author | Puneeth Chaganti <punchagan@fossee.in> |
Wed, 28 Oct 2009 15:46:21 +0530 | |
changeset 225 | a83f7be6f16c |
parent 224 | 668f88f20218 |
child 226 | 0995e8f32913 |
permissions | -rw-r--r-- |
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%Tutorial slides on Python. |
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% Author: FOSSEE |
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% Copyright (c) 2009, FOSSEE, IIT Bombay |
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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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% Modified from: generic-ornate-15min-45min.de.tex |
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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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% Macros |
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\setbeamercolor{emphbar}{bg=blue!20, fg=black} |
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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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% showstringspaces=false, |
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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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%% 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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\frametitle{Outline} |
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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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|
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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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|
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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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|
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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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\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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$f(x+h)=f(x)+h.f^{'}(x)$ &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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243 |
\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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253 |
\end{frame} |
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254 |
|
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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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|
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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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284 |
\end{tabular} |
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285 |
\end{center}} |
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286 |
\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 []: 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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304 |
\end{lstlisting} |
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\end{frame} |
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|
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|
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308 |
\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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315 |
|
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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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319 |
\end{lstlisting} |
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Try Plotting the position, velocity \& acceleration. |
221
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321 |
\end{frame} |
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|
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\section{Quadrature} |
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|
325 |
\begin{frame}[fragile] |
|
326 |
\frametitle{Quadrature} |
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\begin{itemize} |
|
328 |
\item We wish to find area under a curve |
|
329 |
\item Area under $(sin(x) + x^2)$ in $(0,1)$ |
|
330 |
\item scipy has functions to do that |
|
331 |
\end{itemize} |
|
221
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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} |
347 |
||
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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} |
380 |
||
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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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388 |
\item Look at \typ{dblquad} for Double integrals |
202 | 389 |
\item Use \typ{tplquad} for Triple integrals |
390 |
\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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396 |
\item Interpolation |
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397 |
\item Differentiation |
202 | 398 |
\item Functions |
399 |
\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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408 |
|
202 | 409 |
\end{document} |
410 |