An Iterative Method to Compute Zeros of Quaternion Polynomials (original) (raw)

An Iterative Method to Compute the Dominant Zero of a Quaternionic Unilateral Polynomial

Advances in Applied Clifford Algebras, 2018

The aim of this paper is to propose an iterative method to compute the dominant zero of a quaternionic unilateral polynomial. We prove that the method is convergent in the sense that it generates a sequence of quaternions that converges to the dominant zero of the polynomial. The idea subjacent to this method is the well known Sebastião e Silva's method, proposed in "Sur une méthode d'approximation semblableà celle de Gräffe", Portugaliae Mathematica, 1941, to approximate the dominant zero of complex polynomials.

A Note on the Computation of All Zeros of Simple Quaternionic Polynomials

Siam Journal on Numerical Analysis, 2010

Polynomials with quaternionic coefficients located on only one side of the powers (we call them simple polynomials) may have two different types of zeros: isolated and spherical zeros. We will give a new characterization of the types of the zeros and, based on this characterization, we will present an algorithm for producing all zeros including their types without using an iteration process which requires convergence. The main tool is the representation of the powers of a quaternion as a real, linear combination of the quaternion and the number one (as introduced by Pogorui and Shapiro [Complex Var. and Elliptic Funct., 49 (2004), pp. 379-389]) and the use of a real companion polynomial which already was introduced for the first time by Niven [Amer. Math. Monthly, 48 (1941), pp. 654-661]. There are several examples. Key words. zeros of quaternionic polynomials, structure of zeros of quaternionic polynomials AMS subject classifications. 11R52, 12E15, 12Y05, 65H05

The classification and the computation of the zeros of quaternionic, two-sided polynomials

Numerische Mathematik, 2010

Already for a long time it is known that quaternionic polynomials whose coefficients are located only at one side of the powers, may have two classes of zeros: isolated zeros and spherical zeros. Only recently a classification of the two types of zeros and a means to compute all zeros of such polynomials have been developed. In this investigation we consider quaternionic polynomials whose coefficients are located at both sides of the powers, and we show that there are three more classes of zeros defined by the rank of a certain real (4 × 4) matrix. This information can be used to find all zeros in the same class if only one zero in that class is known. The essential tool is the description of the polynomial p by a matrix equation P(z) := A(z)z + B(z), where A(z) is a real (4 × 4) matrix determined by the coefficients of the given polynomial p and P, z, B are real column vectors with four rows. This representation allows also to include two-sided polynomials which contain several terms of the same degree. We applied Newton’s method to P(z) = 0. This method turned out to be a very effective tool in finding the zeros. This method allowed also to prove, that the essential number of zeros of a quaternionic, two-sided polynomial p of degree n is, in general, not bounded by n. We conjecture that the bound is 2n. There are various examples.

Zeros of Unilateral Quaternionic 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.

Zeros of one class of quaternionic polynomials

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.

Roots of Quaternion Standard 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.

On the Multiplicity of Zeroes of Polynomials with Quaternionic Coefficients

Milan Journal of Mathematics, 2008

Regular polynomials with quaternionic coefficients admit only isolated zeroes and spherical zeroes. In this paper we prove a factorization theorem for such polynomials. Specifically, we show that every regular polynomial can be written as a product of degree one binomials and special second degree polynomials with real coefficients. The degree one binomials are determined (but not uniquely) by the knowledge of the isolated zeroes of the original polynomial, while the second degree factors are uniquely determined by the spherical zeroes. We also show that the number of zeroes of a polynomial, counted with their multiplicity as defined in this paper, equals the degree of the polynomial. While some of these results are known in the general setting of an arbitrary division ring, our proofs are based on the theory of regular functions of a quaternionic variable, and as such they are elementary in nature and offer explicit constructions in the quaternionic setting.

On the zeros of a quaternionic polynomial: An extension of the Eneström-Kakeya theorem

Czechoslovak Mathematical Journal, 2023

We present some results on the location of zeros of regular polynomials of a quaternionic variable. We derive new bounds of Eneström-Kakeya type for the zeros of these polynomials by virtue of a maximum modulus theorem and the structure of the zero sets of a regular product established in the newly developed theory of regular functions and polynomials of a quaternionic variable. Our results extend some classical results from complex to the quaternionic setting as well.

The Eneström–Kakeya Theorem for polynomials of a quaternionic variable

Journal of Approximation Theory, 2019

The well-known Eneström-Kakeya Theorem states that a polynomial with real, nonnegative, monotone increasing coefficients has all its complex zeros in the closed unit disk in the complex plane. In this paper, we extend this result by showing that all quaternionic zeros of such a polynomial lie in the unit sphere in the quaternions. We also extend related results from the complex to quaternionic setting.