Eliminating Iterative Methods: A Closed-Form Solution to Multivariate Quaternionic Least Squares

These Volumes present the generalized form of the cubic equation proposed by Li-Ping Huang and Wasin So for solving quaternionic quadratic equations. Utilizing a natural transformation from the standard orthogonal basis ${\vec{i},\vec{j},\vec{k}}$ to ${\vec{\lambda},\vec{\mu},\vec{\nu}}$, which maintains quaternion multiplication rules, we derive the general and depressed forms of the quadratic equation.The general form is expressed as $\vec{f}=\vec{x}^2+\vec{x}\vec{b}+\vec{a}\vec{x}$, whilst the depressed form takes the structure $\vec{c}=\vec{t}^{2}-\vec{t}\vec{v}+\vec{v}\vec{t}$, where $\vec{c}=\vec{f}+\vec{a}\vec{b}$ and $\vec{u}=\frac{1}{2}(\vec{a}+\vec{b})$, $\vec{v}=\frac{1}{2}(\vec{a}-\vec{b})$, and $\vec{t}=\vec{x}-\vec{u}$. Note that $\vec{u}$ and $\vec{v}$ are the Vector Mean Sum and Difference of $\vec{a}$ and $\vec{b}$, a crucial observation that also allows the definition of logarithms of quaternionic logics (logics that are associative, but not commutative).Considering complex roots of the cubic equation yields additional solutions in the form of complex quaternions (biquaternions), resulting in six unique solutions to $\vec{c}=\vec{t}^{2}-\vec{t}\vec{v}+\vec{v}\vec{t}$.Furthermore, we derive the Closed Form Solution to Quaternionic Least Squares, both for real and complex quaternions. We also extend our analysis to Tessarine Least Squares, Euler's Formula for Reflectors, and establish the well-defined properties such as the conjugate, reciprocal, and logarithm of tessarines.The inaccuracies surrounding tessarines, quaternions, and other hypercomplex algebras in existing literature \textbf{necessitate} this Introductory Volume to rectify these misconceptions. Consequently, the introductory volume is rather lengthy and narrative-driven. In contrast, the subsequent volumes are relatively short, primarily presenting Theorems and Proofs with minimal narrative.