On Euler-like methods for the simultaneous approximation of polynomial zeros

Abstract

In this paper we consider some iterative methods of higher order for the simultaneous determination of polynomial zeros. The proposed methods are based on Euler's third order method for finding a zero of a given function and involve Weierstrass' correction in the case of simple zeros. We prove that the presented methods have the order of convergence equal to four or more. Based on a fixed-point relation of Euler's type, two inclusion methods are derived. Combining the proposed methods in floating-point arithmetic and complex interval arithmetic, an efficient hybrid method with automatic error bounds is constructed. Computational aspect and the implementation of the presented algorithms on parallel computers are given.