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−%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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−%Tutorial slides on Python.
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−%
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−% Author: FOSSEE 
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−% Copyright (c) 2009, FOSSEE, IIT Bombay
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−%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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−
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−% Modified from: generic-ornate-15min-45min.de.tex
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−%\usepackage{times}
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−\usepackage{amsmath}
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−
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−% Taken from Fernando's slides.
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−\usepackage{ae,aecompl}
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−\usepackage{listings}
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−\lstset{language=Python,
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−
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−\newcommand{\kwrd}[1]{ \texttt{\textbf{\color{blue}{#1}}}  }
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−
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−%%% This is from Fernando's setup.
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−% \usepackage{color}
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−% \definecolor{orange}{cmyk}{0,0.4,0.8,0.2}
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−% % Use and configure listings package for nicely formatted code
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−% \usepackage{listings}
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−% \lstset{
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−%    language=Python,
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−%    breaklines=true,
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−%    postbreak = \space\dots
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−% }
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−
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−
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−%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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−% Title page
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−\title[Calculus]{Python for Science and Engg: Interpolation, Differentiation and Integration}
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−
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−\author[FOSSEE] {FOSSEE}
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−
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−\institute[IIT Bombay] {Department of Aerospace Engineering\\IIT Bombay}
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−\date[] {31, October 2009\\Day 1, Session 5}
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−%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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−
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−%\pgfdeclareimage[height=0.75cm]{iitmlogo}{iitmlogo}
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−%\logo{\pgfuseimage{iitmlogo}}
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−
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−
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−%% Delete this, if you do not want the table of contents to pop up at
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−%% the beginning of each subsection:
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−\AtBeginSubsection[]
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−{
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−  \begin{frame}<beamer>
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−    \frametitle{Outline}
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−    \tableofcontents[currentsection,currentsubsection]
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−  \end{frame}
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−}
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−
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−\AtBeginSection[]
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−{
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−  \begin{frame}<beamer>
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−   \frametitle{Outline}
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−   \tableofcontents[currentsection,currentsubsection]
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−  \end{frame}
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−}
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−
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−% If you wish to uncover everything in a step-wise fashion, uncomment
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−% the following command: 
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−%\beamerdefaultoverlayspecification{<+->}
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−
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−%\includeonlyframes{current,current1,current2,current3,current4,current5,current6}
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−
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−%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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−% DOCUMENT STARTS
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−\begin{document}
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−
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−\begin{frame}
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−  \titlepage
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−\end{frame}
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−
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−%% \begin{frame}
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−%%   \frametitle{Outline}
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−%%   \tableofcontents
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−%% %  \pausesections
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−%% \end{frame}
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−
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−\section{\typ{loadtxt}}
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−
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−\begin{frame}[fragile]
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−  \frametitle{Array slicing}
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−  \begin{lstlisting}
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−In []: A = array([[ 1,  1,  2, -1],
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−                  [ 2,  5, -1, -9],
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−                  [ 2,  1, -1,  3],
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−                  [ 1, -3,  2,  7]])
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−
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−In []: A[:,0]
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−Out[]: array([ 1,  2,  2, 1])
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−
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−In []: A[1:3,1:3]
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−Out[]: 
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−array([[ 5, -1],
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−       [ 1, -1]])
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−\end{lstlisting}
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−\end{frame}
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−
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−\begin{frame}[fragile]
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−  \frametitle{\typ{loadtxt}}
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−  \begin{itemize}
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−  \item Load data from a text file.
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−  \item Each row must have same number of values.
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−  \end{itemize}
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−\begin{lstlisting}
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−In []: data = loadtxt('pendulum.txt')
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−In []: x = data[:, 0]
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−In []: y = data[:, 1]
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−\end{lstlisting}
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−\end{frame}
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−
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−%% \begin{frame}[fragile]
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−%%   \frametitle{\typ{loadtxt}}
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−%% \end{frame}
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−
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−\section{Interpolation}
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−\begin{frame}[fragile]
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−\frametitle{Loading data (revisited)}
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−\begin{itemize}
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−  \item Given data file \typ{points.txt}.
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−  \item It contains x,y position of particle.
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−  \item Plot the given points.
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−%%  \item Interpolate the missing region.
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−\end{itemize}
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−\begin{lstlisting}
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−In []: x, y = loadtxt('points.txt',
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−                       unpack = True)
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−In []: plot(x, y, '.')
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−\end{lstlisting}
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−\end{frame}
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−
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−\begin{frame}
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−  \frametitle{Plot}
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−  \begin{center}
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−    \includegraphics[height=2in, interpolate=true]{data/missing_points}
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−  \end{center}
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−\end{frame}
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−%% \begin{frame}[fragile]
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−%% \frametitle{Interpolation \ldots}
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−%% \begin{small}
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−%%   \typ{In []: from scipy.interpolate import interp1d}
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−%% \end{small}
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−%% \begin{itemize}
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−%% \item The \typ{interp1d} function returns a function
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−%% \begin{lstlisting}
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−%%   In []: f = interp1d(L, T)
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−%% \end{lstlisting}
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−%% \item Functions can be assigned to variables 
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−%% \item This function interpolates between known data values to obtain unknown
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−%% \end{itemize}
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−%% \end{frame}
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−
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−%% \begin{frame}[fragile]
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−%% \frametitle{Interpolation \ldots}
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−%% \begin{lstlisting}
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−%% In []: Ln = arange(0.1,0.99,0.005)
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−%% # Interpolating! 
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−%% # The new values in range of old data
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−%% In []: plot(L, T, 'o', Ln, f(Ln), '-')
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−%% In []: f = interp1d(L, T, kind='cubic')
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−%% # When kind not specified, it's linear
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−%% # Others are ...
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−%% # 'nearest', 'zero', 
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−%% # 'slinear', 'quadratic'
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−%% \end{lstlisting}
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−%% \end{frame}
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−
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−\begin{frame}[fragile]
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−\frametitle{Spline Interpolation}
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−\begin{small}
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−\begin{lstlisting}
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−In []: from scipy.interpolate import splrep
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−In []: from scipy.interpolate import splev
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−\end{lstlisting}
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−\end{small}
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−\begin{itemize}
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−\item Involves two steps
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−  \begin{enumerate}
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−  \item Find out the spline curve, coefficients
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−  \item Evaluate the spline at new points
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−  \end{enumerate}
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−\end{itemize}
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−\end{frame}
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−
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−\begin{frame}[fragile]
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−\frametitle{\typ{splrep}}
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−To find the spline curve
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−\begin{lstlisting}
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−In []: tck = splrep(x, y)
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−\end{lstlisting}
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−\typ{tck} contains parameters required for representing the spline curve!
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−\end{frame}
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−
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−\begin{frame}[fragile]
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−\frametitle{\typ{splev}}
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−To Evaluate a spline and it's derivatives
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−\begin{lstlisting}
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−In []: Xnew = arange(0.01,3,0.02)
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−In []: Ynew = splev(Xnew, tck)
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−
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−In []: y.shape
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−Out[]: (40,)
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−
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−In []: Ynew.shape
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−Out[]: (150,)
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− 
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−In []: plot(Xnew, Ynew)
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−\end{lstlisting}
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−
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−\end{frame}
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−
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−%% \begin{frame}[fragile]
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−%% \frametitle{Interpolation \ldots}
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−%% \begin{itemize}
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−%% \item 
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−%% \end{itemize}
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−%% \end{frame}
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−
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−\begin{frame}
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−  \frametitle{Plot}
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−  \begin{center}
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−    \includegraphics[height=2in, interpolate=true]{data/interpolate}
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−  \end{center}
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−\end{frame}
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−
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−\section{Differentiation}
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−
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−\begin{frame}[fragile]
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−\frametitle{Numerical Differentiation}
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−\begin{itemize}
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−\item Given function $f(x)$ or data points $y=f(x)$
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−\item We wish to calculate $f^{'}(x)$ at points $x$
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−\item Taylor series - finite difference approximations
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−\end{itemize}
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−\begin{center}
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−\begin{tabular}{l l}
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−$f(x+h)=f(x)+hf^{'}(x)$ &Forward \\
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−$f(x-h)=f(x)-hf^{'}(x)$ &Backward
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−\end{tabular}
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−\end{center}
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−\end{frame}
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−
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−\begin{frame}[fragile]
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−\frametitle{Forward Difference}
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−\begin{lstlisting}
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−In []: x = linspace(0, 2*pi, 100)
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−In []: y = sin(x)
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−In []: deltax = x[1] - x[0]
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−\end{lstlisting}
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−Obtain the finite forward difference of y
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−\end{frame}
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−
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−\begin{frame}[fragile]
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−\frametitle{Forward Difference \ldots}
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−\begin{lstlisting}
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−In []: fD = (y[1:] - y[:-1]) / deltax
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−In []: print len(fD)
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−Out[]: 99
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−In []: plot(x, y) 
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−In []: plot(x[:-1], fD)
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−\end{lstlisting}
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−\vspace{-.2in}
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−\begin{center}
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−  \includegraphics[height=1.8in, interpolate=true]{data/fwdDiff}
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−\end{center}
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−\end{frame}
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−
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−\begin{frame}[fragile]
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−\frametitle{Example}
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−\begin{itemize}
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−\item Given x, y positions of a particle in \typ{pos.txt}
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−\item Find velocity \& acceleration in x, y directions
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−\end{itemize}
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−\small{
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−\begin{center}
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−\begin{tabular}{| c | c | c |}
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−\hline
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−$X$ & $Y$ \\ \hline
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−0.     &  0.\\ \hline
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−0.25   &  0.47775\\ \hline
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−0.5    &  0.931\\ \hline
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−0.75   &  1.35975\\ \hline
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−1.     &  1.764\\ \hline
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−1.25   &  2.14375\\ \hline
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−\vdots & \vdots\\ \hline
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−\end{tabular}
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−\end{center}}
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−\end{frame}
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−
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−\begin{frame}[fragile]
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−\frametitle{Example \ldots}
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−\begin{itemize}
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−\item Read the file
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−\item Obtain an array of X, Y
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−\item Obtain velocity and acceleration
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−\item use \typ{deltaT = 0.05}
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−\end{itemize}
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−\begin{lstlisting}
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−In []: data = loadtxt('pos.txt')
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−In []: X,Y = data[:,0], data[:,1]
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−In []: S = array([X, Y])
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−\end{lstlisting}
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−\end{frame}
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−
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−
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−\begin{frame}[fragile]
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−\frametitle{Example \ldots}
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−\begin{lstlisting}
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−In []: deltaT = 0.05
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−
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−In []: v = (S[:,1:]-S[:,:-1])/deltaT
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−
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−In []: a = (v[:,1:]-v[:,:-1])/deltaT
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−\end{lstlisting}
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−\end{frame}
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−
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−\begin{frame}[fragile]
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−\frametitle{Example \ldots}
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−Plotting Y, $v_y$, $a_y$
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−\begin{lstlisting}
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−In []: plot(Y)
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−In []: plot(v[1,:])
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−In []: plot(a[1,:])
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−\end{lstlisting}
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−\begin{center}
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−  \includegraphics[height=1.8in, interpolate=true]{data/pos_vel_accel}  
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−\end{center}
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−\end{frame}
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−
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−\section{Quadrature}
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−
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−\begin{frame}[fragile]
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−\frametitle{Quadrature}
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−
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−\emphbar{$\int_0^1(sin(x) + x^2)$}
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−
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−\typ{In []: from scipy.integrate import quad}
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−
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−\begin{itemize}
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−\item Inputs - function to integrate, limits
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−\end{itemize}
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−\begin{lstlisting}
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−In []: quad(sin(x)+x**2, 0, 1)
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−NameError: name 'x' is not defined
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−In []: x = 0
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−In []: quad(sin(x)+x**2, 0, 1)
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−\end{lstlisting}
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−\begin{small}
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−\alert{\typ{error:}}
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−\typ{First argument must be a callable function.}
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−\end{small}
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−\end{frame}
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−
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−\begin{frame}[fragile]
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−\frametitle{Functions - Definition}
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−We have been using them all along. Now let's see how to define them.
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−\begin{lstlisting}
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−In []: def f(x):
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−           return sin(x)+x**2
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−In []: quad(f, 0, 1)
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−\end{lstlisting}
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−\begin{itemize}
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−\item \typ{def}
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−\item name
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−\item arguments
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−\item \typ{return}
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−\end{itemize}
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−\end{frame}
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−
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−\begin{frame}[fragile]
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−\frametitle{Functions - Calling them}
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−\begin{lstlisting}
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−In [15]: f()
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−---------------------------------------
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−\end{lstlisting}
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−\alert{\typ{TypeError:}}\typ{f() takes exactly 1 argument}
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−\typ{(0 given)}
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−\begin{lstlisting}
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−In []: f(0)
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−Out[]: 0.0
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−In []: f(1)
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−Out[]: 1.8414709848078965
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−\end{lstlisting}
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−More on Functions later \ldots
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−\end{frame}
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−
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−\begin{frame}[fragile]
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−\frametitle{Quadrature \ldots}
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−\begin{lstlisting}
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−In []: quad(f, 0, 1)
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−\end{lstlisting}
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−Returns the integral and an estimate of the absolute error in the result.
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−\begin{itemize}
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−\item \typ{dblquad}, \typ{tplquad} are available
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−\end{itemize}
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−\end{frame}
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−
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−\begin{frame}
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−  \frametitle{Things we have learned}
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−  \begin{itemize}
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−  \item Interpolation
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−  \item Differentiation
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−  \item Functions
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−    \begin{itemize}
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−    \item Definition
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−    \item Calling
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−    \item Default Arguments
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−    \item Keyword Arguments
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−    \end{itemize}
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−  \item Quadrature
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−  \end{itemize}
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−\end{frame}
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−
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−\end{document}
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−