Algorithms and Polynomiography for Solving Quaternion Quadratic Equations

TL;DR

This paper explores algorithms and visualizations for solving quaternion quadratic equations, simplifying the process through polynomial reduction, analyzing iterative methods like Newton and Halley, and creating polynomiography to illustrate convergence behaviors.

Contribution

It introduces a simplified approach to solving quaternion quadratic equations and develops 2D polynomiography based on Newton and Halley methods, analyzing their convergence properties.

Findings

Zeros of quaternion quadratics can be found via real quartic equations.

Polynomiography visualizes iterative convergence of Newton and Halley methods.

Measured rendering times reveal relative speeds of convergence.

Abstract

Solving a quadratic equation $P(x)=ax^2+bx+c=0$ with real coefficients is known to middle school students. Solving the equation over the quaternions is not straightforward. Huang and So \cite{Huang} give a complete set of formulas, breaking it into several cases depending on the coefficients. From a result of the second author in \cite{kalQ}, zeros of $P(x)$ can be expressed in terms of the zeros of a real quartic equation. This drastically simplifies solving a quadratic equation. Here we also consider solving $P(x)=0$ iteratively via Newton and Halley methods developed in \cite{kalQ}. We prove a property of the Jacobian of Newton and Halley methods and describe several 2D polynomiography based on these methods. The images not only encode the outcome of the iterative process, but by measuring the time taken to render them we find the relative speed of convergence for the methods.