day1/session5.tex
author Puneeth Chaganti <punchagan@fossee.in>
Wed, 28 Oct 2009 15:45:57 +0530
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child 226 0995e8f32913
permissions -rw-r--r--
Added differentiation example.
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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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\documentclass[14pt,compress]{beamer}
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%\documentclass[draft]{beamer}
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%\documentclass[compress,handout]{beamer}
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%\usepackage{pgfpages} 
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%\pgfpagesuselayout{2 on 1}[a4paper,border shrink=5mm]
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% Modified from: generic-ornate-15min-45min.de.tex
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\mode<presentation>
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{
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  \usetheme{Warsaw}
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  \useoutertheme{split}
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  \setbeamercovered{transparent}
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}
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\usepackage[english]{babel}
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\usepackage[latin1]{inputenc}
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%\usepackage{times}
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\usepackage[T1]{fontenc}
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\usepackage{amsmath}
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% Taken from Fernando's slides.
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\usepackage{ae,aecompl}
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\usepackage{mathpazo,courier,euler}
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\usepackage[scaled=.95]{helvet}
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\definecolor{darkgreen}{rgb}{0,0.5,0}
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\usepackage{listings}
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\lstset{language=Python,
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    basicstyle=\ttfamily\bfseries,
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    commentstyle=\color{red}\itshape,
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  stringstyle=\color{darkgreen},
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  showstringspaces=false,
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  keywordstyle=\color{blue}\bfseries}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Macros
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\setbeamercolor{emphbar}{bg=blue!20, fg=black}
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\newcommand{\emphbar}[1]
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{\begin{beamercolorbox}[rounded=true]{emphbar} 
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      {#1}
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 \end{beamercolorbox}
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}
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\newcounter{time}
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\setcounter{time}{0}
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\newcommand{\inctime}[1]{\addtocounter{time}{#1}{\tiny \thetime\ m}}
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\newcommand{\typ}[1]{\lstinline{#1}}
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\newcommand{\kwrd}[1]{ \texttt{\textbf{\color{blue}{#1}}}  }
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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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%    basicstyle=\small\ttfamily,
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%    commentstyle=\ttfamily\color{blue},
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%    stringstyle=\ttfamily\color{orange},
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%    showstringspaces=false,
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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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% Title page
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\title[Basic Python]{Interpolation, Differentiation and Integration}
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\author[FOSSEE] {FOSSEE}
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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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%\pgfdeclareimage[height=0.75cm]{iitmlogo}{iitmlogo}
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%\logo{\pgfuseimage{iitmlogo}}
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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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\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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% 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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%\includeonlyframes{current,current1,current2,current3,current4,current5,current6}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% DOCUMENT STARTS
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\begin{document}
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\begin{frame}
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  \titlepage
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\end{frame}
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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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\section{Interpolation}
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\begin{frame}[fragile]
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\frametitle{Interpolation}
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\begin{itemize}
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\item Let us begin with interpolation
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\item Let's use the L and T arrays and interpolate this data to obtain data at new points
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\end{itemize}
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\begin{lstlisting}
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In []: L = []
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In []: T = []
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In []: for line in open('pendulum.txt'):
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           l, t = line.split()
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           L.append(float(l))
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           T.append(float(t))
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In []: L = array(L)
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In []: T = array(T)
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\end{lstlisting}
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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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%% \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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\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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\begin{frame}[fragile]
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\frametitle{\typ{splrep}}
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To find the B-spline representation
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\begin{lstlisting}
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In []: tck = splrep(L, T)
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\end{lstlisting}
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Returns a tuple containing 
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\begin{enumerate}
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\item the vector of knots, 
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\item the B-spline coefficients 
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\item the degree of the spline (default=3)
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\end{enumerate}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{\typ{splev}}
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To Evaluate a B-spline and it's derivatives
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\begin{lstlisting}
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In []: Lnew = arange(0.1,1,0.005)
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In []: Tnew = splev(Lnew, tck)
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#To obtain derivatives of the spline
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#use der=1, 2,.. for 1st, 2nd,.. order
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In []: Tnew = splev(Lnew, tck, der=1)
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\end{lstlisting}
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\end{frame}
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224
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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}
221
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\section{Differentiation}
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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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$f(x+h)=f(x)+h.f^{'}(x)$ &Forward \\
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$f(x-h)=f(x)-h.f^{'}(x)$ &Backward
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\end{tabular}
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\end{center}
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\end{frame}
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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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\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 []: plot(x, y, x[:-1], fD)
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\end{lstlisting}
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\begin{center}
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  \includegraphics[height=2in, interpolate=true]{data/fwdDiff}
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\end{center}
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\end{frame}
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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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\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 []: X = []
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In []: Y = []
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In []: for line in open('location.txt'):
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  ....     points = line.split()
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  ....     X.append(float(points[0]))
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  ....     Y.append(float(points[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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\begin{frame}[fragile]
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\frametitle{Example \ldots}
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\begin{itemize}
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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 []: deltaT = 0.05
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In []: v = (S[:,1:]-S[:,:-1])/deltaT
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In []: a = (v[:,1:]-v[:,:-1])/deltaT
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\end{lstlisting}
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Try Plotting the position, velocity \& acceleration.
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\end{frame}
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\section{Quadrature}
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\begin{frame}[fragile]
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\frametitle{Quadrature}
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\begin{itemize}
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\item We wish to find area under a curve
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\item Area under $(sin(x) + x^2)$ in $(0,1)$
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\item scipy has functions to do that
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\end{itemize}
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\begin{small}
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  \typ{In []: from scipy.integrate import quad}
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\end{small}
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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 []: 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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\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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\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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\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 Look at \typ{dblquad} for Double integrals
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\item Use \typ{tplquad} for Triple integrals
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\end{itemize}
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\end{frame}
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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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\end{document}
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