1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

     2 % Tutorial slides on Python.

     3 %

     4 % Author: Prabhu Ramachandran <prabhu at aero.iitb.ac.in>

     5 % Copyright (c) 2005-2009, Prabhu Ramachandran

     6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

     7 

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    27 % Taken from Fernando's slides.

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    58 %%% This is from Fernando's setup.

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    72 

    73 

    74 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    75 % Title page

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

    77 

    78 \author[FOSSEE Team] {FOSSEE}

    79 

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

    81 \date[] {11, October 2009}

    82 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    83 

    84 %\pgfdeclareimage[height=0.75cm]{iitmlogo}{iitmlogo}

    85 %\logo{\pgfuseimage{iitmlogo}}

    86 

    87 

    88 %% Delete this, if you do not want the table of contents to pop up at

    89 %% the beginning of each subsection:

    90 \AtBeginSubsection[]

    91 {

    92   \begin{frame}<beamer>

    93     \frametitle{Outline}

    94     \tableofcontents[currentsection,currentsubsection]

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    96 }

    97 

    98 \AtBeginSection[]

    99 {

   100   \begin{frame}<beamer>

   101     \frametitle{Outline}

   102     \tableofcontents[currentsection,currentsubsection]

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   104 }

   105 

   106 % If you wish to uncover everything in a step-wise fashion, uncomment

   107 % the following command: 

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   109 

   110 %\includeonlyframes{current,current1,current2,current3,current4,current5,current6}

   111 

   112 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

   113 % DOCUMENT STARTS

   114 \begin{document}

   115 

   116 \begin{frame}

   117   \maketitle

   118 \end{frame}

   119 

   120 \section{Advanced Numpy}

   121 \begin{frame}[fragile]

   122   \frametitle{Broadcasting}

   123   \begin{lstlisting}

   124     >>> a = arange(4)

   125     >>> b = arange(5)

   126     >>> a+b #Does this work?

   127     >>> a+3

   128     >>> c = array([3])

   129     >>> a+c #Works!

   130     >>> b+c #But how?

   131     >>> a.shape, b.shape, c.shape

   132   \end{lstlisting}

   133   \begin{itemize}

   134     \item Enter Broadcasting!

   135   \end{itemize}

   136 \end{frame}

   137 

   138 \begin{frame}[fragile]

   139   \frametitle{Broadcasting}

   140   \begin{columns}

   141     \column{0.65\textwidth}

   142     \hspace*{-1.5in}

   143     \begin{lstlisting}

   144       >>> a = arange(4)

   145       >>> a+3

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

   147     \end{lstlisting}

   148     \column{0.35\textwidth}

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

   150   \end{columns}

   151 \end{frame}

   152 

   153 \begin{frame}[fragile]

   154   \frametitle{Broadcasting in 3D}

   155     \begin{lstlisting}

   156       >>> x = ones((3, 5, 1))

   157       >>> y = ones(8)

   158       >>> (x + y).shape

   159       (3, 5, 8)

   160     \end{lstlisting}

   161     \begin{figure}

   162       \begin{center}

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

   164       \end{center}

   165     \end{figure}

   166 \end{frame}

   167 

   168 \begin{frame}[fragile]

   169   \frametitle{Copies \& Views}

   170   \vspace{-0.1in}

   171   \begin{lstlisting}

   172     >>> a = arange(1,9); a.shape=3,3

   173     >>> b = a

   174     >>> b is a

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

   176     >>> c = a.view()

   177     >>> c is a

   178     >>> c.base is a

   179     >>> c.flags.owndata

   180     >>> d = a.copy()

   181     >>> d.base is a

   182     >>> d.flags.owndata

   183   \end{lstlisting}

   184 \end{frame}

   185 

   186 \begin{frame}[fragile]

   187   \frametitle{Copies \& Views}

   188   \vspace{-0.1in}

   189   \begin{lstlisting}

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

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

   192     >>> a.flags.owndata

   193     >>> b.flags.owndata

   194     >>> b.base

   195     >>> c.base is a

   196   \end{lstlisting}

   197   \begin{itemize}

   198   \item Slicing and Striding just reference the same memory

   199   \item They produce views of the data, not copies

   200   \end{itemize}

   201 \end{frame}

   202 

   203 \begin{frame}[fragile]

   204   \frametitle{Copies contd \ldots}

   205   \begin{lstlisting}

   206     >>> a = arange(1, 10, 2)

   207     >>> b = a[array([0,2,3])]

   208     >>> b.flags.owndata

   209     >>> abool=a>5

   210     >>> c = a[abool]

   211     >>> c.flags.owndata

   212   \end{lstlisting}

   213   \begin{itemize}

   214   \item Indexing arrays or Boolean arrays produce copies

   215   \end{itemize}

   216 \inctime{15}

   217 \end{frame}

   218 

   219 \section{SciPy}

   220 \subsection{Introduction}

   221 \begin{frame}

   222     {Intro to SciPy}

   223   \begin{itemize}

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

   225   \item Open source scientific libraries for Python

   226   \item Based on NumPy

   227     \end{itemize}

   228 \end{frame}

   229 

   230 \begin{frame}

   231   \frametitle{SciPy}

   232   \begin{itemize}

   233   \item Provides:

   234     \begin{itemize}

   235     \item Linear algebra

   236     \item Numerical integration

   237     \item Fourier transforms

   238     \item Signal processing

   239     \item Special functions

   240     \item Statistics

   241     \item Optimization

   242     \item Image processing

   243     \item ODE solvers

   244     \end{itemize}

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

   246   \end{itemize}

   247 \end{frame}

   248 

   249 \begin{frame}[fragile]

   250   \frametitle{SciPy - Functions \& Submodules}

   251   \begin{itemize}

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

   253     \item Domain specific functions organized into subpackages

   254     \item Subpackages need to be imported separately

   255   \end{itemize}

   256   \begin{lstlisting}

   257     >>> from scipy import linalg

   258   \end{lstlisting}

   259 \end{frame}

   260 

   261 \subsection{Linear Algebra}

   262 \begin{frame}[fragile]

   263   \frametitle{Linear Algebra}

   264   \begin{lstlisting}

   265     >>> import scipy as sp

   266     >>> from scipy import linalg

   267     >>> A = sp.array(sp.arange(1,10))

   268     >>> A.shape = 3,3

   269     >>> linalg.inv(A)

   270     >>> linalg.det(A)

   271     >>> linalg.norm(A)

   272     >>> linalg.expm(A) #logm

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

   274   \end{lstlisting}

   275 \end{frame}

   276 

   277 \begin{frame}[fragile]

   278   \frametitle{Linear Algebra ...}

   279   \begin{lstlisting}

   280     >>> A = sp.array(sp.arange(1,10))

   281     >>> A.shape = 3,3

   282     >>> linalg.lu(A)

   283     >>> linalg.eig(A)

   284     >>> linalg.eigvals(A)

   285   \end{lstlisting}

   286 \end{frame}

   287 

   288 \begin{frame}[fragile]

   289   \frametitle{Solving Linear Equations}

   290   \vspace{-0.2in}

   291   \begin{align*}

   292     3x + 2y - z  & = 1 \\

   293     2x - 2y + 4z  & = -2 \\

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

   295   \end{align*}

   296   To Solve this, 

   297   \begin{lstlisting}

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

   299                   ,[-1,1/2,-1]])

   300     >>> b = sp.array([1,-2,0])

   301     >>> x = linalg.solve(A,b)

   302     >>> Ax = sp.dot(A,x)

   303     >>> sp.allclose(Ax, b)

   304   \end{lstlisting}

   305 \inctime{15}

   306 \end{frame}

   307 

   308 \subsection{Integration}

   309 \begin{frame}[fragile]

   310   \frametitle{Integrate}

   311   \begin{itemize}

   312     \item Integrating Functions given function object

   313     \item Integrating Functions given fixed samples

   314     \item Numerical integrators of ODE systems

   315   \end{itemize}

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

   317   \begin{lstlisting}

   318     >>> def f(x):

   319             return sin(x)+x**2

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

   321   \end{lstlisting}

   322 \end{frame}

   323 

   324 \begin{frame}[fragile]

   325   \frametitle{Integrate \ldots}

   326   Numerically solve ODEs\\

   327   \begin{align*}

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

   329            x&=2 \quad at \ t=0

   330   \end{align*}

   331   \begin{lstlisting}

   332 >>> def dx_dt(x,t):

   333         return -exp(-t)*x**2

   334 >>> t = linspace(0,2,100)

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

   336 >>> plt.plot(x,t)

   337   \end{lstlisting}

   338 \inctime{10}

   339 \end{frame}

   340 

   341 \subsection{Interpolation}

   342 \begin{frame}[fragile]

   343   \frametitle{Interpolation}

   344   \begin{lstlisting}

   345 >>> from scipy import interpolate

   346 >>> interpolate.interp1d?

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

   348 >>> y = sin(x)

   349 >>> fl = interpolate.interp1d(

   350             x,y,kind='linear')

   351 >>> fc = interpolate.interp1d(

   352              x,y,kind='cubic')

   353 >>> fl(pi/3)

   354 >>> fc(pi/3)

   355   \end{lstlisting}

   356 \end{frame}

   357 

   358 \begin{frame}[fragile]

   359   \frametitle{Interpolation - Splines}

   360   Plot the Cubic Spline of $sin(x)$

   361   \begin{lstlisting}

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

   363 >>> xs = arange(0,2*pi,pi/50)

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

   365 >>> plt.plot(x,y,'o',x,y,xs,ys)

   366 >>> plt.show()

   367   \end{lstlisting}

   368 \inctime{10}

   369 \end{frame}

   370 

   371 \subsection{Signal Processing}

   372 \begin{frame}[fragile]

   373   \frametitle{Signal \& Image Processing}

   374     \begin{itemize}

   375      \item Convolution

   376      \item Filtering

   377      \item Filter design

   378      \item IIR filter design

   379      \item Linear Systems

   380      \item LTI Representations

   381      \item Window functions

   382     \end{itemize}

   383 \end{frame}

   384 

   385 \begin{frame}[fragile]

   386   \frametitle{Signal \& Image Processing}

   387   Applying a simple median filter

   388   \begin{lstlisting}

   389 >>> from scipy import signal, ndimage

   390 >>> from scipy import lena

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

   392 >>> B = signal.medfilt2d(A)

   393 >>> imshow(B)

   394   \end{lstlisting}

   395   Zooming an array - uses spline interpolation

   396   \begin{lstlisting}

   397 >>> b = ndimage.zoom(A,0.5)

   398 >>> imshow(b)

   399 >>> b = ndimage.zoom(A,2)

   400   \end{lstlisting}

   401     \inctime{5}

   402 \end{frame}

   403 

   404 \begin{frame}[fragile]

   405   \frametitle{Problems}

   406   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:

   407   \begin{equation*}

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

   409   \end{equation*}

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

   411 \inctime{30}

   412 \end{frame}

   413 \begin{frame}{Summary}

   414   \begin{itemize}

   415     \item Advanced NumPy

   416     \item SciPy

   417       \begin{itemize}

   418         \item Linear Algebra

   419         \item Integration

   420         \item Interpolation

   421         \item Signal and Image processing

   422       \end{itemize}

   423   \end{itemize}

   424 \end{frame}

   425 \end{document}

   426 

   427 - Numpy arrays (30 mins)

   428     - Matrices

   429     - random number generation.

   430     - Image manipulation: jigsaw puzzle.

   431     - Monte-carlo integration.