Asymptotic properties of the QR factorization of banded Hessenberg-Toeplitz matrices

We consider Givens QR factorization of banded Hessenberg-Toeplitz matrices of large order and relatively small bandwidth. We investigate the asymptotic behaviour of the R factor and Givens rotation when the order of the matrix goes to infinity, and present some interesting convergence properties. These properties can lead to savings in the computation of the exact QR factorization and give insight into the approximate QR factorizations of interest in preconditioning. The properties also reveal the relation between the limit of the main diagonal elements of R and the largest absolute root of a polynomial.