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Newton method in the context of quaternion analysis
Falcão, M. I.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.
Modified quaternion Newton methods
Falcão, M. I.; Miranda, FernandoWe 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.
Monogenic pseudo-complex power functions and their applications
Cruz, Carla; Falcão, M. I.; Malonek, H. R.The use of a non-commutative algebra in hypercomplex function theory requires a large variety of different representations of polynomials suitably adapted to the solution of different concrete problems. Naturally arises the question of their relationships and the advantages or disadvantages of different types of polynomials. In this sense, the present paper investigates the intrinsic relationship between two di...
On numerical aspects of pseudo-complex powers in R^3
Cruz, Carla; Falcão, M. I.; Malonek, H. R.In this paper we consider a particularly important case of 3D monogenic polynomials that are isomorphic to the integer powers of one complex variable (called pseudo-complex powers or pseudo-complex polynomials, PCP). The construction of bases for spaces of monogenic polynomials in the framework of Clifford Analysis has been discussed by several authors and from different points of view. Here our main concern ar...
A note on totally regular variables and Appell sequences in hypercomplex functi...
Cruz, Carla; Falcão, M. I.; Malonek, H. R.Series title : Lecture notes in computer science, vol. 7971, ISSN 0302-9743 ; The aim of our contribution is to call attention to the relationship between totally regular variables, introduced by R. Delanghe in 1970, and Appell sequences with respect to the hypercomplex derivative. Under some natural normalization condition the set of all paravector valued totally regular variables defined in the three dimensi...
About Pascal’s tetrahedron with hypercomplex entries
Cruz, Carla; Falcão, M. I.; Malonek, H. R."11th International Conference of Numerical Analysis and Applied Mathematics, 21 - 27 September 2013" ; It is evident, that the properties of monogenic polynomials in $(n+1)-$real variables significantly depend on the generators $e_1, e_2, \dots, e_n$ of the underlying $2^n$-dimensional Clifford algebra $Cl_{0,n}$ over $\mathbb{R}$ and their interactions under multiplication. The case of $n=3$ is studied thro...
Matrix representations of a special polynomial sequence in arbitrary dimension
Cação, I.; Falcão, M. I.; Malonek, H. R.This paper provides an insight into different structures of a special polynomial sequence of binomial type in higher dimensions with values in a Clifford algebra. The elements of the special polynomial sequence are homogeneous hypercomplex differentiable (monogenic) functions of different degrees and their matrix representation allows to prove their recursive construction in analogy to the complex power functio...
On paravector valued homogeneous monogenic polynomials with binomial expansion
Malonek, H. R.; Falcão, M. I.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...
A note on a one-parameter family of non-symmetric number triangles
Falcão, M. I.; Malonek, H. R.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 o...
On the structure of generalized Appell sequences of paravector valued homogeneo...
Cruz, Carla; Falcão, M. I.; Malonek, Helmuth R.The fact that generalized Appell sequences of monogenic polynomials in the setting of hypercomplex function theory also satisfy a corresponding binomial type theorem allows to obtain their explicit structure. Recently it has been obtained a complete characterization in the case of paravector valued homogeneous polynomials of three real variables. The aim of this contribution is the study of paravector valued ho...
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