Pseudozeros of polynomials and pseudospectra of companion matrices

It is well known that the zeros of a polynomial p are equal to the eigenvalues of the associated companion matrix A. In this paper we take a geometric view of the conditioning of these two problems and of the stability of algorithms for polynomial zerofinding. The -pseudozero setZ (p) is the set of zeros of all polynomials p̂ obtained by coefficientwise perturbations of p of size ≤ ; this is a subset of the complex plane considered earlier by Mosier, and is bounded by a certain generalized lemniscate. The -pseudospectrum Λ (A) is another subset of C defined as the set of eigenvalues of matrices  = A + E with ‖E‖ ≤ ; it is bounded by a level curve of the resolvent of A. We find that if A is first balanced in the usual EISPACK sense, then Z ‖p‖(p) and Λ ‖A‖(A) are usually quite close to one another. It follows that the Matlab ROOTS algorithm of balancing the companion matrix, then computing its eigenvalues, is a stable algorithm for polynomial zerofinding. Experimental comparisons with the Jenkins-Traub (IMSL) and Madsen-Reid (Harwell) Fortran codes confirm that these three algorithms have roughly similar stability properties.