307 \end{frame}

   308 

   309 \section{Least Squares Fit}

   310 \begin{frame}[fragile]

   311 \frametitle{Least Squares Fit}

   312 \vspace{-0.15in}

   313 \begin{figure}

   314 \includegraphics[width=4in]{data/L-Tsq-Line.png}

   315 \end{figure}

   316 \end{frame}

   317 

   318 \begin{frame}[fragile]

   319 \frametitle{Least Squares Fit}

   320 \vspace{-0.15in}

   321 \begin{figure}

   322 \includegraphics[width=4in]{data/L-Tsq-points.png}

   323 \end{figure}

   324 \end{frame}

   325 

   326 \begin{frame}[fragile]

   327 \frametitle{Least Squares Fit}

   328 \vspace{-0.15in}

   329 \begin{figure}

   330 \includegraphics[width=4in]{data/least-sq-fit.png}

   331 \end{figure}

   332 \end{frame}

   333 

   334 \begin{frame}

   335 \frametitle{Least Square Fit Curve}

   336 \begin{itemize}

   337 \item $T^2$ and $L$ have a linear relationship

   338 \item Hence, Least Square Fit Curve is a line

   339 \item we shall use the \typ{lstsq} function

   340 \end{itemize}

   341 \end{frame}

   342 

   343 \begin{frame}[fragile]

   344 \frametitle{\typ{lstsq}}

   345 \begin{itemize}

   346 \item We need to fit a line through points for the equation $T^2 = m \cdot L+c$

   347 \item The equation can be re-written as $T^2 = A \cdot p$

   348 \item where A is   

   349   $\begin{bmatrix}

   350   L_1 & 1 \\

   351   L_2 & 1 \\

   352   \vdots & \vdots\\

   353   L_N & 1 \\

   354   \end{bmatrix}$

   355   and p is 

   356   $\begin{bmatrix}

   357   m\\

   358   c\\

   359   \end{bmatrix}$

   360 \item We need to find $p$ to plot the line

   361 \end{itemize}

   362 \end{frame}

   363 

   364 \begin{frame}[fragile]

   365 \frametitle{Generating $A$}

   366 \begin{lstlisting}

   367 In []: A = array([L, ones_like(L)])

   368 In []: A = A.T

   369 \end{lstlisting}

   370 %% \begin{itemize}

   371 %% \item A is also called a Van der Monde matrix

   372 %% \item It can also be generated using \typ{vander}

   373 %% \end{itemize}

   374 %% \begin{lstlisting}

   375 %% In []: A = vander(L, 2)

   376 %% \end{lstlisting}

   377 \end{frame}

   378 

   379 \begin{frame}[fragile]

   380 \frametitle{\typ{lstsq} \ldots}

   381 \begin{itemize}

   382 \item Now use the \typ{lstsq} function

   383 \item Along with a lot of things, it returns the least squares solution

   384 \end{itemize}

   385 \begin{lstlisting}

   386 In []: coef, res, r, s = lstsq(A,TSq)

   387 \end{lstlisting}

   388 \end{frame}

   389 

   390 \subsection{Plotting}

   391 \begin{frame}[fragile]

   392 \frametitle{Least Square Fit Line \ldots}

   393 We get the points of the line from \typ{coef}

   394 \begin{lstlisting}

   395 In []: Tline = coef[0]*L + coef[1]

   396 \end{lstlisting}

   397 \begin{itemize}

   398 \item Now plot Tline vs. L, to get the Least squares fit line. 

   399 \end{itemize}

   400 \begin{lstlisting}

   401 In []: plot(L, Tline)

   402 \end{lstlisting}

   575   \item Least Square Curve fitting