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Note that since $A'$ is upper triangular, its inverse can be easily calculated.   \subsection{Additions to SRIF}  I have implemented 2 main additions to the standard SRIF. An inherent downside to the SRIF is the number of operations necessary to generate the Householder matrix $H$ grows like $mn^2$. $n*m^2$.  My implementation of the SRIF runs the matrix triangularization simultaneously on multiple cores, which results in a significant speed-up. This can be done in a thread-safe manner because each Householder operation occurs on a column-by-column basis. \par The second addition I have made to the standard SRIF is the inclusion of a secondary $\chi^2$ minimization for the scaling of $\vec{\delta{x}}$. So   \[Q'' = Q(\vec{x} + \alpha \vec{\delta x})\] is minimized. This minimization is done with a grid search over 6 decades. The additional minimization adds a trivial additional computation cost to the overall minimization of $\chi^2$, but allows for faster convergence, and the possibility of skipping over local minima in the $\chi^2$-space.