Solving Cubic Equations By the Quadratic Formula

TL;DR

This paper introduces a novel method for solving cubic equations by leveraging the Voronoi property of critical points and a specific recurrence sequence, providing an alternative to classical solutions.

Contribution

It characterizes the Voronoi property of critical points in cubic polynomials and develops a new iterative algorithm based on this property for solving cubic equations.

Findings

At least one critical point has the Voronoi property.

The sequence B_m converges to a root of p(z).

The convergence speed depends on root proximity ratios.

Abstract

Let $p(z)$ be a monic cubic complex polynomial with distinct roots and distinct critical points. We say a critical point has the {\it Voronoi property} if it lies in the Voronoi cell of a root $\theta$, $V(\theta)$, i.e. the set of points that are closer to $\theta$ than to the other roots. We prove at least one critical point has the Voronoi property and characterize the cases when both satisfy this property. It is known that for any $\xi \in V(\theta)$, the sequence $B_m(\xi) =\xi - p(\xi) d_{m-2}/d_{m-1}$ converges to $\theta$, where $d_m$ satisfies the recurrence $d_m =p'(\xi)d_{m-1}-0.5 p(\xi)p''(\xi)d_{m-2} +p^2(\xi)d_{m-3}$, $d_0 =1, d_{-1}=d_{-2}=0$. Thus by the Voronoi property, there is a solution $c$ of $p'(z)=0$ where $B_m(c)$ converges to a root of $p(z)$. The speed of convergence is dependent on the ratio of the distances between $c$ and the closest and the second closest roots of $p(z)$. This results in a different algorithm for solving a cubic equation than the classical methods. We give polynomiography for an example.