Abstract :
The expressions for the componentwise and normwise condition numbers, View the MathML source and View the MathML source respectively, of a complex root View the MathML source of a polynomial p(x) that have been developed and used extensively assume that View the MathML source can be considered in isolation, independently of a neighboring root View the MathML source. 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 View the MathML source and View the MathML source are close, the condition numbers View the MathML source and View the MathML source are functions of ϵ. This paper considers the componentwise and normwise condition numbers of a simple root View the MathML source when there exists a neighboring root View the MathML source 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.
