Condition numbers of a nearly singular simple root of a polynomial

The expressions for the componentwise and normwise condition numbers, kappa (c)(z(0)) and kappa (n)(z(0)) respectively, of a complex root z(0) of a polynomial p(x) that have been developed and used extensively assume that z(0) can be considered in isolation, independently of a neighboring root z(0) +/- epsilon. This assumption is adequate for roots that are well-separated, corresponding to a large value of \ epsilon \, but if epsilon is small such that these two roots are close but distinct, this assumed independence may not be appropriate. In these circumstances, it is reasonable to consider the existence of a 'region of influence', such that if z(0) and z(0) + epsilon are close, the condition numbers kappa (c) (z(0)) and kappa (n) (z(0)) are functions of epsilon. This paper considers the componentwise and normwise condition numbers of a simple root z(0) when there exists a neighboring root z(0) + epsilon where epsilon is small. It is shown that these revised expressions for the condition numbers reduce to the established formulae for a double root when epsilon = 0, but if \ epsilon \ is large, they reduce to the formulae for a simple root. The existing formulae for the condition numbers of a simple and double root are therefore particular instances of the more general expressions that are developed in this paper. (C) 2001 IMACS. Published by Elsevier Science B.V All rights reserved.