Jim, My biggest brother, sent me a copy of a presentation that he planned to give to a bunch of Mentally Proficient people about a little mathematical trick he created which allows a computer to do a certain calculation hundreds or thousands of times quicker than using the normal way to do a Quaternion thingie. I have pasted the abstract of his presentation below so that everyone can review it and point out to him where he went wrong. Good luck.
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An Object-Oriented Operator-Overloaded Quaternion Toolbox
By James D. Turner, Ph.D
Texas A&M University, Department of Aerospace Engineering, College Station Texas, 77843
Abstract Follows:
A Fortran 95 Object-Oriented Operator-Overloaded Toolbox is presented for automating engineering and scientific applications involving quaternion algebra. The Toolbox supports basic math operations (+,-,*,/,**), standard library functions (sin, asin, exp, ln, .), linear equations, quadratic polynomials, and matrix-vector operations for solving linear matrix equations, matrix inversion, and eigenvalue/eigenvector operations. Extensive use is made operator-overloading and generic operators for automating the often tedious calculations required for manipulating quaternion objects. A further advantage of this approach is that conventional programming language constructs are used for developing math models, which reduces programming errors. Two strategies are presented for enabling matrix capabilities: (1) re-writing core algorithms to reflect the noncommunitive behavior of quaternions, and (2) mapping quaternion matrices to higher-dimensional complex-valued matrices. The aim of the package is to facilitate and encourage further research relying on quaternion algebraic computations for applications in dynamics, control, and optimization. Numerical examples are presented for demonstrating the basic computational capabilities of the software.
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Go Jim.
If anyone has any corrections to the above or would like to read the whole presentation. Please let me know.