Semilocal convergence of two iterative methods for simultaneous computation of polynomial zeros

TL;DR

This paper establishes new semilocal convergence theorems with error bounds for Ehrlich's and Nourein's iterative methods, improving previous results in the simultaneous computation of polynomial zeros.

Contribution

The paper introduces generalized and improved semilocal convergence theorems for Ehrlich's and Nourein's methods, along with a new condition for simple zeros.

Findings

New convergence theorems with error bounds for Ehrlich's and Nourein's methods.

Generalization and improvement of previous convergence results.

A new sufficient condition for identifying simple zeros.

Abstract

In this paper we study some iterative methods for simultaneous approximation of polynomial zeros. We give new semilocal convergence theorems with error bounds for Ehrlich's and Nourein's iterations. Our theorems generalize and improve recent results of Zheng and Huang [J. Comput. Math. 18 (2000), 113--122], Petkovi\'c and Herceg [J. Comput. Appl. Math. 136 (2001), 283--307] and Nedi\'c [Novi Sad J. Math. 31 (2001), 103--111]. We also present a new sufficient condition for simple zeros of a polynomial.