A general semilocal convergence theorem for simultaneous methods for polynomial zeros and its applications to Ehrlich's and Dochev-Byrnev's methods

TL;DR

This paper presents a general semilocal convergence theorem for simultaneous polynomial root-finding methods, improving existing results and establishing the equivalence of Dochev-Byrnev's and Prešić-Tanabe's methods.

Contribution

It introduces a new convergence theorem with verifiable initial conditions and applies it to enhance understanding of Ehrlich's and Dochev-Byrnev's methods.

Findings

New semilocal convergence results for Ehrlich's and Dochev-Byrnev's methods

Improved convergence conditions over previous studies

Proof of equivalence between Dochev-Byrnev's and Prešić-Tanabe's methods

Abstract

In this paper, we establish a general semilocal convergence theorem (with computationally verifiable initial conditions and error estimates) for iterative methods for simultaneous approximation of polynomial zeros. As application of this theorem, we provide new semilocal convergence results for Ehrlich's and Dochev-Byrnev's root-finding methods. These results improve the results of Petkovi\'c, Herceg and Ili\'c [Numer. Algorithms 17 (1998) 313--331] and Proinov [C.~R. Acad. Bulg. Sci. 59 (2006) 705--712]. We also prove that Dochev-Byrnev's method (1964) is identical to Pre{\v s}i\'c-Tanabe's method (1972).