Backward stability of polynomial root-finding using Fiedler companion matrices

Abstract

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. In other words, given a polynomial p(z), the question is to determine whether the whole set of computed eigenvalues of the companion matrix, obtained with a backward stable algorithm for the standard eigenvalue problem, are the set of roots of a nearby polynomial or not. We show that, if the coefficients of p(z) are bounded in absolute value by a moderate number, then algorithms for polynomial root-finding using Fiedler matrices are backward stable, and Fiedler matrices are as good as the Frobenius companion matrices. This allows us to use Fiedler companion matrices with favorable structures in the polynomial root-finding problem. However, when some of the coefficients of the polynomial are large, Fiedler companion matrices may produce larger backward errors than Frobenius companion matrices, although in this case neither Frobenius nor Fiedler matrices lead to backward stable computations. To prove this we obtain explicit expressions for the change, to first order, of the characteristic polynomial coefficients of Fielder matrices under small perturbations. We show that, for all Fiedler matrices except the Frobenius ones, this change involves quadratic terms in the coefficients of the characteristic polynomial of the original matrix, while for the Frobenius matrices it only involves linear terms. We present extensive numerical experiments that support these theoretical results. The effect of balancing these matrices is also investigated.

Fiedler-comrade and Fiedler--Chebyshev pencils

“…[30], have been studied in the monomial basis for many years since their invention (for the monic case) by Fiedler [13]. Applications include, to name but a few, rootfinding [11], polynomial eigenvalue problems [18], and the design of structured linearizations [4]. We hope that Fiedler pencils in the Chebyshev and related bases, here introduced, will lead to a similarly fruitful research line in the next future.…”

“…One family of particular interest is Fiedler pencils (and Fiedler matrices), introduced in [13] and since then deeply studied and generalized in many directions, see for example [2,4,7,11,30] and the references therein. Among Fiedler pencils we find, for instance, companion linearizations (the monomial analogues of the colleague), the particular Fiedler pencil analyzed in [18] (particularly advantageous for the QZ algorithm), and pentadiagonal linearizations (also potentially advantageous numerically, although currently lacking an algorithm capable to fully exploit the small bandwidth).…”

Linearizations of matrix polynomials in Newton bases

“…In the last 14 years Fiedler pencils for scalar and matrix polynomials expressed in the standard basis have been extensively studied [3,4,8,9,10,11,13,19,31]; this includes work [10,11,19] concerned especially with the numerical properties of Fiedler pencils, and with algorithms for solving polynomial eigenvalue problems based on these pencils. In this section we extend the notion of Fiedler pencils and show how to adapt them to polynomials expressed in a Newton basis.…”

Matrix structures and applications

“…Among the most popular companion matrices we recall the first and second Frobenius forms F 1 and F 2 given by ( [36])…”