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.