On a family of Weierstrass-type root-finding methods with accelerated convergence

TL;DR

This paper introduces a family of Weierstrass-type root-finding methods with increasing convergence order, providing semilocal convergence results and verifiable initial conditions for all methods in the family.

Contribution

It extends existing local convergence results to semilocal convergence for a whole family of Weierstrass-type methods, with practical initial condition criteria.

Findings

Methods have convergence order N+1 for the Nth method.

Semilocal convergence results are established.

Verifiable initial conditions and error estimates are provided.

Abstract

Kyurkchiev and Andreev (1985) constructed an infinite sequence of Weierstrass-type iterative methods for approximating all zeros of a polynomial simultaneously. The first member of this sequence of iterative methods is the famous method of Weierstrass (1891) and the second one is the method of Nourein (1977). For a given integer $N \ge 1$, the $N$th method of this family has the order of convergence ${N+1}$. Currently in the literature, there are only local convergence results for these methods. The main purpose of this paper is to present semilocal convergence results for the Weierstrass-type methods under computationally verifiable initial conditions and with computationally verifiable a posteriori error estimates.