An efficient QR algorithm for a Hessenberg submatrix of a unitary matrix

Abstract

We describe an efficient procedure for implementing the Hessenberg QR algorithm on a class of matrices that we refer to as subunitary matrices. This class includes the set of Szego-Hessenberg matrices, whose characteristic polynomials are Szego polynomials, i.e., polynomials orthogonal with respect to a measure on the unit circle in the complex plane. Computing the zeros of Szego polynomials is important in time series analysis and in the design and implementation of digital filters. For example, these zeros are the poles of autoregressive filters. © Springer-Verlag Berlin Heidelberg 2005.