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

We consider the Givens QR factorization of banded Hessenberg-Toeplitz matrices of large order and relatively small bandwidth. We investigate the asymptotic behavior of the R factor and the 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 for 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. Copyright c © 2000 John Wiley & Sons, Ltd.