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

Abstract

SUMMARYWe 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 inÿnity, 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.

A Sphere Decoder with Approximate QR Decomposition for Frequency-Selective Channels

“…the sequences Q (n) (m) converge in m to the 0-matrix [9]. The AQRD proposed in this section makes use of the above convergence results to obtain the FQRD of H in the following three steps: 1) Initially, a FQRD is performed on the submatrix of H resulting from selecting its first M B columns, with L < B < T .…”