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In this paper we survey several recent results that highlight an interplay between a relatively new class of quasiseparable matrices and 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,… (More)

We consider the problem of completion of a matrix with a specified lower triangular part to a unitary matrix. In this paper we obtain the necessary and sufficient conditions of existence of a unitary completion without any additional constraints and give a general formula for this completion. The paper is mainly focused on matrices with the specified lower… (More)

In this paper we address the problem of efficiently computing all the eigenvalues of a large N × N Hermitian matrix modified by a possibly non Her-mitian perturbation of low rank. Previously proposed fast adaptations of the QR algorithm are considerably simplified by performing a preliminary transformation of the matrix by similarity into an upper… (More)

In this paper we derive a fast O(n 2) algorithm for solving linear systems where the coefficient matrix is a polynomial-Vandermonde matrix V R (x) = [r j−1 (x i)] with polynomials {r k (x)} related to a Hessenberg quasiseparable matrix. The result generalizes the well-known Björck-Pereyra algorithm for classical Vandermonde systems involving monomials. It… (More)

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