On Ill-conditioned Eigenvalues, Multiple Roots of Polynomials, and Their Accurate Computations

Algebraic eigenvalues with associated left and right eigenvectors (nearly) orthogonal, and polynomial roots that are multiple, have been known to be sensitive to perturbations in numerical computation and thereby ill-conditioned. By constructing an extended polynomial system, it is proved that, under certain conditions, traditionally deened ill-conditioned eigenvalues and polynomial roots can be remarkably stable and insensitive to perturbations, if they are computed as numerical multiple eigenvalues/roots. Furthermore, using the extended polynomial system which is easy to solve, those numerical multiple eigenvalues can be computed with high forward accuracy, while backward errors are estimated as by-products through auxiliary variables of the extended system.