New simultaneous methods for finding all zeros of a polynomial

TL;DR

This paper introduces three novel simultaneous methods for finding all simple zeros of polynomials, combining classical and derivative-free approaches, with proven convergence and numerical comparisons.

Contribution

The paper presents three new methods for simultaneous polynomial zero-finding, integrating the Weierstrass and Newton's methods with derivative-free techniques, and proves their convergence.

Findings

Methods successfully find all simple zeros of polynomials.

Numerical comparisons demonstrate the efficiency of the new methods.

Convergence of the proposed methods is theoretically established.

Abstract

The purpose of this paper is to present three new methods for finding all simple zeros of polynomials simultaneously. First, we give a new method for finding simultaneously all simple zeros of polynomials constructed by applying the Weierstrass method to the zero in the trapezoidal Newton's method, and prove the convergence of the method. We also present two modified Newton's methods combined with the derivative-free method, which are constructed by applying the derivative-free method to the zero in the trapezoidal Newton's method and the midpoint Newton's method, respectively. Finally, we give a numerical comparison between various simultaneous methods for finding zeros of a polynomial.