We discuss formal orthogonal polynomials with respect to a moment matrix that has no structure whatsoever. In the classical case the moment matrix is often a Hankel or a Toeplitz matrix. We link this to block factorization of the moment matrix and its inverse, the block Hessenberg matrix of the recurrence relation, the computation of successive Schur complements and general subspace iterative m...

We consider a standard matrix ow on the set of unitary upper Hessenberg matrices with nonnegative subdiagonal elements. The Schur parametrization of this set of matrices leads to ordinary diier-ential equations for the weights and the parameters that are analogous with the Toda ow as identiied with a ow on Jacobi matrices. We derive explicit diierential equations for the ow on the Schur paramet...

Various strategies have been proposed for arriving at block algorithms for reducing a general matrix to Hessenberg form by means of orthogonal similarity transformations. This paper reviews and systematicallycategorizes the various strategies and discusses their computational characteristics.

We show that a unitary upper Hessenberg matrix with positive subdiago-nal elements is uniquely determined by its eigenvalues and the eigenvalues of a modiied principal submatrix. This provides an analog of a well-known result for Jacobi matrices.

In this paper we derive a fast O(n) algorithm for solving linear systems where the coefficient matrix is a polynomial-Vandermonde matrix VR(x) = [rj−1(xi)] with polynomials {rk(x)} related to a Hessenberg quasiseparable matrix. The result generalizes the well-known Björck-Pereyra algorithm for classical Vandermonde systems involving monomials. It also generalizes the algorithms of [RO91] for VR...

This paper proposes a combination of a hybrid CPU–GPU and a pure GPU software implementation of a direct algorithm for solving shifted linear systems (A− σI)X = B with large number of complex shifts σ and multiple right-hand sides. Such problems often appear e.g. in control theory when evaluating the transfer function, or as a part of an algorithm performing interpolatory model reduction, as we...

The eigenvalues of a matrix A are the zeros of its characteristic polynomial fiX) = dtt[A XI]. With Hyman's method of determinant evaluation, a new homotopy continuation method, homotopy-determinant method, is developed in this paper for finding all eigenvalues of a real upper Hessenberg matrix. In contrast to other homotopy continuation methods, the homotopy-determinant method calculates eigen...

This paper proposes a new type of iteration based on a structured rank factorization for computing eigenvalues of semiseparable and semiseparable plus diagonal matrices. Also the case of higher order semiseparability ranks is included. More precisely, instead of the traditional QR-iteration, a QH-iteration will be used. The QH-factorization is characterized by a unitary matrix Q and a Hessenber...

It has been shown in [4, 5, 6, 31] that the Hessenberg iterates of a companion matrix under the QR iterations have low off-diagonal rank structures. Such invariant rank structures were exploited therein to design fast QR iteration algorithms for finding eigenvalues of companion matrices. These algorithms require only O(n) storage and run in O(n) time where n is the dimension of the matrix. In t...