Computations with quasiseparable polynomials and matrices

In this paper, we survey several recent results that highlight an interplay between a relatively new class of quasiseparable matrices and univariate polynomials. Quasiseparable matrices generalize two classical matrix classes, Jacobi (tridiagonal) matrices and unitary Hessenberg matrices that are known to correspond to real orthogonal polynomials and Szegö polynomials, respectively. The latter two polynomial families arise in a wide variety of applications, and their short recurrence relations are the basis for a number of efficient algorithms. For historical reasons, algorithm development is more advanced for real orthogonal polynomials. Recent variations of these algorithms tend to be valid only for the Szegö polynomials; they are analogues and not generalizations of the original algorithms. Herein, we survey several recent results for the "superclass" of quasiseparable matrices, which includes both Jacobi and unitary Hessenberg matrices as special cases. The interplay between quasiseparable matrices and their associated polynomial sequences (which contain both real orthogonal and Szegö polynomials) allows one to obtain true generalizations of several algorithms. Specifically, we discuss the Björck-Pereyra algorithm, the Traub algorithm, certain new digital filter structures, as well as QR and divide and conquer eigenvalue algorithms.