On the convergence of Gander’s type family of iterative methods for simultaneous approximation of polynomial zeros

Abstract

In this paper, we propose a fifth-order family of iterative methods for approximation of all zeros of a polynomial simultaneously. The new family is developed by combining Gander’s third-order family of iterative methods with the second-order Weierstrass root-finding method. The aim of the paper is to state initial conditions that provide local and semilocal convergence of the proposed methods as well as a priori and a posteriori error estimates. In the case of semilocal convergence the initial conditions and error estimates are computationally verifiable which is of practical importance.