Modified Quaternion Newton Methods

Miranda, Fernando; Falcão, M. I.

doi:10.1007/978-3-319-09144-0_11

Cited by 6 publications

(5 citation statements)

References 19 publications

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“…In [25] and references therein the characterization of the zeros of quaternionic polynomials was studied, based on Niven's algorithm. Other numerical methods based on Newton, Weierstrass and based on Sebastião e Silva's methods have been given in [15,16,28].…”

Section: Introductionmentioning

confidence: 99%

Bounds for the zeros of unilateral octonionic polynomials

Serôdio

Beites

Vitória

2021

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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In the present work it is proved that the zeros of a unilateral octonionic polynomial belong to the conjugacy classes of the latent roots of an appropriate lambda-matrix. This allows the use of matricial norms, and matrix norms in particular, to obtain upper and lower bounds for the zeros of unilateral octonionic polynomials. Some results valid for complex and/or matrix polynomials are extended to octonionic polynomials.

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Section: Introductionmentioning

confidence: 99%

Bounds for the zeros of unilateral octonionic polynomials

Serôdio

Beites

Vitória

2021

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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“…In fact, it was only in the year 2001 that Serôdio, Pereira and Vitória [23] proposed an efficient procedure to replace the first part of Niven's method, presenting what can be considered as the first really usable algorithm for determining the zeros of quaternionic polynomials. After this first paper, the interest in the development of rootfinding methods for quaternionic polynomials has called the attention of many researchers; see e.g., [4,8,18,21,25]. Most of the methods available to compute the roots of a given polynomial P make use of the so-called companion polynomial of P to replace the first part of Niven's procedure, i.e.…”

Section: Introductionmentioning

confidence: 99%

The number of zeros of unilateral polynomials over coquaternions revisited

Falcão

Miranda

Severino

et al. 2018

Linear and Multilinear Algebra

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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.

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“…The literature on quaternion polynomial root-finding reveals a recent growing interest on this subject; see e.g. [7,8,12,13,16,20,25,28,27].…”

Section: Introductionmentioning

confidence: 99%

Polynomials over Quaternions and Coquaternions: A Unified Approach

Falcão

Miranda

Severino

et al. 2017

Computational Science and Its Applications – ICCSA 2017

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This paper aims to present, in a unified manner, results which are valid on both the algebras of quaternions and coquaternions and, simultaneously, call the attention to the main differences between these two algebras. The rings of one-sided polynomials over each of these algebras are studied and some important differences in what concerns the structure of the set of their zeros are remarked. Examples illustrating this different behavior of the zero-sets of quaternionic and coquaternionic polynomials are also presented.

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Modified Quaternion Newton Methods

Abstract: We revisit the quaternion Newton method for computing roots of a class of quaternion valued functions and propose modified algorithms for finding multiple roots of simple polynomials. We illustrate the performance of these new methods by presenting several numerical experiments.

Cited by 6 publications

References 19 publications

Bounds for the zeros of unilateral octonionic polynomials

Bounds for the zeros of unilateral octonionic polynomials

The number of zeros of unilateral polynomials over coquaternions revisited

Polynomials over Quaternions and Coquaternions: A Unified Approach

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