(\ell ^2\). The finitely representable if \(a_k=0\) \(k<-m\) and \(k>n\), given \(m,n>0\), has finite number nonzero entries. problem numerically computing eigenpairs QT investigated, i.e., pairs \((\lambda ,\varvec{v})\) such that \(A\varvec{v}=\lambda \varvec{v}\), \(\lambda \in \(\varvec{v}=(v_j)_{j\in {Z}^+}\), \(\varvec{v}\ne 0\), \({\sum }_{j=1}^\infty |v_j|^2<\infty\). It shown reduced to nonlinear eigenvalue kind \(WU(\lambda )\varvec{\beta }=0\), W constant U depends on \(\lambda\) can be terms either Vandermonde or companion matrix. Algorithms relying Newton’s method applied equation det )=0\) are analyzed. Numerical experiments show effectiveness this approach. algorithms have been included CQT-Toolbox [Numer. 81 (2019), no. 2, 741–769].