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%Tutorial slides on Python.
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% Author: The FOSSEE Group
% Copyright (c) 2009, The FOSSEE Group, IIT Bombay
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% Title page
\title[Basic Python]{Matrices, Solution of equations and Integration\\}

\author[FOSEE Team] {The FOSSEE Group}

\institute[IIT Bombay] {Department of Aerospace Engineering\\IIT Bombay}
\date[] {31, October 2009\\Day 1, Session 4}
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%% Delete this, if you do not want the table of contents to pop up at
%% the beginning of each subsection:
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  \begin{frame}<beamer>
    \frametitle{Outline}
    \tableofcontents[currentsection,currentsubsection]
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\begin{document}

\begin{frame}
  \titlepage
\end{frame}

\begin{frame}
  \frametitle{Outline}
  \tableofcontents
%  \pausesections
\end{frame}

\section{Matrices}
\subsection{Initializing}
\begin{frame}[fragile]
\frametitle{Matrices: Initializing}
\begin{lstlisting}
  In []: a = matrix([[1,2,3],[4,5,6],[7,8,9]])

  In []: a
  Out[]: 
  matrix([[1, 2, 3],
         [4, 5, 6],
         [7, 8, 9]])
\end{lstlisting}
\end{frame}

\subsection{Basic Operations}
\begin{frame}[fragile]
\frametitle{Inverse of a Matrix}
\begin{lstlisting}
  In []: linalg.inv(a)
  Out[]: 
  matrix([[  3.15221191e+15,  -6.30442381e+15,   3.15221191e+15],
          [ -6.30442381e+15,   1.26088476e+16,  -6.30442381e+15],
          [  3.15221191e+15,  -6.30442381e+15,   3.15221191e+15]])
\end{lstlisting}
\end{frame}

\begin{frame}[fragile]
\frametitle{Determinant}
\begin{lstlisting}
  In []: linalg.det(a)
  Out[]: -9.5171266700777579e-16
\end{lstlisting}
\end{frame}

\begin{frame}[fragile]
\frametitle{Computing Norms}
\begin{lstlisting}
  In []: linalg.norm(a)
  Out[]: 16.881943016134134
  
  In []: linalg.norm?
\end{lstlisting}
\end{frame}

\begin{frame}[fragile]
\frametitle{Eigen Values and Eigen Matrix}
\begin{lstlisting}
  In []: linalg.eigvals(a)
  Out[]: array([  1.61168440e+01,  -1.11684397e+00,  -1.22196337e-15])

  In []: linalg.eig(a)
  Out[]: 
  (array([  1.61168440e+01,  -1.11684397e+00,  -1.22196337e-15]),
   matrix([[-0.23197069, -0.78583024,  0.40824829],
          [-0.52532209, -0.08675134, -0.81649658],
          [-0.8186735 ,  0.61232756,  0.40824829]]))
\end{lstlisting}
\end{frame}

\section{Solving linear equations}
\begin{frame}[fragile]
\frametitle{Solution of equations}
Example problem: Consider the set of equations
  \begin{align*}
    3x + 2y - z  & = 1 \\
    2x - 2y + 4z  & = -2 \\
    -x + \frac{1}{2}y -z & = 0
  \end{align*}

  To Solve this, 
  \begin{lstlisting}
    In []: A = array([[3,2,-1],[2,-2,4],[-1, 0.5, -1]])
    In []: b = array([1, -2, 0])
    In []: x = linalg.solve(A, b)
    In []: Ax = dot(A, x)
    In []: allclose(Ax, b)
    Out[]: True
  \end{lstlisting}
\end{frame}

\section{Integration}

\subsection{ODEs}

\begin{frame}[fragile]
\frametitle{ODE Integration}
We shall use the simple ODE of a simple pendulum. 
\begin{equation*}
\ddot{\theta} = -\frac{g}{L}sin(\theta)
\end{equation*}
\begin{itemize}
\item This equation can be written as a system of two first order ODEs
\end{itemize}
\begin{align}
\dot{\theta} &= \omega \\
\dot{\omega} &= -\frac{g}{L}sin(\theta) \\
 \text{At}\ t &= 0 : \nonumber \\
 \theta = \theta_0\quad & \&\quad  \omega = 0 \nonumber
\end{align}
\end{frame}

\begin{frame}[fragile]
\frametitle{Solving ODEs using SciPy}
\begin{itemize}
\item We use the \typ{odeint} function from scipy to do the integration
\item Define a function as below
\end{itemize}
\begin{lstlisting}
In []: def pend_int(unknown, t, p):
  ....     theta, omega = unknown
  ....     g, L = p
  ....     f=[omega, -(g/L)*sin(theta)]
  ....     return f
  ....
\end{lstlisting}
\end{frame}

\begin{frame}[fragile]
\frametitle{Solving ODEs using SciPy \ldots}
\begin{itemize}
\item \typ{t} is the time variable \\ 
\item \typ{p} has the constants \\
\item \typ{initial} has the initial values
\end{itemize}
\begin{lstlisting}
In []: t = linspace(0, 10, 101)
In []: p=(-9.81, 0.2)
In []: initial = [10*2*pi/360, 0]
\end{lstlisting}
\end{frame}

\begin{frame}[fragile]
\frametitle{Solving ODEs using SciPy \ldots}

\small{\typ{In []: from scipy.integrate import odeint}}
\begin{lstlisting}
In []: pend_sol = odeint(pend_int, 
                         initial,t, 
                         args=(p,))
\end{lstlisting}
\end{frame}

\subsection{Quadrature}

\begin{frame}[fragile]
\frametitle{Quadrature}
Calculate the area under $(sin(x) + x^2)$ in the range $(0,1)$
\small{\typ{In []: from scipy.integrate import quad}}
  \begin{lstlisting}
In []: f(x):
        return sin(x)+x**2
In []: integrate.quad(f, 0, 1)
  \end{lstlisting}
\end{frame}

\end{document}