Newton-Ellipsoid Method and its Polynomiography

TL;DR

This paper introduces the Newton-Ellipsoid method, a novel iterative root-finding algorithm for complex polynomials, combining properties of Newton's method and the Ellipsoid method, with computational results and visualizations called polynomiography.

Contribution

It presents a new root-finding method inspired by optimization techniques, integrating bounds on zeros and plane-cutting properties, extending to higher-order methods like Halley's.

Findings

Successful computational examples demonstrating the method.

Generation of polynomiography visualizations.

Extension to higher-order methods like Halley's.

Abstract

We introduce a new iterative root-finding method for complex polynomials, dubbed {\it Newton-Ellipsoid} method. It is inspired by the Ellipsoid method, a classical method in optimization, and a property of Newton's Method derived in \cite{kalFTA}, according to which at each complex number a half-space can be found containing a root. Newton-Ellipsoid method combines this property, bounds on zeros, together with the plane-cutting properties of the Ellipsoid Method. We present computational results for several examples, as well as corresponding polynomiography. Polynomiography refers to algorithmic visualization of root-finding. Newton's method is the first member of the infinite family of iterations, the {\it basic family}. We also consider general versions of this ellipsoid approach where Newton's method is replaced by a higher-order member of the family such as Halley's method.