Pure Imaginary Roots of Quaternion Standard Polynomials

Journal of Pure and Applied Mathematics: Advances and Applications, 2015

The fundamental theorem of algebra over the quaternion skew field has gained little attention. It is of less importance than that over the complex number field, though some problems may reduce to it. In this work, we use the vector representation of quaternions in order to obtain simplified forms of the solutions of the quaternionic equation 0 2

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

2011

The aim of this paper is to propose an iterative method to compute the dominant zero of a quaternion polynomial. We prove that the method is convergent in the sense that it generates a sequence of quaternions that converges to the domi- nant zero of the polynomial. The idea subjacent to the proposed method is the well known method proposed by Sebastiao e Silva in "Sur une methode d'approximation semblable a celle de Graffe", Portugaliae Mathematica, 1941, to compute approxi- mately the zeros of complex polynomials.

2006

The purpose of this paper is to show how the problem of finding the zeros of unilateral n-order quaternionic polynomials can be solved by determining the eigenvectors of the corresponding companion matrix. This approach, probably superfluous in the case of quadratic equations for which a closed formula can be given, becomes truly useful for (unilateral) n-order polynomials. To understand the strength of this method, it is compared with the Niven algorithm and it is shown where this (full) matrix approach improves previous methods based on the use of the Niven algorithm. For convenience of the readers, some examples of second and third order unilateral quaternionic polynomials are explicitly solved. The leading idea of the practical solution method proposed in this work can be summarized in the following three steps: translating the quaternionic polynomial in the eigenvalue problem for its companion matrix, finding its eigenvectors, and, finally, giving the quaternionic solution of the unilateral polynomial in terms of the components of such eigenvectors. A brief discussion on bilateral quaternionic quadratic equations is also presented.

Real Quaternionic Calculus Handbook, 2013

Axioms

Quaternion is a four-dimensional and an extension of the complex number system. It is often viewed from various fields, such as analysis, algebra, and geometry. Several applications of quaternions are related to an object’s rotation and motion in three-dimensional space in the form of a differential equation. In this paper, we do a systematic literature review on the development of quaternion differential equations. We utilize PRISMA (preferred reporting items for systematic review and meta-analyses) framework in the review process as well as content analysis. The expected result is a state-of-the-art and the gap of concepts or problems that still need to develop or answer. It was concluded that there are still some opportunities to develop a quaternion differential equation using a quaternion function domain.

In this paper we present a solution for any standard quaternion quadratic equation, i.e. an equation of the form z^2+μz+ν=0 where μ and ν belong to some quaternion division algebra Q over some field F, assuming the characteristic of F is 2.

Journal of Symbolic Computation, 2015

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2011

This paper describes new issues of the Mathematica standard package Quaternions for implementing Hamilton's Quaternion Algebra. This work attempts to endow the original package with the ability to perform operations on symbolic expressions involving quaternion-valued functions. A collection of new functions is introduced in order to provide basic mathematical tools necessary for dealing with regular functions in R n+1 , for n ≥ 2. The performance of the package is illustrated by presenting several examples and applications.