

\subsection{\bf 4.2\hspace{1em} Derivative Formalism for $\chi^2$---I}

\begin{slide}
\begin{itemize}
\item One great advantage of the $\chi^2$ or maximum likelihood
problem is that to calculate the second derivative of the function to
be minimized only requires the {\em first\/} derivative of the
theoretical functions.  

\item First consider this in the case of $\chi^2$:
\begin{equation}
\chi^2 = \sum^N_{i=1} \frac{(t_i - e_i)^2}{\sigma^2_i}
\end{equation}

\item where $e_i$ is experimental measurement, $\sigma_i$ is error, and
$t_i$ is theory---a function of $n$ parameters.

\item We take, without loss of generality, $n=1$ and one parameter
$\alpha$ in the following:
\begin{eqnarray*}
\frac{\partial\chi^2}{\partial\alpha} & = & \sum^N_{i=1} 2 (t_i - e_i)
\frac{\partial t_i}{\partial\alpha}/\sigma^2_i \\
\frac{\partial^2 \chi^2}{\partial\alpha^2} & = & S_1 + S_2 \\
S_1 & = & \sum^N_{i=1} 2( t_1 - e_i ) \frac{\partial^2
t_i}{\partial\alpha^2}/\sigma^2_i \\
S_2 & = & 2 \sum^N_{i=1} \frac{\left( \frac{\partial
t_i}{\partial\alpha}\right)^2}{\sigma^2_i}
\end{eqnarray*}
\end{itemize}
\end{slide}

