Relationships between different types of initial conditions for simultaneous root finding methods

TL;DR

This paper explores how various initial conditions influence the convergence of simultaneous root-finding methods, offering a new approach to derive verifiable initial conditions from local convergence results for solving polynomial equations.

Contribution

It introduces a novel method to relate different initial conditions, enabling conversion of local convergence theorems into practically verifiable semilocal convergence results.

Findings

Established relationships between initial conditions and convergence guarantees.

Proposed a new approach to derive semilocal convergence results from local theorems.

Enhanced practical applicability of simultaneous root-finding methods.

Abstract

The construction of initial conditions of an iterative method is one of the most important problems in solving nonlinear equations. In this paper, we obtain relationships between different types of initial conditions that guarantee the convergence of iterative methods for simultaneous finding all zeros of a polynomial. In particular, we show that any local convergence theorem for a simultaneous method can be converted into a convergence theorem with computationally verifiable initial conditions which is of practical importance. Thus, we propose a new approach for obtaining semilocal convergence results for simultaneous methods via local convergence results.