In this paper a new family of companion forms associated to a regular polynomial matrix is presented. Similar results have been presented in a recent paper by M. Fiedler, where the scalar case is considered. It is shown that the new family of companion forms preserves both the finite and infinite elementary divisors structure of the original polynomial matrix, thus all its members can be seen a...

We describe a BDI (Belief, Desire, Intention) goal-oriented architecture for a conversational virtual companion embodied as a child’s Toy, designed to be both entertaining and capable of carrying out collaborative tasks. We argue that the goal-oriented approach supports both structured conversational activities (e.g., story-telling, collaborative games) as well as more “free-flowing” engaging d...

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...

Computing roots of scalar polynomials as the eigenvalues of Frobenius companion matrices using backward stable eigenvalue algorithms is a classical approach. The introduction of new families of companion matrices allows for the use of other matrices in the root-finding problem. In this paper, we analyze the backward stability of polynomial root-finding algorithms via Fiedler companion matrices....

We explore the computation of roots of polynomials via eigenvalue problems. In particular, we look at the case when the leading coeecient is relatively very small. We argue that the companion matrix algorithm (used, for instance, by the Matlab roots function) is inaccurate in this case. The accuracy problem is addressed by using matrix pencils instead. This improvement can be predicted from the...

Matrix methods are increasingly popular for polynomial root-finding. The idea is to approximate the roots as the eigenvalues of the companion or generalized companion matrix associated with an input polynomial. The algorithms also solve secular equation. QR algorithm is the most customary method for eigen-solving, but we explore the inverse Rayleigh quotient iteration instead, which turns out t...

In [1] and [2] a new family of companion forms associated to a regular polynomial matrix T (s) has been presented, using products of permutations of n elementary matrices, generalizing similar results presented in [3] where the scalar case was considered. In this paper, extending this “permuted factors” approach, we present a broader family of companion like linearizations, using products of up...

We obtain the singular value decomposition of multi-companion matrices. We completely characterise the columns of the matrix U and give a simple formula for obtaining the columns of the other unitary matrix, V , from the columns of U . We also obtain necessary and sufficient conditions for the related matrix polynomial to be hyperbolic.