day1/session4.tex
author Puneeth Chaganti <punchagan@fossee.in>
Thu, 05 Nov 2009 13:09:17 +0530
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Added initial slides of Slicing and Striding to session 4.
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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, Least Square Fit, \& 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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\alert{All matrix operations are done using \kwrd{arrays}}
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\end{frame}
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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 = array([[ 1,  1,  2, -1],
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                  [ 2,  5, -1, -9],
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                  [ 2,  1, -1,  3],
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                  [ 1, -3,  2,  7]])
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In []: A
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Out[]: 
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array([[ 1,  1,  2, -1],
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       [ 2,  5, -1, -9],
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       [ 2,  1, -1,  3],
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       [ 1, -3,  2,  7]])
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\end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Accessing elements}
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  \begin{lstlisting}
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In []: C = array([[1,1,2],
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                  [2,4,1],
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                  [-1,3,7]])
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In []: C[1][2]
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Out[]: 1
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In []: C[1,2]
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Out[]: 1
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In []: C[1]
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Out[]: array([2, 4, 1])
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  \end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Changing elements}
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  \begin{small}
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  \begin{lstlisting}
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In []: C[1,1] = -2
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In []: C
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Out[]: 
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array([[ 1,  1,  2],
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       [ 2, -2,  1],
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       [-1,  3,  7]])
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In []: C[1] = [0,0,0]
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In []: C
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Out[]: 
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array([[ 1,  1,  2],
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       [ 0,  0,  0],
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       [-1,  3,  7]])
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  \end{lstlisting}
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  \end{small}
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How to change one \alert{column}?
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Slicing}
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\begin{small}
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  \begin{lstlisting}
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In []: C[:,1]
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Out[]: array([1, 0, 3])
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In []: C[1,:]
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Out[]: array([0, 0, 0])
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In []: C[0:2,:]
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Out[]: 
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array([[1, 1, 2],
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       [0, 0, 0]])
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In []: C[1:3,:]
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Out[]: 
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array([[ 0,  0,  0],
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       [-1,  3,  7]])
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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{Slicing \ldots}
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\begin{small}
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  \begin{lstlisting}
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In []: C[:2,:]
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Out[]: 
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array([[1, 1, 2],
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       [0, 0, 0]])
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In []: C[1:,:]
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Out[]: 
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array([[ 0,  0,  0],
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       [-1,  3,  7]])
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In []: C[1:,:2]
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Out[]: 
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array([[ 0,  0],
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       [-1,  3]])
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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{Striding}
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  \begin{lstlisting}
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  \end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Slicing \& Striding Exercises}
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  \begin{lstlisting}
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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 []: A.T
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Out[]:
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array([[ 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 []: sum(A)
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Out[]: 12
214
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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 = array([[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 []: A + B
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Out[]: 
263
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array([[  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{Elementwise Multiplication}
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\begin{lstlisting}
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In []: A*B
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Out[]: 
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array([[  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{Matrix Multiplication}
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\begin{lstlisting}
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In []: dot(A,B)
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Out[]: 
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array([[ -6. ,   6. ,  -6. ,  -3. ],
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       [-64. ,  38.5, -44. ,  35. ],
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       [ 36. , -13.5,  24. ,  35. ],
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       [ 58. , -26. ,  34. , -15. ]])
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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{lstlisting}
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In []: inv(A)
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\end{lstlisting}
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\begin{small}
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\begin{lstlisting}
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Out[]: 
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array([[-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[]: 80.0
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\end{lstlisting}
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\end{frame}
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263
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%%use S=array(X,Y)
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\begin{frame}[fragile]
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\frametitle{Eigenvalues and Eigen Vectors}
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\begin{small}
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\begin{lstlisting}
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In []: E = array([[3,2,4],[2,0,2],[4,2,3]])
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263
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In []: eig(E)
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Out[]: 
253
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(array([-1.,  8., -1.]),
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 array([[-0.74535599,  0.66666667, -0.1931126 ],
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        [ 0.2981424 ,  0.33333333, -0.78664085],
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        [ 0.59628479,  0.66666667,  0.58643303]]))
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263
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In []: eigvals(E)
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Out[]: array([-1.,  8., -1.])
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\end{lstlisting}
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\end{small}
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\end{frame}
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272
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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 []: norm(E)
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%% Out[]: 8.1240384046359608
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%% \end{lstlisting}
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%% \end{frame}
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272
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%% \begin{frame}[fragile]
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%%   \frametitle{Singular Value Decomposition}
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%%   \begin{small}
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%%   \begin{lstlisting}
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%% In []: svd(E)
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%% Out[]: 
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%% (array(
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%% [[ -6.66666667e-01,  -1.23702565e-16,   7.45355992e-01],
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%%  [ -3.33333333e-01,  -8.94427191e-01,  -2.98142397e-01],
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%%  [ -6.66666667e-01,   4.47213595e-01,  -5.96284794e-01]]),
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%%  array([ 8.,  1.,  1.]),
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%%  array([[-0.66666667, -0.33333333, -0.66666667],
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%%         [-0.        ,  0.89442719, -0.4472136 ],
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%%         [-0.74535599,  0.2981424 ,  0.59628479]]))
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%%   \end{lstlisting}
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%%   \end{small}
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%% \inctime{15}
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%% \end{frame}
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268
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\section{Least Squares Fit}
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\begin{frame}[fragile]
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\frametitle{$L$ vs. $T^2$}
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\vspace{-0.15in}
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   386
\begin{figure}
272
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\includegraphics[width=4in]{data/L-Tsq-points.png}
268
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   388
\end{figure}
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   389
\end{frame}
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   390
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   391
\begin{frame}[fragile]
272
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\frametitle{$L$ vs. $T^2$}
268
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   393
\vspace{-0.15in}
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   394
\begin{figure}
272
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\includegraphics[width=4in]{data/L-Tsq-Line.png}
268
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   396
\end{figure}
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   397
\end{frame}
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   398
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   399
\begin{frame}[fragile]
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\frametitle{Least Squares Fit}
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   401
\vspace{-0.15in}
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   402
\begin{figure}
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\includegraphics[width=4in]{data/least-sq-fit.png}
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   404
\end{figure}
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   405
\end{frame}
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   406
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   407
\begin{frame}
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\frametitle{Least Square Fit Curve}
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\begin{itemize}
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   410
\item $T^2$ and $L$ have a linear relationship
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   411
\item Hence, Least Square Fit Curve is a line
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   412
\item we shall use the \typ{lstsq} function
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   413
\end{itemize}
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   414
\end{frame}
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   415
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   416
\begin{frame}[fragile]
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\frametitle{\typ{lstsq}}
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   418
\begin{itemize}
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   419
\item We need to fit a line through points for the equation $T^2 = m \cdot L+c$
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   420
\item The equation can be re-written as $T^2 = A \cdot p$
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   421
\item where A is   
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   422
  $\begin{bmatrix}
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  L_1 & 1 \\
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  L_2 & 1 \\
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   425
  \vdots & \vdots\\
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   426
  L_N & 1 \\
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   427
  \end{bmatrix}$
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   428
  and p is 
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   429
  $\begin{bmatrix}
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   430
  m\\
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   431
  c\\
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   432
  \end{bmatrix}$
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   433
\item We need to find $p$ to plot the line
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   434
\end{itemize}
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   435
\end{frame}
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   436
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   437
\begin{frame}[fragile]
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   438
\frametitle{Generating $A$}
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   439
\begin{lstlisting}
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In []: A = array([L, ones_like(L)])
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   441
In []: A = A.T
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   442
\end{lstlisting}
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%% \begin{itemize}
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%% \item A is also called a Van der Monde matrix
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   445
%% \item It can also be generated using \typ{vander}
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   446
%% \end{itemize}
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   447
%% \begin{lstlisting}
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%% In []: A = vander(L, 2)
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%% \end{lstlisting}
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   450
\end{frame}
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   451
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   452
\begin{frame}[fragile]
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   453
\frametitle{\typ{lstsq} \ldots}
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   454
\begin{itemize}
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   455
\item Now use the \typ{lstsq} function
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   456
\item Along with a lot of things, it returns the least squares solution
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   457
\end{itemize}
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   458
\begin{lstlisting}
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   459
In []: result = lstsq(A,TSq)
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   460
In []: coef = result[0]
268
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   461
\end{lstlisting}
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   462
\end{frame}
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   463
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   464
\subsection{Plotting}
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   465
\begin{frame}[fragile]
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   466
\frametitle{Least Square Fit Line \ldots}
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   467
We get the points of the line from \typ{coef}
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   468
\begin{lstlisting}
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   469
In []: Tline = coef[0]*L + coef[1]
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   470
\end{lstlisting}
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   471
\begin{itemize}
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   472
\item Now plot Tline vs. L, to get the Least squares fit line. 
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   473
\end{itemize}
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   474
\begin{lstlisting}
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   475
In []: plot(L, Tline)
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   476
\end{lstlisting}
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   477
\end{frame}
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   478
197
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   479
\section{Solving linear equations}
172
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   480
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   481
\begin{frame}[fragile]
197
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   482
\frametitle{Solution of equations}
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diff changeset
   483
Consider,
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   484
  \begin{align*}
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   485
    3x + 2y - z  & = 1 \\
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   486
    2x - 2y + 4z  & = -2 \\
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diff changeset
   487
    -x + \frac{1}{2}y -z & = 0
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diff changeset
   488
  \end{align*}
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diff changeset
   489
Solution:
8c42ffdaaeec Updated session 4 slides based on review.
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   490
  \begin{align*}
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diff changeset
   491
    x & = 1 \\
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   492
    y & = -2 \\
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   493
    z & = -2
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   494
  \end{align*}
178
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diff changeset
   495
\end{frame}
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diff changeset
   496
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diff changeset
   497
\begin{frame}[fragile]
197
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   498
\frametitle{Solving using Matrices}
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diff changeset
   499
Let us now look at how to solve this using \kwrd{matrices}
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   500
  \begin{lstlisting}
263
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   501
    In []: A = array([[3,2,-1],
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   502
                      [2,-2,4],                   
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   503
                      [-1, 0.5, -1]])
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   504
    In []: b = array([[1], [-2], [0]])
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   505
    In []: x = solve(A, b)
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   506
    In []: Ax = dot(A,x)
197
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diff changeset
   507
  \end{lstlisting}
172
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   508
\end{frame}
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diff changeset
   509
178
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   510
\begin{frame}[fragile]
197
8c42ffdaaeec Updated session 4 slides based on review.
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parents: 185
diff changeset
   511
\frametitle{Solution:}
178
8a3a9d98fa84 Changed functions section and some minor edits.
Puneeth Chaganti <punchagan@fossee.in>
parents: 172
diff changeset
   512
\begin{lstlisting}
197
8c42ffdaaeec Updated session 4 slides based on review.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 185
diff changeset
   513
In []: x
8c42ffdaaeec Updated session 4 slides based on review.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 185
diff changeset
   514
Out[]: 
8c42ffdaaeec Updated session 4 slides based on review.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 185
diff changeset
   515
array([[ 1.],
8c42ffdaaeec Updated session 4 slides based on review.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 185
diff changeset
   516
       [-2.],
8c42ffdaaeec Updated session 4 slides based on review.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 185
diff changeset
   517
       [-2.]])
232
b9748af050c4 Updated the session 4, day 1 slides.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 214
diff changeset
   518
\end{lstlisting}
b9748af050c4 Updated the session 4, day 1 slides.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 214
diff changeset
   519
\end{frame}
127
44c2f614e321 Updated Integration section in Session 4.
Puneeth Chaganti <punchagan@fossee.in>
parents: 124
diff changeset
   520
232
b9748af050c4 Updated the session 4, day 1 slides.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 214
diff changeset
   521
\begin{frame}[fragile]
b9748af050c4 Updated the session 4, day 1 slides.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 214
diff changeset
   522
\frametitle{Let's check!}
b9748af050c4 Updated the session 4, day 1 slides.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 214
diff changeset
   523
\begin{lstlisting}
197
8c42ffdaaeec Updated session 4 slides based on review.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 185
diff changeset
   524
In []: Ax
8c42ffdaaeec Updated session 4 slides based on review.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 185
diff changeset
   525
Out[]: 
263
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   526
array([[  1.00000000e+00],
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   527
       [ -2.00000000e+00],
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   528
       [  2.22044605e-16]])
127
44c2f614e321 Updated Integration section in Session 4.
Puneeth Chaganti <punchagan@fossee.in>
parents: 124
diff changeset
   529
\end{lstlisting}
232
b9748af050c4 Updated the session 4, day 1 slides.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 214
diff changeset
   530
\begin{block}{}
b9748af050c4 Updated the session 4, day 1 slides.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 214
diff changeset
   531
The last term in the matrix is actually \alert{0}!\\
b9748af050c4 Updated the session 4, day 1 slides.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 214
diff changeset
   532
We can use \kwrd{allclose()} to check.
b9748af050c4 Updated the session 4, day 1 slides.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 214
diff changeset
   533
\end{block}
b9748af050c4 Updated the session 4, day 1 slides.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 214
diff changeset
   534
\begin{lstlisting}
b9748af050c4 Updated the session 4, day 1 slides.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 214
diff changeset
   535
In []: allclose(Ax, b)
b9748af050c4 Updated the session 4, day 1 slides.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 214
diff changeset
   536
Out[]: True
b9748af050c4 Updated the session 4, day 1 slides.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 214
diff changeset
   537
\end{lstlisting}
253
e446ed7287d7 Updated session 2 and session 4 slides of day 1.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 240
diff changeset
   538
\inctime{15}
232
b9748af050c4 Updated the session 4, day 1 slides.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 214
diff changeset
   539
\end{frame}
b9748af050c4 Updated the session 4, day 1 slides.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 214
diff changeset
   540
233
c435d5846ef5 Updated Session 4 day 1 slides.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 232
diff changeset
   541
\subsection{Exercises}
235
8eab0fee0fc2 Updated session 4 day 1.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 233
diff changeset
   542
232
b9748af050c4 Updated the session 4, day 1 slides.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 214
diff changeset
   543
\begin{frame}[fragile]
263
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   544
\frametitle{Problem 1}
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   545
Given the matrix:\\
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   546
\begin{center}
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   547
$\begin{bmatrix}
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   548
-2 & 2 & 3\\
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   549
 2 & 1 & 6\\
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   550
-1 &-2 & 0\\
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   551
\end{bmatrix}$
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   552
\end{center}
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   553
Find:
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   554
\begin{itemize}
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   555
  \item[i] Transpose
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   556
  \item[ii]Inverse
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   557
  \item[iii]Determinant
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   558
  \item[iv] Eigenvalues and Eigen vectors
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   559
  \item[v] Singular Value decomposition
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   560
\end{itemize}
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   561
\end{frame}
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   562
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   563
\begin{frame}[fragile]
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   564
\frametitle{Problem 2}
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   565
Given 
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   566
\begin{center}
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   567
A = 
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   568
$\begin{bmatrix}
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   569
-3 & 1 & 5 \\
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   570
1 & 0 & -2 \\
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   571
5 & -2 & 4 \\
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   572
\end{bmatrix}$
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   573
, B = 
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   574
$\begin{bmatrix}
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   575
0 & 9 & -12 \\
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   576
-9 & 0 & 20 \\
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   577
12 & -20 & 0 \\
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   578
\end{bmatrix}$
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   579
\end{center}
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   580
Find:
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   581
\begin{itemize}
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   582
  \item[i] Sum of A and B
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   583
  \item[ii]Elementwise Product of A and B
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   584
  \item[iii] Matrix product of A and B
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   585
\end{itemize}
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   586
\end{frame}
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   587
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   588
\begin{frame}[fragile]
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   589
\frametitle{Solution}
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   590
Sum: 
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   591
$\begin{bmatrix}
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   592
-3 & 10 & 7 \\
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   593
-8 & 0 & 18 \\
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   594
17 & -22 & 4 \\
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   595
\end{bmatrix}$
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   596
,\\ Elementwise Product:
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   597
$\begin{bmatrix}
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   598
0 & 9 & -60 \\
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   599
-9 & 0 & -40 \\
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   600
60 & 40 & 0 \\
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   601
\end{bmatrix}$
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   602
,\\ Matrix product:
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   603
$\begin{bmatrix}
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   604
51 & -127 & 56 \\
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   605
-24 & 49 & -12 \\
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   606
66 & -35 & -100 \\
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   607
\end{bmatrix}$
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   608
\end{frame}
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   609
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   610
\begin{frame}[fragile]
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   611
\frametitle{Problem 3}
235
8eab0fee0fc2 Updated session 4 day 1.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 233
diff changeset
   612
Solve the set of equations:
233
c435d5846ef5 Updated Session 4 day 1 slides.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 232
diff changeset
   613
\begin{align*}
c435d5846ef5 Updated Session 4 day 1 slides.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 232
diff changeset
   614
  x + y + 2z -w & = 3\\
c435d5846ef5 Updated Session 4 day 1 slides.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 232
diff changeset
   615
  2x + 5y - z - 9w & = -3\\
c435d5846ef5 Updated Session 4 day 1 slides.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 232
diff changeset
   616
  2x + y -z + 3w & = -11 \\
c435d5846ef5 Updated Session 4 day 1 slides.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 232
diff changeset
   617
  x - 3y + 2z + 7w & = -5\\
c435d5846ef5 Updated Session 4 day 1 slides.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 232
diff changeset
   618
\end{align*}
253
e446ed7287d7 Updated session 2 and session 4 slides of day 1.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 240
diff changeset
   619
\inctime{10}
e446ed7287d7 Updated session 2 and session 4 slides of day 1.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 240
diff changeset
   620
\end{frame}
e446ed7287d7 Updated session 2 and session 4 slides of day 1.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 240
diff changeset
   621
e446ed7287d7 Updated session 2 and session 4 slides of day 1.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 240
diff changeset
   622
\begin{frame}[fragile]
e446ed7287d7 Updated session 2 and session 4 slides of day 1.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 240
diff changeset
   623
\frametitle{Solution}
263
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   624
Use \kwrd{solve()}
253
e446ed7287d7 Updated session 2 and session 4 slides of day 1.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 240
diff changeset
   625
\begin{align*}
e446ed7287d7 Updated session 2 and session 4 slides of day 1.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 240
diff changeset
   626
  x & = -5\\
e446ed7287d7 Updated session 2 and session 4 slides of day 1.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 240
diff changeset
   627
  y & = 2\\
e446ed7287d7 Updated session 2 and session 4 slides of day 1.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 240
diff changeset
   628
  z & = 3\\
e446ed7287d7 Updated session 2 and session 4 slides of day 1.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 240
diff changeset
   629
  w & = 0\\
e446ed7287d7 Updated session 2 and session 4 slides of day 1.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 240
diff changeset
   630
\end{align*}
127
44c2f614e321 Updated Integration section in Session 4.
Puneeth Chaganti <punchagan@fossee.in>
parents: 124
diff changeset
   631
\end{frame}
44c2f614e321 Updated Integration section in Session 4.
Puneeth Chaganti <punchagan@fossee.in>
parents: 124
diff changeset
   632
213
ce62706cf870 Resolved conflicts manually.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 203
diff changeset
   633
\section{Summary}
185
e59ab9ab1a89 Manual Merge with Mainline.
Puneeth Chaganti <punchagan@fossee.in>
parents: 179
diff changeset
   634
\begin{frame}
263
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   635
  \frametitle{What did we learn??}
185
e59ab9ab1a89 Manual Merge with Mainline.
Puneeth Chaganti <punchagan@fossee.in>
parents: 179
diff changeset
   636
  \begin{itemize}
213
ce62706cf870 Resolved conflicts manually.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 203
diff changeset
   637
  \item Matrices
214
19592f802dde Added matrix operations.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 213
diff changeset
   638
    \begin{itemize}
263
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   639
      \item Accessing elements
214
19592f802dde Added matrix operations.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 213
diff changeset
   640
      \item Transpose
19592f802dde Added matrix operations.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 213
diff changeset
   641
      \item Addition
19592f802dde Added matrix operations.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 213
diff changeset
   642
      \item Multiplication
19592f802dde Added matrix operations.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 213
diff changeset
   643
      \item Inverse of a matrix
19592f802dde Added matrix operations.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 213
diff changeset
   644
      \item Determinant
263
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   645
      \item Eigenvalues and Eigen vector
214
19592f802dde Added matrix operations.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 213
diff changeset
   646
      \item Norms
263
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 253
diff changeset
   647
      \item Singular Value Decomposition
214
19592f802dde Added matrix operations.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 213
diff changeset
   648
    \end{itemize}
268
f978ddc90960 Moved least square fitting to session 4; removed vander function.
Puneeth Chaganti <punchagan@fossee.in>
parents: 263
diff changeset
   649
  \item Least Square Curve fitting
213
ce62706cf870 Resolved conflicts manually.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents: 203
diff changeset
   650
  \item Solving linear equations
185
e59ab9ab1a89 Manual Merge with Mainline.
Puneeth Chaganti <punchagan@fossee.in>
parents: 179
diff changeset
   651
  \end{itemize}
e59ab9ab1a89 Manual Merge with Mainline.
Puneeth Chaganti <punchagan@fossee.in>
parents: 179
diff changeset
   652
\end{frame}
e59ab9ab1a89 Manual Merge with Mainline.
Puneeth Chaganti <punchagan@fossee.in>
parents: 179
diff changeset
   653
118
c7f85ba59af3 Added session4.tex.
Santosh G. Vattam <vattam.santosh@gmail.com>
parents:
diff changeset
   654
\end{document}