day1/session4.tex
author Madhusudan.C.S <madhusudancs@gmail.com>
Wed, 28 Oct 2009 20:32:52 +0530
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child 253 e446ed7287d7
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Fixed typos and hanging words in Session 1 day 2.
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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{infolines}
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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[Matrices \& Equations]{Python for Science and Engg: Matrices \& Solution of equations}
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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{Matrices}
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\begin{frame}
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\frametitle{Matrices: Introduction}
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We looked at the Van der Monde matrix in the previous session,\\ 
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let us now look at matrices in a little more detail.
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\end{frame}
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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,  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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In []: A
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Out[]: 
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matrix([[ 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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\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{Transpose of a Matrix}
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\begin{lstlisting}
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In []: linalg.transpose(A)
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Out[]:
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matrix([[ 1,  2,  2,  1],
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        [ 1,  5,  1, -3],
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        [ 2, -1, -1,  2],
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        [-1, -9,  3,  7]])
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\end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Sum of all elements}
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  \begin{lstlisting}
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In []: linalg.sum(A)
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Out[]: 12
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  \end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Matrix Addition}
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  \begin{lstlisting}
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In []: B = matrix([[3,2,-1,5],
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                   [2,-2,4,9],
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                   [-1,0.5,-1,-7],
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                   [9,-5,7,3]])
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In []: linalg.add(A,B)
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Out[]: 
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matrix([[  4. ,   3. ,   1. ,   4. ],
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        [  4. ,   3. ,   3. ,   0. ],
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        [  1. ,   1.5,  -2. ,  -4. ],
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        [ 10. ,  -8. ,   9. ,  10. ]])
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  \end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Matrix Multiplication}
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\begin{lstlisting}
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In []: linalg.multiply(A, B)
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Out[]: 
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matrix([[  3. ,   2. ,  -2. ,  -5. ],
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        [  4. , -10. ,  -4. , -81. ],
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        [ -2. ,   0.5,   1. , -21. ],
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        [  9. ,  15. ,  14. ,  21. ]])
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\end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Inverse of a Matrix}
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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([[-0.5 ,  0.55, -0.15,  0.7 ],
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        [ 0.75, -0.5 ,  0.5 , -0.75],
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        [ 0.5 , -0.15, -0.05, -0.1 ],
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        [ 0.25, -0.25,  0.25, -0.25]])
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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 []: det(A)
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Out[66]: 80.0
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\end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Eigen Values and Eigen Matrix}
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\begin{small}
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\begin{lstlisting}
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In []: linalg.eig(A)
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Out[]: 
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(array([ 11.41026155,   3.71581643,  -0.81723144,  -2.30884654]),
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 matrix([[ 0.12300187, -0.53899627,  0.63269982,  0.56024583],
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        [ 0.8225266 , -0.67562403, -0.63919634, -0.20747251],
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        [-0.04763219, -0.47575453, -0.3709497 , -0.80066041],
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        [-0.55321941, -0.16331814, -0.23133374,  0.04497415]]))
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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{Computing Norms}
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\begin{lstlisting}
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In []: linalg.norm(A)
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Out[]: 14.0
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\end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Single Value Decomposition}
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  \begin{small}
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  \begin{lstlisting}
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In []: linalg.svd(A)
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Out[]: 
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(matrix([[-0.08588113,  0.29164297, -0.74892678,  0.58879325],
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        [-0.79093255,  0.39530483, -0.11188116, -0.45347812],
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        [ 0.1523891 ,  0.78799358,  0.51966138,  0.29290907],
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        [ 0.58636823,  0.37113957, -0.39565558, -0.60156827]]),
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 array([ 13.17656506,   3.76954116,   2.79959047,   0.57531379]),
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 matrix([[-0.05893795, -0.42858358,  0.12442679,  0.89295039],
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        [ 0.80364672,  0.51537891,  0.03774111,  0.29514767],
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        [-0.11752501,  0.14226922, -0.96333552,  0.19476145],
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        [-0.58040171,  0.72832696,  0.23468759,  0.27855956]]))
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\end{lstlisting}
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  \end{small}
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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],
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                       [2,-2,4],
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                       [-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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  \end{lstlisting}
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\end{frame}
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178
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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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\end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Let's check!}
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\begin{lstlisting}
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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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\begin{block}{}
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The last term in the matrix is actually \alert{0}!\\
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We can use \kwrd{allclose()} to check.
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\end{block}
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\begin{lstlisting}
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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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\subsection{Exercises}
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\begin{frame}[fragile]
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\frametitle{Problem}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Problem}
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Solve the set of equations:
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\begin{align*}
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  x + y + 2z -w & = 3\\
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  2x + 5y - z - 9w & = -3\\
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  2x + y -z + 3w & = -11 \\
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  x - 3y + 2z + 7w & = -5\\
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\end{align*}
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\end{frame}
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\section{Summary}
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\begin{frame}
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  \frametitle{Summary}
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So what did we learn??
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  \begin{itemize}
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  \item Matrices
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    \begin{itemize}
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      \item Transpose
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      \item Addition
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      \item Multiplication
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      \item Inverse of a matrix
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      \item Determinant
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      \item Eigen values and Eigen matrix
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      \item Norms
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      \item Single Value Decomposition
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    \end{itemize}
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  \item Solving linear equations
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  \end{itemize}
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\end{frame}
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\end{document}