Quasiseparable Matrices and Polynomials

Date of Completion

January 2010






The interplay between structured matrices and corresponding systems of polynomials is a classical topic, and two classical matrices: Jacobi (tridiagonal) and unitary Hessenberg 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. It has been shown recently that a family of low rank structured matrices called quasiseparable include both unitary Hessenberg and tridiagonal matrices thus allowing one to obtain true generalizations of several classical algorithms. Quasiseparable matrices also arise in many applications in linear system theory and control, statistics, mechanics, orthogonal polynomials and others. This justifies why quasiseparable matrices have been among the hottest research topics in Numerical Linear Algebra in recent years. ^ We present several results obtained for quasiseparable matrices and related areas. First, we describe the results of error analysis of several published quasiseparable system solvers that indicate that only one of them is a provably backward stable algorithm while the others are not. To the best of our knowledge, this is the first error analysis result for this type of matrices. Second, we obtained a classification of the subcasses of Hessenberg quasiseparable matrices via the recurrence relations satisfied by their characteristic polynomials and vice versa. This results let us to generalize classical fast Traub-like algorithms for inversion of polynomial Vandermode matrices to more general classes of polynomials. Next, we generalized some already classical results for CMV matrices, important in orthogonal polynomials theory. We also generalize the celebrated Markel-Grey filter signal flow graph structure and Kimura structure. Finally, we study the relation between signal flow graphs, quasiseparable matrices and numerical linear algebra algorithms. ^