Statistical Condition Estimation for the Roots of Polynomials

This paper presents fast and reliable condition estimates for the roots of a real polynomial based on the method of statistical condition estimation (SCE) by Kenney and Laub. Using relative perturbations, we provide both a basic implementation of SCE and an advanced one via the condition of the eigenvalues of the corresponding companion matrix. New results for estimating the condition of eigenvalues are given. Fast structured calculations of the eigenvalues of the companion matrix are used, and fast structured manipulations of the Schur decomposition and the invariant subspaces of the companion matrix are presented. The overall process is based on fast operations for sequentially semiseparable structures with small off-diagonal ranks. The cost of obtaining the condition estimates for all of the roots is $O(n^2)$, where $n$ is the degree of the polynomial. Numerical examples are used to demonstrate the accuracy and efficiency of the proposed methods.