A stable algorithm to compute the roots of polynomials is presented. The roots are found by computing the eigenvalues of the associated companion matrix by Francis’s implicitly shifted QR algorithm. A companion matrix is an upper Hessenberg matrix that is unitary-plus-rankone, that is, it is the sum of a unitary matrix and a rank-one matrix. These properties are preserved by iterations of Franc...

This paper describes two methods for computing the invariant subspace of a matrix. The rst method involves using transformations to interchange the eigenvalues. The matrix is assumed to be in Schur form and transformations are applied to interchange neighboring blocks. The blocks can be either one by one or two by two. The second method involves the construction of an invariant subspace by a di...

The lgrade of a n × n matrix A is the largest rank of any subdiagonal block of a symmetric partition of A. A number of algebraic results on lgrade are given. When A has lgrade d, it can be be approximately decomposed as A = U + V , where U is an upper triangular matrix and V has rank d. If A satisfies GA = N with G and N have lower bandwidths dG and dN , then the decomposition is exact: A = U +...

We describe an efficient procedure for implementing the Hessenberg QR algorithm on a class of matrices that we refer to as subunitary matrices. This class includes the set of Szegő-Hessenberg matrices, whose characteristic polynomials are Szegő polynomials, i.e., polynomials orthogonal with respect to a measure on the unit circle in the complex plane. Computing the zeros of Szegő polynomials is...

Recently a generalization of Francis’s implicitly shifted QR algorithm was proposed, notably widening the class of matrices admitting low-cost implicit QR steps. This unifying framework covered the methods and theory for Hessenberg and inverse Hessenberg matrices and furnished also new, single-shifted, QR-type methods for, e.g., CMVmatrices. Convergence of this approach was only suggested by nu...

Wheel-rail systems can be modelled as mechanical multibody systems with holonomic constraints. The dynamical behaviour is described by a diierential-algebraic system of index 3 in non{Hessenberg form ((10]). We discuss the eecient numerical solution of these model equations. A perturbation analysis suggests the integration of the model equations in the Gear{Gupta{Leimkuhler (GGL) formulation. T...

Abstract Multiple orthogonal polynomials with respect to two weights on the step-line are considered. A connection between different dual spectral matrices, one banded (recursion matrix) and Hessenberg, respectively, Gauss–Borel factorization of moment matrix is given. It shown a hidden freedom exhibited by system related multiple polynomials. Pearson equations discussed, Laguerre–Freud conside...