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\setcounter{section}{4}  \section{Method}  \subsection{The Square Root Information Filter}  The Square Root Information Filter (SRIF) was originally developed by Bierman in 1977 [ref]. The algorithm minimizes $\chi^2$ for time series data with Gaussian errors, and was based on the Kalman filter algorithm. SRIF is more stable, more accurate, and faster than the current algorithm used in \textbf{shape}. SRIF is also more numerically stable (and, in some cases faster) than a standard Jacobi-based steepest descent $\chi^2$-minimzation routine (SDR). routine.  My implmentation of SRIF includes some changes to the original algorithm, which will be discussed in section blah.blah. \par The fundamental difference between SRIF algorithm and a classic SDR is the use of matrix square roots and Householder operations for increased numerical stability.  \subsection{Steepest Descent Routine}  A classical steepest descent routine (SDR) minimizes the weighted residuals between a model and data with Gaussian noise by determining the direction in parameter space in which the $\chi^2$ is decreasing fastest. Specifically, if one has a set of observables, $\vec{z}$ , and a model $M(x_1,x_2,...,x_n)$  The \emph{characteristic polynomial} $\chi(\lambda)$ of the  $3 \times 3$~matrix