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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}

\section{Advanced Numpy}

\begin{frame}[fragile]

  \frametitle{Broadcasting}

  Try it!

  \begin{lstlisting}

    >>> a = np.arange(4)

    >>> b = np.arange(5)

    >>> a+3

    >>> c=np.array([3])

  \end{lstlisting}

  \begin{itemize}

    \item Enter Broadcasting!

  \end{itemize}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Broadcasting}

  \begin{columns}

    \column{0.65\textwidth}

    \hspace*{-1.5in}

    \begin{lstlisting}

      >>> a = np.arange(4)

      >>> a+3

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

    \end{lstlisting}

    \column{0.35\textwidth}

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

  \end{columns}

  \begin{itemize}

    \item Allows functions to take inputs not of the same shape

    \item 2 rules -

      \begin{enumerate}

      \item 1 is (repeatedly) prepended to shapes of smaller arrays

      \item Size 1 in a dimension changed to Largest size in that dimension

      \end{enumerate}

  \end{itemize}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Broadcasting in 3D}

    \begin{lstlisting}

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

      >>> y = np.ones(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}

  Try it!

  \begin{lstlisting}

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

    >>> b = a

    >>> b is a

    >>> b[0,0]=0; print a

    >>> c = a.view()

    >>> c is a

    >>> c.base is a

    >>> c.flags.owndata

    >>> d = a.copy()

    >>> d.base is a

    >>> d.flags.owndata

  \end{lstlisting}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Copies \& Views}

  Try it!

  \begin{lstlisting}

    >>> a = np.arange(1,9)

    >>> a.shape=3,3

    >>> b = a[0,1:3]

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

    >>> a.flags.owndata

    >>> b.flags.owndata

    >>> b.base

    >>> c.base is a

  \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}

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

    array([[1, 2, 3],

           [4, 5, 6],

           [7, 8, 9]])

    >>> b.flags.owndata

    >>> abool=np.greater(a,2)

    >>> c = a[abool]

    >>> c.flags.owndata

  \end{lstlisting}

  \begin{itemize}

  \item Indexing arrays or Boolean arrays produce copies

  \end{itemize}

\inctime{15}

\end{frame}

\section{SciPy}

\subsection{Introduction}

\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{SciPy - Functions \& Submodules}

  \begin{itemize}

    \item All \typ{numpy} functions are in \typ{scipy} namespace

    \item Domain specific functions organized into subpackages

    \item Subpackages need to be imported separately

  \end{itemize}

  \begin{lstlisting}

    >>> from scipy import linalg

  \end{lstlisting}

\end{frame}

\subsection{Linear Algebra}

\begin{frame}[fragile]

  \frametitle{Linear Algebra}

  Try it!

  \begin{lstlisting}

    >>> import scipy as sp

    >>> from scipy import linalg

    >>> A=sp.mat(np.arange(1,10))

    >>> A.shape=3,3

    >>> linalg.inv(A)

    >>> linalg.det(A)

    >>> linalg.norm(A)

    >>> linalg.expm(A) #logm

    >>> linalg.sinm(A) #cosm, tanm, ...

  \end{lstlisting}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Linear Algebra ...}

  Try it!

  \begin{lstlisting}

    >>> A = sp.mat(np.arange(1,10))

    >>> A.shape=3,3

    >>> linalg.lu(A)

    >>> linalg.eig(A)

    >>> linalg.eigvals(A)

  \end{lstlisting}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Solving Linear 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}

    >>> A = sp.mat([[3,2,-1],[2,-2,4]

                  ,[-1,1/2,-1]])

    >>> B = sp.mat([[1],[-2],[0]])

    >>> linalg.solve(A,B)

  \end{lstlisting}

\inctime{15}

\end{frame}

\subsection{Integration}

\begin{frame}[fragile]

  \frametitle{Integrate}

  \begin{itemize}

    \item Integrating Functions given function object

    \item Integrating Functions given fixed samples

    \item Numerical integrators of ODE systems

  \end{itemize}

  Calculate the area under $(sin(x) + x^2)$ in the range $(0,1)$

  \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\\ 

           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}

\subsection{Interpolation}

\begin{frame}[fragile]

  \frametitle{Interpolation}

  Try it!

  \begin{lstlisting}

>>> from scipy import interpolate

>>> interpolate.interp1d?

>>> x = np.arange(0,2*np.pi,np.pi/4)

>>> y = np.sin(x)

>>> fl = interpolate.interp1d(

            x,y,kind='linear')

>>> fc = interpolate.interp1d(

             x,y,kind='cubic')

>>> fl(np.pi/3)

>>> fc(np.pi/3)

  \end{lstlisting}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Interpolation - Splines}

  Plot the Cubic Spline of $sin(x)$

  \begin{lstlisting}

>>> x = np.arange(0,2*np.pi,np.pi/4)

>>> y = np.sin(x)

>>> tck = interpolate.splrep(x,y)

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

>>> Y = interpolate.splev(X,tck,der=0)

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

>>> plt.show()

  \end{lstlisting}

\inctime{10}

\end{frame}

\subsection{Signal Processing}

\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)

  \end{lstlisting}

    \inctime{5}

\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*}

  Make a plot of $\frac{dx}{dt}$ vs. $x$.

\inctime{30}

\end{frame}

\begin{frame}{Summary}

  \begin{itemize}

    \item Advanced NumPy

    \item SciPy

      \begin{itemize}

        \item Linear Algebra

        \item Integration

        \item Interpolation

        \item Signal and Image processing

      \end{itemize}

  \end{itemize}

\end{frame}

\end{document}

- Numpy arrays (30 mins)

    - Matrices

    - random number generation.

    - Image manipulation: jigsaw puzzle.

    - Monte-carlo integration.