Condition numbers of a nearly singular simple root of a polynomial

Abstract The expressions for the componentwise and normwise condition numbers, κ c (z 0 ) and κ 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 +ϵ . This assumption is adequate for roots that are well-separated, corresponding to a large value of |ϵ| , but if ϵ 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 +ϵ are close, the condition numbers κ c (z 0 ) and κ n (z 0 ) are functions of  ϵ . This paper considers the componentwise and normwise condition numbers of a simple root z 0 when there exists a neighboring root z 0 +ϵ where ϵ is small. It is shown that these revised expressions for the condition numbers reduce to the established formulae for a double root when ϵ=0 , but if |ϵ| 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.