Ehrlich-type Methods with King's Correction for the Simultaneous Approximation of Polynomial Complex Zeros

Roselaine Neves Machado,Luiz Guerreiro Lopes

doi:10.13189/ms.2019.070406

Abstract

There are many simultaneous iterative methods for approximating complex polynomial zeros, from more traditional numerical algorithms, such as the well-known third order Ehrlich–Aberth method, to the more recent ones. In this paper, we present a new family of combined iterative methods for the simultaneous determination of simple complex zeros of a polynomial, which uses the Ehrlich iteration and a correction based on King's family of iterative methods for nonlinear equations. The use of King's correction allows increasing the convergence order of the basic method from three to six. Some numerical examples are given to illustrate the convergence behaviour and effectiveness of the proposed sixth order Ehrlich-like family of combined iterative methods for the simultaneous approximation of simple complex polynomial zeros.

Highlights

The importance of the problem of finding polynomial zeros in the different branches of Science and Engineering has led to the development of many different methods for their numerical determination
The family of simultaneous iterative methods here proposed is obtained by combining the scheme of the third order Ehrlich iteration (6) with an iterative correction term obtained from King’s fourth order iteration (7), given by
In order to achieve this, and aiming to improve the convergence rate and efficiency of the iterative scheme due to Ehrlich, we propose a family of simultaneous iterative methods constructed on the basis of the third order Ehrlich iteration, combined with a correction based on the optimal fourth order twostep King’s method for nonlinear equations [10]

Summary

Introduction

The importance of the problem of finding polynomial zeros in the different branches of Science and Engineering has led to the development of many different methods for their numerical determination. There is a significant interest in the development of new and efficient numerical iterative methods for determining the zeros of real and complex polynomials. These methods can approximate the zeros of a polynomial either in a sequential or simultaneous manner. 130 Ehrlich-type Methods with King’s Correction for the Simultaneous Approximation of Polynomial Complex Zeros. The family of simultaneous iterative methods here proposed is obtained by combining the scheme of the third order Ehrlich iteration (6) with an iterative correction term obtained from King’s fourth order iteration (7), given by CK (zj ). Using the King approximation zj − CK (zj) in (6) instead of zj, we obtain a new one-parameter family of Ehrlich-type simultaneous iterative methods with King’s correction, defined by. Where CK (zj) is the iterative correction appearing in (8)

Order of convergence of the new family of combined methods

Ehrlich-type simultaneous methods with King’s correction

Numerical examples

Conclusion