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% Author: Prabhu Ramachandran <prabhu at aero.iitb.ac.in>

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% Title page

\title[]{Numerical Computing with Numpy \& Scipy}

\author[FOSSEE Team] {Asokan Pichai\\Prabhu Ramachandran}

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

\date[] {11, October 2009}

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% DOCUMENT STARTS

\begin{document}

\begin{frame}

  \maketitle

\end{frame}

\begin{frame}[fragile]

  \frametitle{Broadcasting}

  \begin{itemize}

    \item Used so that functions can take inputs that are not of the same shape.

    \item 2 rules -

      \begin{enumerate}

      \item 1 (repeatedly) pre-pended to shapes of smaller arrays

      \item Size 1 in a dimension -> Largest size in that dimension

      \end{enumerate}

  \end{itemize}

  \begin{columns}

    \column{0.65\textwidth}

    \hspace*{-1.5in}

    \begin{lstlisting}

      >>> x = np.arange(4)

      >>> x+3

      array([3, 4, 5, 6])

    \end{lstlisting}

    \column{0.35\textwidth}

    \includegraphics[height=0.7in, interpolate=true]{data/broadcast_scalar}

  \end{columns}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Broadcasting in 3D}

    \begin{lstlisting}

      >>> x = np.zeros((3, 5))

      >>> y = np.zeros(8)

      >>> (x[..., None] + y).shape

      (3, 5, 8)

    \end{lstlisting}

    \begin{figure}

      \begin{center}

      \includegraphics[height=1.5in, interpolate=true]{data/array_3x5x8}        

      \end{center}

    \end{figure}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Copies \& Views}

  \begin{lstlisting}

    >>> a = array([[1,2,3], [4,5,6],     

                   [7,8,9]])

    >>> a[0,1:3]

    array([2, 3])

    >>> a[0::2,0::2]

    array([[1, 3],

          [7, 9]])

  \end{lstlisting}

  \begin{itemize}

  \item Slicing and Striding just reference the same memory

  \item They produce views of the data, not copies

  \end{itemize}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Copies contd \ldots}

  \begin{lstlisting}

    >>> a[np.array([0,1,2])]

    array([[1, 2, 3],

           [4, 5, 6],

           [7, 8, 9]])

  \end{lstlisting}

  \begin{itemize}

  \item Index arrays or Boolean arrays produce copies

  \end{itemize}

\inctime{15}

\end{frame}

\begin{frame}

  \frametitle{More Numpy Functions \& Methods}

  More functions

  \begin{itemize}

    \item \typ{take}

    \item \typ{choose}

    \item \typ{where}

    \item \typ{compress}

    \item \typ{concatenate}

  \end{itemize}

  Ufunc methods 

  \begin{itemize}

    \item \typ{reduce}

    \item \typ{accumulate}

    \item \typ{outer}

    \item \typ{reduceat}

  \end{itemize}

\inctime{5}

\end{frame}

\begin{frame}

    {Intro to SciPy}

  \begin{itemize}

  \item \url{http://www.scipy.org}

  \item Open source scientific libraries for Python

  \item Based on NumPy

    \end{itemize}

\end{frame}

\begin{frame}

  \frametitle{SciPy}

  \begin{itemize}

  \item Provides:

    \begin{itemize}

    \item Linear algebra

    \item Numerical integration

    \item Fourier transforms

    \item Signal processing

    \item Special functions

    \item Statistics

    \item Optimization

    \item Image processing

    \item ODE solvers

    \end{itemize}

  \item Uses LAPACK, QUADPACK, ODEPACK, FFTPACK etc. from netlib

  \end{itemize}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Linear Algebra}

  \typ{>>> from scipy import linalg}

  \begin{itemize}

    \item \typ{linalg.det, linalg.norm}

    \item \typ{linalg.eig, linalg.lu}

    \item \typ{linalg.expm, linalg.logm}

    \item \typ{linalg.sinm, linalg.sinhm}

  \end{itemize}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Linear Algebra \ldots}

  \begin{align*}

    3x + 2y - z  & = 1 \\

    2x - 2y + 4z  & = -2 \\

    -x + \frac{1}{2}y -z & = 0

  \end{align*}

  \begin{lstlisting}

    >>> linalg.solve(A,B)

  \end{lstlisting}

\inctime{15}

\end{frame}

\begin{frame}[fragile]

  \begin{itemize}

    \item Integrating Functions given function object

    \item Integrating Functions given fixed samples

    \item Numerical integrators of ODE systems

  \end{itemize}

  \frametitle{Integrate}

  Calculate $\int^1_0sin(x) + x^2$

  \begin{lstlisting}

    >>> def f(x):

            return np.sin(x)+x**2

    >>> integrate.quad(f, 0, 1)

  \end{lstlisting}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Integrate \ldots}

  Numerically solve ODEs\\

  \begin{align*}

  \frac{dx}{dt}&=-e^{(-t)}x^2(t)\\ 

           x(0)&=2    

  \end{align*}

  \begin{lstlisting}

    def dx_dt(x,t):

        return -np.exp(-t)*x**2

    x=integrate.odeint(dx_dt, 2, t)

    plt.plot(x,t)

  \end{lstlisting}

\inctime{10}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Interpolation}

  \begin{itemize}

    \item \typ{interpolate.interp1d, ...}

    \item \typ{interpolate.splrep, splev}

  \end{itemize}

  Cubic Spline of $sin(x)$

  \begin{lstlisting}

    x = np.arange(0,2*np.pi,np.pi/8)

    y = np.sin(x)

    t = interpolate.splrep(x,y,s=0)

    X = np.arange(0,2*np.pi,np.pi/50)

    Y = interpolate.splev(X,t,der=0)

    plt.plot(x,y,'o',x,y,X,Y)

    plt.show()

  \end{lstlisting}

\inctime{10}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Signal \& Image Processing}

    \begin{itemize}

     \item Convolution

     \item B-splines

     \item Filtering

     \item Filter design

     \item IIR filter design

     \item Linear Systems

     \item LTI Reresentations

     \item Waveforms

     \item Window functions

     \item Wavelets

    \end{itemize}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Signal \& Image Processing}

  Applying a simple median filter

  \begin{lstlisting}

    from scipy import signal, ndimage

    from scipy import lena

    A=lena().astype('float32')

    B=signal.medfilt2d(A)

    imshow(B)

  \end{lstlisting}

  Zooming an array - uses spline interpolation

  \begin{lstlisting}

    b=ndimage.zoom(A,0.5)

    imshow(b)

    \inctime{5}

  \end{lstlisting}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Problems}

  The Van der Pol oscillator is a type of nonconservative oscillator with nonlinear damping. It evolves in time according to the second order differential equation:

  \begin{equation*}

  \frac{d^2x}{dt^2}+\mu(x^2-1)\frac{dx}{dt}+x= 0

  \end{equation*}

\inctime{25}

\end{frame}

\end{document}

- Numpy arrays (30 mins)

    - Matrices

    - random number generation.

    - Image manipulation: jigsaw puzzle.

    - Monte-carlo integration.