This note is concerned with the following problem: For a given matrix A E C’Ix” and a vector a E C”, does there exist a mapping 2’ assigning to each manic polynomial f of degree n a vector X(f) E C” such that the matrix B := A a. X(f)’ is a companion matrix off, i.e., the characteristic polynomial of B is t l)“ftf’ The classes of suitable matrices A and vectors a are characterized, and some pro...

Hamkins and Löwe proved that the modal logic of forcing is S4.2. In this paper, we consider its modal companion, the intermediate logic KC and relate it to the fatal Heyting algebra HZFC of forcing persistent sentences. This Heyting algebra is equationally generic for the class of fatal Heyting algebras. Motivated by these results, we further analyse the class of fatal Heyting algebras.

Several matrix norms of the classical Frobenius companion matrices of a monic polynomial p(z) have been used in the literature to obtain simple lower and upper bounds on the absolute values of the roots λ of p(z). Recently, M. Fiedler has introduced a new family of companion matrices of p(z) (Lin. Alg. Appl., 372 (2003) 325-331) that has received considerable attention and it is natural to inve...

Computing roots of a monic polynomial may be done by computing the eigenvalues of the corresponding companion matrix using for instance the well-known QR-algorithm. We know this algorithm to be backward stable since it computes exact eigenvalues of a slightly modified matrix. But it may yield very poor backward errors in the coefficients of the polynomial. In this paper we investigate what can ...

In this note, we investigate the relationship between the finite and infinite frequency structure of a regular polynomial matrix and that of a particular linearization, called the generalised companion matrix. A special resolvent decomposition of the regular polynomial matrix is proposed which is based on the Weierstrass canonical form of this generalised companion matrix and the solution of a ...

In this note, we investigate the relationship between the finite and infinite frequency structure of a regular polynomial matrix and that of a particular linearization, called the generalised companion matrix. A special resolvent decomposition of the regular polynomial matrix is proposed which is based on the Weierstrass canonical form of this generalised companion matrix and the solution of a ...

In this paper we present a novel matrix method for polynomial rootfinding. We approximate the roots by computing the eigenvalues of a permuted version of the companion matrix associated with the polynomial. This form, referred to as a lower staircase form of the companion matrix in the literature, has a block upper Hessenberg shape with possibly nonsquare subdiagonal blocks. It is shown that th...

This paper presents fast and reliable condition estimates for the roots of a real polynomial based on the method of statistical condition estimation (SCE) by Kenney and Laub. Using relative perturbations, we provide both a basic implementation of SCE and an advanced one via the condition of the eigenvalues of the corresponding companion matrix. New results for estimating the condition of eigenv...