A New Family of High-Order Ehrlich-Type Iterative Methods

Petko D Proinov,Maria T Vasileva

doi:10.3390/math9161855

Abstract

One of the famous third-order iterative methods for finding simultaneously all the zeros of a polynomial was introduced by Ehrlich in 1967. In this paper, we construct a new family of high-order iterative methods as a combination of Ehrlich’s iteration function and an arbitrary iteration function. We call these methods Ehrlich’s methods with correction. The paper provides a detailed local convergence analysis of presented iterative methods for a large class of iteration functions. As a consequence, we obtain two types of local convergence theorems as well as semilocal convergence theorems (with computer verifiable initial condition). As special cases of the main results, we study the convergence of several particular iterative methods. The paper ends with some experiments that show the applicability of our semilocal convergence theorems.

Highlights

In 1967, Ehrlich [1] introduced one of the most famous iterative methods for calculating all zeros of a polynomial simultaneously
We provide a local and semilocal convergence of the iterative methods of the new family
We present many convergence results (Corollaries 1–10) for the iterative methods listed in Definition 2

Summary

Introduction

In 1967, Ehrlich [1] introduced one of the most famous iterative methods for calculating all zeros of a polynomial simultaneously. It has a third order of convergence (if all zeros of the polynomial are simple). In 1977, Nourein [4] constructed a fourth-order improved Ehrlich method by combining Ehrlich’s iterative function with Newton’s iterative function. Nowadays this method is known as Ehrlich’s method with Newton’s correction. The latest convergence results for Ehrlich’s method with Newton’s correction can be seen in [9]

Objectives

Methods

Conclusion