On the stability of computing polynomial roots via confederate linearizations

A common way of computing the roots of a polynomial is to find the eigenvalues of a linearization, such as the companion (when the polynomial is expressed in the monomial basis), colleague (Chebyshev basis) or comrade matrix (general orthogonal polynomial basis). For the monomial case, many studies exist on the stability of linearization-based rootfinding algorithms. By contrast, little seems to be known for other polynomial bases. This paper studies the stability of algorithms that compute the roots via linearization in nonmonomial bases, and has three goals. First we prove normwise stability when the polynomial is properly scaled and the QZ algorithm (as opposed to the more commonly used QR algorithm) is applied to a comrade pencil associated with a Jacobi orthogonal polynomial. Second, we extend a result by Arnold that leads to a first-order expansion of the backward error when the eigenvalues are computed via QR, which shows that the method can be unstable. Finally, we focus on the special case of Chebyshev basis, in particular the Chebfun rootfinder: we discuss its stability and describe an optional functionality, made available for improved stability, for computing the roots of a general continuous function f(x), implemented in the recently updated version 5. The main message is that to guarantee backward stability QZ applied to a properly scaled pencil is necessary.