day1/session4.tex
author Puneeth Chaganti <punchagan@fossee.in>
Tue, 27 Oct 2009 13:05:40 +0530
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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]{Matrices, Solution of equations 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 4}
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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{Solving linear equations}
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\begin{frame}[fragile]
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\frametitle{Solution of equations}
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Consider,
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  \begin{align*}
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    3x + 2y - z  & = 1 \\
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    2x - 2y + 4z  & = -2 \\
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    -x + \frac{1}{2}y -z & = 0
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  \end{align*}
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Solution:
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  \begin{align*}
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    x & = 1 \\
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    y & = -2 \\
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    z & = -2
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  \end{align*}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Solving using Matrices}
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Let us now look at how to solve this using \kwrd{matrices}
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  \begin{lstlisting}
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    In []: A = matrix([[3,2,-1],[2,-2,4],[-1, 0.5, -1]])
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    In []: b = matrix([[1], [-2], [0]])
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    In []: x = linalg.solve(A, b)
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    In []: Ax = dot(A, x)
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    In []: allclose(Ax, b)
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    Out[]: True
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  \end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Solution:}
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\begin{lstlisting}
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In []: x
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Out[]: 
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array([[ 1.],
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       [-2.],
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       [-2.]])
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In []: Ax
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Out[]: 
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matrix([[  1.00000000e+00],
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        [ -2.00000000e+00],
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        [  2.22044605e-16]])
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\end{lstlisting}
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\end{frame}
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\section{Matrices}
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\subsection{Initializing}
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\begin{frame}[fragile]
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\frametitle{Matrices: Initializing}
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\begin{lstlisting}
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  In []: a = matrix([[1,2,3],
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                     [4,5,6],
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                     [7,8,9]])
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  In []: a
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  Out[]: 
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  matrix([[1, 2, 3],
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         [4, 5, 6],
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         [7, 8, 9]])
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\end{lstlisting}
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\end{frame}
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\subsection{Basic Operations}
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\begin{frame}[fragile]
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\frametitle{Inverse of a Matrix}
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131
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\begin{small}
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\begin{lstlisting}
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In []: linalg.inv(A)
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Out[]: 
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matrix([[ 0.07734807,  0.01657459,  0.32044199],
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        [ 0.09944751, -0.12154696, -0.01657459],
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        [-0.02762431, -0.07734807,  0.17127072]])
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118
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\end{lstlisting}
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\end{small}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Determinant}
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\begin{lstlisting}
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  In []: linalg.det(a)
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  Out[]: -9.5171266700777579e-16
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\end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Computing Norms}
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\begin{lstlisting}
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  In []: linalg.norm(a)
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  Out[]: 16.881943016134134
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\end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Eigen Values and Eigen Matrix}
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\begin{small}
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\begin{lstlisting}
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  In []: linalg.eigvals(a)
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  Out[]: array([1.61168440e+01, -1.11684397e+00, -1.22196337e-15])
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  In []: linalg.eig(a)
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  Out[]: 
179
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  (array([ 1.61168440e+01, -1.11684397e+00, -1.22196337e-15]),
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   matrix([[-0.23197069, -0.78583024,  0.40824829],
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          [-0.52532209, -0.08675134, -0.81649658],
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          [-0.8186735 ,  0.61232756,  0.40824829]]))
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\end{lstlisting}
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\end{small}
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\end{frame}
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172
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127
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\section{Integration}
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172
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\subsection{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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\small{\typ{In []: from scipy.integrate import quad}}
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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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\alert{\typ{error:}}
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\typ{First argument must be a callable function.}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Functions - Definition}
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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 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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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Functions - Default Arguments}
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\begin{lstlisting}
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In []: def f(x=1):
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           return sin(x)+x**2
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In []: f(10)
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Out[]: 99.455978889110625
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In []: f(1)
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Out[]: 1.8414709848078965
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In []: f()
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Out[]: 1.8414709848078965
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\end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Functions - Keyword Arguments}
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\begin{lstlisting}
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In []: def f(x=1, y=pi):
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           return sin(y)+x**2
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In []: f()
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Out[]: 1.0000000000000002
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In []: f(2)
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Out[]: 4.0
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In []: f(y=2)
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Out[]: 1.9092974268256817
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In []: f(y=pi/2,x=0)
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Out[]: 1.0
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\end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{More on functions}
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  \begin{itemize}
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  \item Scope of variables in the function is local
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  \item Mutable items are \alert{passed by reference}
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  \item First line after definition may be a documentation string
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    (\alert{recommended!})
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  \item Function definition and execution defines a name bound to the
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    function
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  \item You \emph{can} assign a variable to a function!
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  \end{itemize}
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\end{frame}
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178
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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 Use \typ{dblquad} for Double integrals
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\item Use \typ{tplquad} for Triple integrals
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\end{itemize}
178
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\end{frame}
172
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127
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\subsection{ODEs}
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123
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\begin{frame}[fragile]
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\frametitle{ODE Integration}
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We shall use the simple ODE of a simple pendulum. 
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\begin{equation*}
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\ddot{\theta} = -\frac{g}{L}sin(\theta)
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\end{equation*}
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\begin{itemize}
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\item This equation can be written as a system of two first order ODEs
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\end{itemize}
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\begin{align}
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\dot{\theta} &= \omega \\
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\dot{\omega} &= -\frac{g}{L}sin(\theta) \\
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 \text{At}\ t &= 0 : \nonumber \\
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 \theta = \theta_0\quad & \&\quad  \omega = 0 \nonumber
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\end{align}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Solving ODEs using SciPy}
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\begin{itemize}
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\item We use the \typ{odeint} function from scipy to do the integration
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\item Define a function as below
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\end{itemize}
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\begin{lstlisting}
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In []: def pend_int(unknown, t, p):
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  ....     theta, omega = unknown
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  ....     g, L = p
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  ....     f=[omega, -(g/L)*sin(theta)]
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  ....     return f
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  ....
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\end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Solving ODEs using SciPy \ldots}
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\begin{itemize}
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\item \typ{t} is the time variable \\ 
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\item \typ{p} has the constants \\
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\item \typ{initial} has the initial values
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\end{itemize}
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\begin{lstlisting}
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In []: t = linspace(0, 10, 101)
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In []: p=(-9.81, 0.2)
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In []: initial = [10*2*pi/360, 0]
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\end{lstlisting}
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\end{frame}
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127
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\begin{frame}[fragile]
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\frametitle{Solving ODEs using SciPy \ldots}
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\small{\typ{In []: from scipy.integrate import odeint}}
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\begin{lstlisting}
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In []: pend_sol = odeint(pend_int, 
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                         initial,t, 
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                         args=(p,))
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\end{lstlisting}
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\end{frame}
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185
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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
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  \item
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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 Integration
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    \begin{itemize}
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      \item Quadrature
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      \item ODEs
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    \end{itemize}
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  \end{itemize}
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\end{frame}
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\end{document}
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