# Weierstrass method for quaternionic polynomial root-finding

## TL;DR

This paper introduces a quaternionic Weierstrass-like method for efficiently finding all zeros of unilateral quaternionic polynomials, supported by convergence analysis and numerical experiments.

## Contribution

It presents the first quaternionic Weierstrass method with convergence analysis and numerical validation for polynomial root-finding.

## Key findings

- The method converges reliably for quaternionic polynomials.
- Numerical examples demonstrate the method's effectiveness.
- The approach extends complex root-finding techniques to quaternions.

## Abstract

Quaternions, introduced by Hamilton in 1843 as a generalization of complex numbers, have found, in more recent years, a wealth of applications in a number of different areas which motivated the design of efficient methods for numerically approximating the zeros of quaternionic polynomials. In fact, one can find in the literature recent contributions to this subject based on the use of complex techniques, but numerical methods relying on quaternion arithmetic remain scarce. In this paper we propose a Weierstrass-like method for finding simultaneously {\sl all} the zeros of unilateral quaternionic polynomials. The convergence analysis and several numerical examples illustrating the performance of the method are also presented.