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 po lynomial 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 suc h that these two roots are close but distinct, this assumed independence ma y 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) + epsi lon are close, the condition numbers kappa (c) (z(0)) and kappa (n) (z(0)) are functions of epsilon. This paper considers the componentwise and normwi se 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 fo r 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 t he more general expressions that are developed in this paper. (C) 2001 IMAC S. Published by Elsevier Science B.V All rights reserved.