High Order Terms for Condition Estimation of Univariate Polynomials

The established expressions for the condition numbers of a root of a polynomial assume that the perturbations in its coefficients are small, such that only the lowest order terms in a Taylor series need be considered. This restriction to the lowest order terms is considered in this paper, and it is shown that there exists a well-defined class of polynomial for which it is not valid. In particular, if a root $x_{0}$ of a polynomial $p(x)$ is close to but distinct from its neighboring roots $x_{i},i=1,\dots,r$, it is shown that it is necessary to consider higher order terms in the Taylor series. These additional terms imply that a closed form expression for the condition number of $x_{0}$ cannot be obtained, and numerical methods must therefore be employed. In this paper, a high order Taylor series is used to compute the region in the complex plane in which the perturbed root $x_{0} + \delta x_{0}$ lies, and this enables the variation of the condition number of $x_{0}$ with the distances to its neighboring roots to be established. It is shown that this extra information, which cannot be obtained if only the lowest order terms are used, is significant.