Chebyshev rootfinding via computing eigenvalues of colleague matrices: when is it stable?

Computing the roots of a scalar polynomial, or the eigenvalues of a matrix polynomial, expressed in the Chebyshev basis $\{ T_k(x)\}$ is a fundamental problem that arises in many applications. In this work, we analyze the backward stability of the polynomial rootfinding problem solved with colleague matrices. In other words, given a scalar polynomial $p(x)$ or a matrix polynomial $P(x)$ expressed in the Chebyshev basis, the question is to determine whether the whole set of computed eigenvalues of the colleague matrix, obtained with a backward stable algorithm, like the QR algorithm, are the set of roots of a nearby polynomial or not. In order to do so, we derive a first order backward error analysis of the polynomial rootfinding algorithm using colleague matrices adapting the geometric arguments in [A. Edelman and H. Murakami, \emph{Polynomial roots for companion matrix eigenvalues}, Math. Comp. 210, 763--776, 1995] to the Chebyshev basis. We show that, if the absolute value of the coefficients of $p(x)$ (respectively, the norm of the coefficients of $P(x)$) are bounded by a moderate number, computing the roots of $p(x)$ (respectively, the eigenvalues of $P(x)$) via the eigenvalues of its colleague matrix using a backward stable eigenvalue algorithm is backward stable. This backward error analysis also expands on the very recent work [Y. Nakatsukasa and V. Noferini, \emph{On the stability of computing polynomial roots via confederate linearizations}, To appear in Math. Comp.] that already showed that this algorithm is not backward normwise stable if the coefficients of the polynomial $p(x)$ do not have moderate norms.