Algorithms for quaternion polynomial root-finding

Abstract

In 1941 Niven pioneered root-finding for a quaternion polynomial P(x), proving the fundamental theorem of algebra (FTA) and proposing an algorithm, practical if the norm and trace of a solution are known. We present novel results on theory, algorithms and applications of quaternion root-finding. Firstly, we give a new proof of the FTA resulting in explicit formulas for both exact and approximate quaternion roots of P(x) in terms of exact and approximate complex roots of the real polynomial F(x)=P(x)P¯(x), where P¯(x) is the conjugate polynomial. In particular, if |F(c)|≤ϵ, then for a computable quaternion conjugate q of c, |P(q)|≤ϵ. Consequences of these include relevance of root-finding methods for complex polynomials, computation of bounds on zeros, and algebraic solution of special quaternion equations. Secondly, working directly in the quaternion space, we develop Newton and Halley methods and analyze their local behavior. Surprisingly, even for a quadratic quaternion polynomial Newton’s method may not converge locally. Finally, we derive an analogue of the Bernoulli method in the quaternion space for computing the dominant root in certain cases. This requires the development of an independent theory for the solution of quaternion homogeneous linear recurrence relations. These results also lay a foundation for quaternion polynomiography.