Newton method in the context of quaternion analysis

In this paper we propose a version of Newton method for finding zeros of a quaternion function of a quaternion variable, based on the concept of quaternion radial derivative. Several numerical examples involving elementary functions are presented

Exercícios de álgebra linear

Estes textos contêm uma seleção de exercícios para apoio às aulas de Álgebra Linear para cursos de Engenharia e Ciências, bem como as resoluções de alguns exercícios. Apresentam-se também algumas provas de avaliação para esses cursos e as respetivas resoluções

Modified quaternion Newton methods

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.The research was partially supported by the Research Centre of Mathematics of the University of Minho with the Portuguese Funds from the " Fundcao para a Ciencia e a Tecnologia", through the Project PEstOE/ MAT/ UI0013/ 2014

Complementos de análise numérica: valores e vectores próprios

Texto de apoio à disciplina de Complementos de Análise Numérica

Linking Clifford analysis and combinatorics through bijective methods

The application of Clifford Analysis methods in Combinatorics has some peculiarities due to the use of noncommutative algebras. But it seems natural to expect from here some new results different from those obtained by using approaches based on several complex variable. For instances, the fact that in Clifford Analysis the point-wise multiplication of monogenic functions as well as their composition are not algebraically closed in this class of generalized holomorphic functions causes serious problems. Indeed, this is one of the reasons why in polynomial approximation almost every problem needs the construction of specially adapted polynomial bases. Our aim is to show that the analysis and comparison of different representations of the same polynomial or entire function allow to link Clifford Analysis and Combinatorics by means of bijective methods. In this context we also stress the central role of the hypercomplex derivative for power series representations in connection with the concept of Appell sequences as analytic tools for establishing this link.Fundação para a Ciência e a Tecnologia (FCT

3D-Mappings using monogenic functions

Conformal mappings of plane domains are realized by holomorphic functions with non vanishing derivative. Therefore complex differentiability plays an important role in all questions related to fundamental properties of such mapping. In contrast to the planar case, in higher dimensions the set of conformal mappings consists only of M¨obius transformations. But unfortunately M¨obius transformations are not monogenic functions and therefore also not hypercomplex differentiable. However the equivalence between both concepts - hypercomplex differentiability in the sense of [9], [11] and monogenicity - suggests the question whether monogenic functions can play or not a special role for other types of 3D-mappings, for instance, for quasi-conformal ones. Our goal is to present a case study of an approach to 3D-mappings by using particularly easy to handle monogenic homogeneous polynomials as basis for approximating the mapping function. Thereby we extend significantly the results obtained in [3]. From the numerical point of view we apply ideas from complex numerical analysis realizing the approximation via polynomials of a small real parameter

Special monogenic polynomials - properties and applications

AIP conference proceedings, vol. 936In Clifford Analysis several different methods have been developed for constructing monogenic functions as series with respect to properly chosen homogeneous monogenic polynomials. Almost all these methods rely on sets of orthogonal polynomials with their origin in classical (real) Harmonic Analysis in order to obtain the desired basis of homogeneous polynomials. We use a direct and elementary approach to this problem and construct a set of homogeneous polynomials involving only products of a hypercomplex variable and its hypercomplex conjugate. The obtained set is an Appell set of monogenic polynomials with respect to the hypercomplex derivative. Its intrinsic properties and some applications are presented.The research of the first author was partially supported by the R&D unit Matematica e Aplicacoes (UIMA) of the University of Aveiro, through the Portuguese Foundation for Science and Technology (FCT). The research of the second author was partially supported by the Portuguese Foundation for Science and Technology (FCT) through the research program POCTI

On paravector valued homogeneous monogenic polynomials with binomial expansion

The aim of this note is to study a set of paravector valued homogeneous monogenic polynomials that can be used for a construction of sequences of generalized Appell polynomials in the context of Clifford analysis. Therefore, we admit a general form of the vector part of the first degree polynomial in the Appell sequence. This approach is different from the one presented in recent papers on this subject. We show that in the case of paravector valued polynomials of three real variables, there exist essentially two different types of such polynomials together with two other trivial types of polynomials. The proof indicates a way of obtaining analogous results in the case of polynomials of more than three variables.Fundação para a Ciência e a Tecnologia (FCT

A note on a one-parameter family of non-symmetric number triangles

The recently growing interest in special Cliff ord Algebra valued polynomial solutions of generalized Cauchy-Riemann systems in (n + 1)-dimensional Euclidean spaces suggested a detailed study of the arithmetical properties of their coe fficients, due to their combinatoric relevance. This concerns, in particular, a generalized Appell sequence of homogeneous polynomials whose coe cient's set can be treated as a one-parameter family of non-symmetric triangles of fractions. The discussion of its properties, similar to those of the ordinary Pascal triangle (which itself does not belong to the family), is carried out in this paper.Fundação para a Ciência e a Tecnologia (FCT

Monogenic generalized Laguerre and Hermite polynomials and related functions

In recent years classical polynomials of a real or complex variable and their generalizations to the case of several real or complex variables have been in a focus of increasing attention leading to new and interesting problems. In this paper we construct higher dimensional analogues to generalized Laguerre and Hermite polynomials as well as some based functions in the framework of Clifford analysis. Our process of construction makes use of the Appell sequence of monogenic polynomials constructed by Falcão/Malonek and stresses the usefulness of the concept of the hypercomplex derivative in connection with the adaptation of the operational approach, developed by Gould et al. in the 60's of the last century and by Dattoli et al. in recent years for the case of the Laguerre polynomials. The constructed polynomials are used to define related functions whose properties show the application of Special Functions in Clifford analysis.FC