Roots of quaternion polynomials: Theory and computation

Abstract

A quaternion polynomial f(t) in the single variable t, is one whose coefficients are in the skew field H of quaternions. In this manuscript an elementary proof is given of the fact that such an f has a root in H. Moreover, an algorithm is proposed for finding all roots ζ of f(t), along with their multiplicities. The algorithm is based on computing the real part of ζ first, and then using the multiplication rule in H, the imaginary part of ζ is computed via a linear quaternion equation. Several numerical examples are also presented to illustrate the performance of the method. © 2019 Elsevier B.V.