Interplay between structured matrices and corresponding systems of polynomials is a classical topic, and two classical matrix classes, Jacobi (tridiagonal) matrices and unitary Hessenberg matrices that are often studied in this context are known to correspond to real orthogonal polynomials and Szegö polynomials, respectively. These two polynomial families arise in a wide variety of applications, and their short recurrence relations are often at the heart of a number of fast algorithms involving them. Historically, algorithms of this type have been developed first for real orthogonal polynomials, however, recently, several important algorithms originally derived for real orthogonal polynomials have subsequently been carried over to the class of Szegö polynomials. Such new algorithms tend to exploit the specific new structure, and thus are valid only for the Szegö polynomials; that is, they are analogues and not generalizations of the original algorithms.
We present several results recently obtained for the "superclass" of quasiseparable matrices, the latter class includes both Jacobi and unitary Hessenberg matrices. Hence the interplay between quasiseparable matrices and their polynomial systems (which contain both real orthogonal and Szegö polynomials) allows one to obtain true generalizations of several algorithms. Included herein are the Björck-Pereyra algorithm, the Traub algorithm, certain new digital filter structures, as well as QR and divide and conquer eigenvalue algorithms.
Other results in structured matrices presented include a result on the possible effects of small, structure-preserving perturbations of a matrix self-adjoint with respect to an indefinite inner product on the so-called canonical Jordan bases of said matrix, and a result regarding Hadamard-Sylvester matrices in the theory of algebraic coding theory.
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