T. Bella

Classifications of Recurrence Relations via Subclasses of (H, m)-quasiseparable Matrices

The results on characterization of orthogonal polynomials and Szego polynomials via tridiagonal matrices and unitary Hessenberg matrices, respectively, are classical. In a recent paper we observed that tridiagonal matrices and unitary Hessenberg matrices both belong to a wider class of \((H,1)\)-quasiseparable matrices and derived a complete characterization of the latter class via polynomials satisfying certain EGO-type recurrence relations. We also established a characterization of polynomials satisfying three-term recurrence relations via \((H,1)\)-well-free matrices and of polynomials satisfying the Szego-type two-term recurrence relations via \((H,1)\)-semiseparable matrices. In this paper we generalize all of these results from \(scalar\) (H,1) to the block (H, m) case. Specifically, we provide a complete characterization of \((H,\,m)\)-quasiseparable matrices via polynomials satisfying \(block\) EGO-type two-term recurrence relations. Further, \((H,\,m)\)-semiseparable matrices are completely characterized by the polynomials obeying \(block\) Szego-type recurrence relations. Finally, we completely characterize polynomials satisfying m-term recurrence relations via a new class of matrices called \((H,\,m)\)-well-free matrices.

A Fast Björck-Pereyra-Type Algorithm for Solving Hessenberg-Quasiseparable-Vandermonde Systems

A fast

Fast Inversion of Polynomial-Vandermonde Matrices for Polynomial Systems Related to Order One Quasiseparable Matrices

\mathcal{O}(n^2)

A Traub-like algorithm for Hessenberg-quasiseparable-Vandermonde matrices of arbitrary order

algorithm is derived for solving linear systems where the coefficient matrix is a polynomial-Vandermonde matrix

Ranks of Hadamard Matrices and Equivalence of Sylvester—Hadamard and Pseudo-Noise Matrices

V_R(x)=\left[r_{j-1}(x_i)\right]

Equivalence of Hadamard matrices and pseudo-noise matrices

with polynomials

Computations with quasiseparable polynomials and matrices

\{r_k(x)\}

A Björck–Pereyra-type algorithm for Szegö–Vandermonde matrices based on properties of unitary Hessenberg matrices

defined by a Hessenberg matrix with quasiseparable structure. The result generalizes the well-known Bjorck-Pereyra algorithm for classical Vandermonde systems involving monomials. It also generalizes the algorithms of Reichel-Opfer for

Lipschitz stability of canonical Jordan bases of H-selfadjoint matrices under structure-preserving perturbations

V_R(x)

The spectral connection matrix for classical orthogonal polynomials of a single parameter

involving Chebyshev polynomials of Higham for