A survey of some recent results on Clifford algebras in ℝ 4

Drahoslava Janovská; Gerhard Opfer

Applications of Mathematics (2023)

  • Volume: 68, Issue: 5, page 571-592
  • ISSN: 0862-7940

Abstract

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We will study applications of numerical methods in Clifford algebras in ℝ 4 , in particular in the skew field of quaternions, in the algebra of coquaternions and in the other nondivision algebras in ℝ 4 . In order to gain insight into the multidimensional case, we first consider linear equations in quaternions and coquaternions. Then we will search for zeros of one-sided (simple) quaternion polynomials. Three different classes of zeros can be distinguished. In general, the quaternionic coefficients can be placed on both sides of the powers. Then there are even five different classes of zeros. All results can be extended to other noncommutative algebras in ℝ 4 . In the paper by R. Lauterbach and G. Opfer (2014), the authors constructed an exact Jacobi matrix for functions defined in noncommutative algebraic systems without the use of any partial derivative. We applied this technique to find the eigenvalues of the companion matrix as zeros of the companion polynomial by Newton’s method.

How to cite

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Janovská, Drahoslava, and Opfer, Gerhard. "A survey of some recent results on Clifford algebras in $\mathbb {R}^4$." Applications of Mathematics 68.5 (2023): 571-592. <http://eudml.org/doc/299341>.

@article{Janovská2023,
abstract = {We will study applications of numerical methods in Clifford algebras in $\mathbb \{R\}^4$, in particular in the skew field of quaternions, in the algebra of coquaternions and in the other nondivision algebras in $\mathbb \{R\}^4$. In order to gain insight into the multidimensional case, we first consider linear equations in quaternions and coquaternions. Then we will search for zeros of one-sided (simple) quaternion polynomials. Three different classes of zeros can be distinguished. In general, the quaternionic coefficients can be placed on both sides of the powers. Then there are even five different classes of zeros. All results can be extended to other noncommutative algebras in $\mathbb \{R\}^4$. In the paper by R. Lauterbach and G. Opfer (2014), the authors constructed an exact Jacobi matrix for functions defined in noncommutative algebraic systems without the use of any partial derivative. We applied this technique to find the eigenvalues of the companion matrix as zeros of the companion polynomial by Newton’s method.},
author = {Janovská, Drahoslava, Opfer, Gerhard},
journal = {Applications of Mathematics},
keywords = {linear equations in quaternions and coquaternions; polynomials over $\mathbb \{R\}^4$ algebras; the algebraic eigenvalue problem over noncommutative algebras; Newton’s method; companion matrix and companion polynomial; Niven’s algorithm},
language = {eng},
number = {5},
pages = {571-592},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A survey of some recent results on Clifford algebras in $\mathbb \{R\}^4$},
url = {http://eudml.org/doc/299341},
volume = {68},
year = {2023},
}

TY - JOUR
AU - Janovská, Drahoslava
AU - Opfer, Gerhard
TI - A survey of some recent results on Clifford algebras in $\mathbb {R}^4$
JO - Applications of Mathematics
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 5
SP - 571
EP - 592
AB - We will study applications of numerical methods in Clifford algebras in $\mathbb {R}^4$, in particular in the skew field of quaternions, in the algebra of coquaternions and in the other nondivision algebras in $\mathbb {R}^4$. In order to gain insight into the multidimensional case, we first consider linear equations in quaternions and coquaternions. Then we will search for zeros of one-sided (simple) quaternion polynomials. Three different classes of zeros can be distinguished. In general, the quaternionic coefficients can be placed on both sides of the powers. Then there are even five different classes of zeros. All results can be extended to other noncommutative algebras in $\mathbb {R}^4$. In the paper by R. Lauterbach and G. Opfer (2014), the authors constructed an exact Jacobi matrix for functions defined in noncommutative algebraic systems without the use of any partial derivative. We applied this technique to find the eigenvalues of the companion matrix as zeros of the companion polynomial by Newton’s method.
LA - eng
KW - linear equations in quaternions and coquaternions; polynomials over $\mathbb {R}^4$ algebras; the algebraic eigenvalue problem over noncommutative algebras; Newton’s method; companion matrix and companion polynomial; Niven’s algorithm
UR - http://eudml.org/doc/299341
ER -

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