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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
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
Summary:
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.
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