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%Tutorial slides on Python.

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% Author: FOSSEE 

% Copyright (c) 2009, FOSSEE, IIT Bombay

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% Title page

\title[Matrices \& Equations]{Python for Science and Engg: Matrices, Least Square Fit, \& Solution of equations}

\author[FOSSEE] {FOSSEE}

\institute[IIT Bombay] {Department of Aerospace Engineering\\IIT Bombay}

\date[] {31, October 2009\\Day 1, Session 4}

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    \frametitle{Outline}

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\begin{document}

\begin{frame}

  \titlepage

\end{frame}

\begin{frame}

  \frametitle{Outline}

  \tableofcontents

%  \pausesections

\end{frame}

\section{Matrices}

\begin{frame}

\frametitle{Matrices: Introduction}

\alert{All matrix operations are done using \kwrd{arrays}}

\end{frame}

\begin{frame}[fragile]

\frametitle{Matrices: Initializing}

\begin{lstlisting}

In []: A = array([[ 1,  1,  2, -1],

                  [ 2,  5, -1, -9],

                  [ 2,  1, -1,  3],

                  [ 1, -3,  2,  7]])

In []: A

Out[]: 

array([[ 1,  1,  2, -1],

       [ 2,  5, -1, -9],

       [ 2,  1, -1,  3],

       [ 1, -3,  2,  7]])

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Accessing elements}

  \begin{lstlisting}

In []: C = array([[1,1,2],

                  [2,4,1],

                  [-1,3,7]])

In []: C[1][2]

Out[]: 1

In []: C[1,2]

Out[]: 1

In []: C[1]

Out[]: array([2, 4, 1])

  \end{lstlisting}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Changing elements}

  \begin{small}

  \begin{lstlisting}

In []: C[1,1] = -2

In []: C

Out[]: 

array([[ 1,  1,  2],

       [ 2, -2,  1],

       [-1,  3,  7]])

In []: C[1] = [0,0,0]

In []: C

Out[]: 

array([[ 1,  1,  2],

       [ 0,  0,  0],

       [-1,  3,  7]])

  \end{lstlisting}

  \end{small}

How to change one \alert{column}?

\end{frame}

\begin{frame}[fragile]

  \frametitle{Slicing}

\begin{small}

  \begin{lstlisting}

In []: C[:,1]

Out[]: array([1, 0, 3])

In []: C[1,:]

Out[]: array([0, 0, 0])

In []: C[0:2,:]

Out[]: 

array([[1, 1, 2],

       [0, 0, 0]])

In []: C[1:3,:]

Out[]: 

array([[ 0,  0,  0],

       [-1,  3,  7]])

  \end{lstlisting}

\end{small}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Slicing \ldots}

\begin{small}

  \begin{lstlisting}

In []: C[:2,:]

Out[]: 

array([[1, 1, 2],

       [0, 0, 0]])

In []: C[1:,:]

Out[]: 

array([[ 0,  0,  0],

       [-1,  3,  7]])

In []: C[1:,:2]

Out[]: 

array([[ 0,  0],

       [-1,  3]])

  \end{lstlisting}

\end{small}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Striding}

  \begin{lstlisting}

  \end{lstlisting}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Slicing \& Striding Exercises}

  \begin{lstlisting}

  \end{lstlisting}

\end{frame}

\subsection{Basic Operations}

\begin{frame}[fragile]

\frametitle{Transpose of a Matrix}

\begin{lstlisting}

In []: A.T

Out[]:

array([[ 1,  2,  2,  1],

       [ 1,  5,  1, -3],

       [ 2, -1, -1,  2],

       [-1, -9,  3,  7]])

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Sum of all elements}

  \begin{lstlisting}

In []: sum(A)

Out[]: 12

  \end{lstlisting}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Matrix Addition}

  \begin{lstlisting}

In []: B = array([[3,2,-1,5],

                  [2,-2,4,9],

                  [-1,0.5,-1,-7],

                  [9,-5,7,3]])

In []: A + B

Out[]: 

array([[  4. ,   3. ,   1. ,   4. ],

       [  4. ,   3. ,   3. ,   0. ],

       [  1. ,   1.5,  -2. ,  -4. ],

       [ 10. ,  -8. ,   9. ,  10. ]])

  \end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Elementwise Multiplication}

\begin{lstlisting}

In []: A*B

Out[]: 

array([[  3. ,   2. ,  -2. ,  -5. ],

       [  4. , -10. ,  -4. , -81. ],

       [ -2. ,   0.5,   1. , -21. ],

       [  9. ,  15. ,  14. ,  21. ]])

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Matrix Multiplication}

\begin{lstlisting}

In []: dot(A,B)

Out[]: 

array([[ -6. ,   6. ,  -6. ,  -3. ],

       [-64. ,  38.5, -44. ,  35. ],

       [ 36. , -13.5,  24. ,  35. ],

       [ 58. , -26. ,  34. , -15. ]])

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Inverse of a Matrix}

\begin{lstlisting}

In []: inv(A)

\end{lstlisting}

\begin{small}

\begin{lstlisting}

Out[]: 

array([[-0.5 ,  0.55, -0.15,  0.7 ],

       [ 0.75, -0.5 ,  0.5 , -0.75],

       [ 0.5 , -0.15, -0.05, -0.1 ],

       [ 0.25, -0.25,  0.25, -0.25]])

\end{lstlisting}

\end{small}

\end{frame}

\begin{frame}[fragile]

\frametitle{Determinant}

\begin{lstlisting}

In []: det(A)

Out[]: 80.0

\end{lstlisting}

\end{frame}

%%use S=array(X,Y)

\begin{frame}[fragile]

\frametitle{Eigenvalues and Eigen Vectors}

\begin{small}

\begin{lstlisting}

In []: E = array([[3,2,4],[2,0,2],[4,2,3]])

In []: eig(E)

Out[]: 

(array([-1.,  8., -1.]),

 array([[-0.74535599,  0.66666667, -0.1931126 ],

        [ 0.2981424 ,  0.33333333, -0.78664085],

        [ 0.59628479,  0.66666667,  0.58643303]]))

In []: eigvals(E)

Out[]: array([-1.,  8., -1.])

\end{lstlisting}

\end{small}

\end{frame}

%% \begin{frame}[fragile]

%% \frametitle{Computing Norms}

%% \begin{lstlisting}

%% In []: norm(E)

%% Out[]: 8.1240384046359608

%% \end{lstlisting}

%% \end{frame}

%% \begin{frame}[fragile]

%%   \frametitle{Singular Value Decomposition}

%%   \begin{small}

%%   \begin{lstlisting}

%% In []: svd(E)

%% Out[]: 

%% (array(

%% [[ -6.66666667e-01,  -1.23702565e-16,   7.45355992e-01],

%%  [ -3.33333333e-01,  -8.94427191e-01,  -2.98142397e-01],

%%  [ -6.66666667e-01,   4.47213595e-01,  -5.96284794e-01]]),

%%  array([ 8.,  1.,  1.]),

%%  array([[-0.66666667, -0.33333333, -0.66666667],

%%         [-0.        ,  0.89442719, -0.4472136 ],

%%         [-0.74535599,  0.2981424 ,  0.59628479]]))

%%   \end{lstlisting}

%%   \end{small}

%% \inctime{15}

%% \end{frame}

\section{Least Squares Fit}

\begin{frame}[fragile]

\frametitle{$L$ vs. $T^2$}

\vspace{-0.15in}

\begin{figure}

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

\end{figure}

\end{frame}

\begin{frame}[fragile]

\frametitle{$L$ vs. $T^2$}

\vspace{-0.15in}

\begin{figure}

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

\end{figure}

\end{frame}

\begin{frame}[fragile]

\frametitle{Least Squares Fit}

\vspace{-0.15in}

\begin{figure}

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

\end{figure}

\end{frame}

\begin{frame}

\frametitle{Least Square Fit Curve}

\begin{itemize}

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

\item Hence, Least Square Fit Curve is a line

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

\end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{\typ{lstsq}}

\begin{itemize}

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

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

\item where A is   

  $\begin{bmatrix}

  L_1 & 1 \\

  L_2 & 1 \\

  \vdots & \vdots\\

  L_N & 1 \\

  \end{bmatrix}$

  and p is 

  $\begin{bmatrix}

  m\\

  c\\

  \end{bmatrix}$

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

\end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{Generating $A$}

\begin{lstlisting}

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

In []: A = A.T

\end{lstlisting}

%% \begin{itemize}

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

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

%% \end{itemize}

%% \begin{lstlisting}

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

%% \end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{\typ{lstsq} \ldots}

\begin{itemize}

\item Now use the \typ{lstsq} function

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

\end{itemize}

\begin{lstlisting}

In []: result = lstsq(A,TSq)

In []: coef = result[0]

\end{lstlisting}

\end{frame}

\subsection{Plotting}

\begin{frame}[fragile]

\frametitle{Least Square Fit Line \ldots}

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

\begin{lstlisting}

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

\end{lstlisting}

\begin{itemize}

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

\end{itemize}

\begin{lstlisting}

In []: plot(L, Tline)

\end{lstlisting}

\end{frame}

\section{Summary}

\begin{frame}

  \frametitle{What did we learn??}

  \begin{itemize}

  \item Matrices

    \begin{itemize}

      \item Accessing elements

      \item Transpose

      \item Addition

      \item Multiplication

      \item Inverse of a matrix

      \item Determinant

      \item Eigenvalues and Eigen vector

      \item Norms

      \item Singular Value Decomposition

    \end{itemize}

  \item Least Square Curve fitting

  \item Solving linear equations

  \end{itemize}