Solving Linear and Quadratic Quaternion Equations
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Mathematical algorithms have been presented for solving linear and quadratic quaternion equations, which can be readily obtained when the problem coefficients are real or complex. The scalar and the vector parts of the solution are analytically uncoupled by the introduction of an exact change-of-variables substitution, resulting in a scalar polynomial equation in a single variable that must be solved. The advantage of this approach is that a 78th order polynomial is replaced with a second-through eighth-order polynomial, depending on the symmetries of the specific application. The four nonlinear equations defining the necessary conditions for the solutions are reduced to a single scalar equation by analytically eliminating the vector part of the solution from the scalar equation. After obtaining the scalar solutions, the vector part of the solution is thereby obtained by introducing the scalar solution into the transformation equation for the vector part of the solution.
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