Zeros of Unilateral Quaternionic Polynomials

2011

In this paper, we present a new method for solving standard quaternion equations. Using this method we reobtain the known formulas for the solution of a quadratic quaternion equation, and provide an explicit solution for the cubic quaternion equation, as long as the equation has at least one pure imaginary root. We also discuss the number of essential pure imaginary roots of a two-sided quaternion polynomial.

arXiv (Cornell University), 2011

In this paper, we present a new method for solving standard quaternion equations. Using this method we reobtain the known formulas for the solution of a quadratic quaternion equation, and provide an explicit solution for the cubic quaternion equation, as long as the equation has at least one pure imaginary root. We also discuss the number of essential pure imaginary roots of a twosided quaternion polynomial.

Newton's method for finding zeros is formally adapted to finding roots of Hamilton's quaternions. Since a derivative in the sense of complex analysis does not exist for quaternion valued functions we compare the resulting formulas with the more classical formulas obtained by using the Jacobian matrix and the Gâteaux derivative. The latter case includes also the so-called damped Newton form. We investigate the convergence behavior and show that under one simple condition all cases introduced, produce the same iteration sequence and have thus the same convergence behavior, namely that of locally quadratic convergence. By introducing an analogue of Taylor's formula for

Filomat

The goal of this paper is to study the properties of zeros of some special quaternionic polynomials with restricted coefficients, namely coefficients whose real and imaginary components satisfy suitable inequalities. We extend the well-known Enestr?m-Kakeya theorem and its various generalizations from complex to the quaternionic setting. The main tools used to derive the bounds for the zeros of these polynomials are the maximum modulus theorem and the structure of the zero sets established in the newly developed theory of regular functions and polynomials of a quaternionic variable.

Applied Mathematics Letters, 2002

In this paper, we derive explicit formulas for computing the roots of a quaternionic quadratic polynomial.

Linear and Multilinear Algebra, 2018

The literature on quaternionic polynomials and, in particular, on methods for determining and classifying their zero-sets, is fast developing and reveals a growing interest on this subject. In contrast, polynomials defined over the algebra of coquaternions have received very little attention from researchers. One of the few exceptions is the very recent paper by Janovská and Opfer [Electronic Transactions on Numerical Analysis, Volume 46, pp. 55-70, 2017], where, among other results, we can find a first attempt to prove that a unilateral coquaternionic polynomial of degree n has, at most, n(2n − 1) zeros. In this paper we present a full proof of the referred result, using a totally different and, from our point of view, much simpler approach. Also, we give a complete characterization of the zero-sets of such polynomials and present a new result giving conditions which guarantee the existence of a special type of zeros. An algorithm to compute and classify all the zeros of a coquaternionic polynomial is proposed and several numerical examples are carefully constructed.

Mathematical Methods in the Applied Sciences, 2007

In this paper we propose a definition of determinant for quaternionic polynomial matrices inspired by the well-known Dieudonné determinant for the constant case. This notion allows to characterize the stability of linear dynamical systems with quaternionic coefficients, yielding results which generalize the ones obtained for the real and complex cases.

Japan Journal of Industrial and Applied Mathematics, 2003

The zeros of a polynomial can be found by computing the eigenvalues of the corresponding companion matrix. However, in the case of multiple zeros, the calculated results are usually not as accurate as those for simple zeros. This problem is discussed and a new method is presented by constructing a new companion matrix which has only simple eigenvalues. By this method, instead of calculating all the zeros simultaneously, we can calculate the distinct zeros and their multiplicities separately. Numerical examples are presented to illustrate the efficiency of the method.

We study the quaternionic linear system which is composed out of terms of the form ln(x) := n p=1 apxbp with quaternionic constants ap, bp and a variable number n of terms. In the first place we investigate one equation in one variable. If n = 2 the corresponding equation, which is normally called Sylvester's equation will be treated completely by using only quaternionic algebra. For larger n a transition to the isomorphic (4 × 4) real matrix case is investigated. Sufficient conditions for non singularity will be obtained by using results from fixed point theorems. Connections to the Kronecker product are presented. The general case of a linear quaternionic system is treated, where each unknown is contained in a sum of the form mentioned above. As a tool the so-called column operator and its properties are used. An analogue of the Kronecker product for quaternionic systems involving terms of the form AXB is given.