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\subsection{Square Root Information Filter}  The Square Root Information Filter (SRIF) gets around the problems inherent in a classical GNR by utilizing matrix square roots and Householder operations to increase the numerical stability when determining $\delta\vec{x}$ . Instead of minimizing $\chi^2$, SRIF minimizes \[Q = (\chi^2)^{\frac{1}{2}} = ||W^{\frac{1}{2}} \vec{R}||\] Then, along similar lines as GNR, a change of $\vec{\delta x}$ is introduced to the parameter vector $\vec{x}$, and $Q' = Q(\vec{x}+\vec{\delta x})$ is minimized over this change. \par  $Q'$ is smallest when \[||(W^{\frac{1}{2}} \vec{R(\vec{x} + \vec{\delta x}}|| \approx ||W^{\frac{1}{2}} \vec{R} + W^{frac{1}{2}} W^{\frac{1}{2}}  A\vec{\delta x}||\] is minimized.