On using symmetric polynomials for constructing root finding methods

TL;DR

This paper introduces a novel approach to polynomial root finding using symmetric polynomials, reconstructing existing methods and proposing modifications to the Durand-Kerner method for improved performance.

Contribution

It presents a new framework based on symmetric polynomials for constructing and modifying simultaneous root-finding algorithms.

Findings

Reconstructed existing root-finding methods within the symmetric polynomial framework

Proposed modifications to the Durand-Kerner method

Demonstrated potential improvements in convergence or stability

Abstract

We propose an approach to constructing iterative methods for finding polynomial roots simultaneously. One feature of this approach is using the fundamental theorem of symmetric polynomials. Within this framework, we reconstruct many of the existing root finding methods. The new results presented in this paper are some modifications of the Durand-Kerner method.