530 \begin{frame}[fragile]

   531 \frametitle{Least Squares Fit}

   532 \vspace{-0.15in}

   533 \begin{figure}

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

   535 \end{figure}

   536 \end{frame}

   537 

   538 \begin{frame}[fragile]

   539 \frametitle{Least Squares Fit}

   540 \vspace{-0.15in}

   541 \begin{figure}

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

   543 \end{figure}

   544 \end{frame}

   545 

   546 \begin{frame}[fragile]

   547 \frametitle{Least Squares Fit}

   548 \vspace{-0.15in}

   549 \begin{figure}

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

   551 \end{figure}

   552 \end{frame}

   553 

   554 \begin{frame}

   555 \frametitle{Least Square Fit Curve}

   556 \begin{itemize}

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

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

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

   560 \end{itemize}

   561 \end{frame}

   562 

   563 \begin{frame}[fragile]

   564 \frametitle{\typ{lstsq}}

   565 \begin{itemize}

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

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

   568 \item where A is   

   569   $\begin{bmatrix}

   570   L_1 & 1 \\

   571   L_2 & 1 \\

   572   \vdots & \vdots\\

   573   L_N & 1 \\

   574   \end{bmatrix}$

   575   and p is 

   576   $\begin{bmatrix}

   577   m\\

   578   c\\

   579   \end{bmatrix}$

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

   581 \end{itemize}

   582 \end{frame}

   583 

   584 \subsection{Van der Monde matrix generation}

   585 \begin{frame}[fragile]

   586 \frametitle{Van der Monde Matrix}

   587 \begin{itemize}

   588 \item A is also called a Van der Monde matrix

   589 \item It can be generated using \typ{vander}

   590 \end{itemize}

   591 \begin{lstlisting}

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

   593 \end{lstlisting}

   594 Gives the required Van der Monde matrix

   595 \begin{equation*}

   596   \begin{bmatrix}

   597     l_1 & 1 \\

   598     l_2 & 1 \\

   599     \vdots & \vdots\\

   600     l_N & 1 \\

   601   \end{bmatrix}

   602 \end{equation*}

   603 

   604 \end{frame}

   605 

   606 \begin{frame}[fragile]

   607 \frametitle{\typ{lstsq} \ldots}

   608 \begin{itemize}

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

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

   611 \end{itemize}

   612 \begin{lstlisting}

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

   614 \end{lstlisting}

   615 \end{frame}

   616 

   617 \subsection{Plotting}

   618 \begin{frame}[fragile]

   619 \frametitle{Least Square Fit Line \ldots}

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

   621 \begin{lstlisting}

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

   623 \end{lstlisting}

   624 \begin{itemize}

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

   626 \end{itemize}

   627 \begin{lstlisting}

   628 In []: plot(L, Tline)

   629 \end{lstlisting}

   630 \end{frame}

   631 

   632 \begin{frame}[fragile]

   633   \frametitle{What did we learn?}

   634   \begin{itemize}

   635    \item Least square fit

   636    \item Van der Monde matrix generation

   637    \item Plotting the least square fit curve

   638   \end{itemize}

   639 \end{frame}

   640