A generalization of the Jenkins-Traub method

Abstract

A class of methods for finding zeros of polynomials is derived which depends upon an arbitrary parameter ρ \rho . The Jenkins-Traub algorithm is a special case, corresponding to the choice ρ = ∞ \rho = \infty . Global convergence is proved for large and small values of ρ \rho and a duality between pairs of members is exhibited. Finally, we show that many members of the class (including the Jenkins-Traub method) converge with R-order at least 2.618..., which improves upon the result obtained by Jenkins and Traub [3].