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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 \[||W^{\frac{1}{2}} \vec{R}(\vec{x} + \vec{\delta x})|| \approx ||W^{\frac{1}{2}}(\vec{R}(\vec{x}) + A\vec{\delta x})||\par  = ||W^{\frac{1}{2}}\vec{R}(\vec{x}) + W^{\frac{1}{2}} A\vec{\delta x}||\] is minimized. \par A matrix $H$ is defined such that $HW^{\frac{1}{2}} A$ is upper triangular. $H$ is orthogonal and can be generated by the product of $m$ Householder operation matrices. Note that the orthogonality of $H$ guarantees that \[|| HW^{\frac{1}{2}}\vec{R}(\vec{x}) + HW^{\frac{1}{2}} A\vec{\delta x} || \\ = ||H(W^{\frac{1}{2}}\vec{R}(\vec{x}) + W^{\frac{1}{2}} A\vec{\delta x})|| \\ = ||W^{\frac{1}{2}}\vec{R}(\vec{x}) + W^{\frac{1}{2}} A\vec{\delta x}||\]   Since $HW^{\frac{1}{2}} A$ is upper triangular, it can be rewritten as   \[ \left( \begin{array}{c}