Sponsored by the Rutgers University Department of Mathematics and the

Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)

**Co-organizers:****Andrew Baxter**, Rutgers University, baxter{at} math [dot] rutgers [dot] edu**Doron Zeilberger**, Rutgers University, zeilberg {at} math [dot] rutgers [dot] edu

Title: Algorithms for Quaternion Polynomial Root-Finding

Speaker: **Bahman Kalantari**, Rutgers University

Date: Thursday, October 28, 2010 5:00pm

Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

In 1941 Niven pioneered quaternion polynomial root-finding by proving the Fundamental Theorem of Algebra (FTA) and proposing an algorithm, practical only if the norm and trace of a solution are known. Here we present several novel results on theory, algorithms and applications of quaternion root-finding. First, we present a new proof of the FTA resulting in explicit exact and approximate formulas for the roots of a given quaternion polynomial in terms of exact and approximate complex roots of a real polynomial. Immediate consequences include, relevance of complex polynomial root-finding algorithms, computing bounds on zeros, and algebraic solution of special quaternion equations. Additionally, we develop Newton and Halley methods in the quaternion space, as well as an analogue of the Bernoulli method for computing dominant roots. The latter is based on the development of a theory for solving quaternion homogeneous linear recurrence relations. Finally, we describe algorithms for the visualization of quaternion polynomials, laying the foundation for quaternion polynomiography.