Eigenvector computation for almost unitary Hessenberg matrices and inversion of Szegő-Vandermonde matrices via discrete transmission lines

Abstract

In this paper we use a discrete transmission line model (known to geophysicists as a layered earth model) to derive several computationally efficient solutions for the following three problems: (i) as is well known, a Hessenberg matrix capturing recurrence relations for Szegő polynomials differs from unitary only by its last column. Hence, the first problem is how to rapidly evaluate the eigenvectors of this almost unitary Hessenberg matrix; (ii) the second problem is to design a fast O( n 2) algorithm for inversion of Szegő-Vandermonde matrices (generalizing the well-known Traub algorithm for inversion of the usual Vandermonde matrices); (iii) finally, the third problem is to extend the well-known Horner rule to evaluate a polynomial represented in the basis of Szegő polynomials. As we shall see, all three problems are closely related, and their solutions can be computed by the same family of fast algorithms. Although all the results can be derived algebraically, here we reveal a connection to system theory to deduce these algorithms via elementary operations on signal flow graphs for digital filter structures, including the celebrated Markel-Gray filter, widely used in speech processing, and certain other filter structures. This choice not only clarifies the derivation and suggests a variety of possible computational schemes, but it also makes an interesting connection to many other results related to Szegő polynomials which have already been interpreted via signal flow graphs for (generalized) lattice filter structures, including the formulas of the Gohberg-Semencul type for inversion of Toeplitz-like matrices, Schur-type and Levinson-type algorithms, etc. For example, this connection allows us to show that moment matrices corresponding to what we called Horner-Szegő polynomials, though not Toeplitz, are quasi-Toeplitz, i.e., they have a certain shift-invariance property.