day1/session6.tex
author Puneeth Chaganti <punchagan@fossee.in>
Thu, 05 Nov 2009 20:47:16 +0530
changeset 280 9bed85f05eb8
parent 273 c378d1ffb1d1
child 281 ce818f645f6b
permissions -rwxr-xr-x
Added save command in session1.
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
133
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
     1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
141
fccf675946bf Made usage of FOSSEE consistent; removed all instances of Team, Group.
Puneeth Chaganti <punchagan@fossee.in>
parents: 137
diff changeset
     2
%Tutorial slides on Python.
133
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
     3
%
142
57e0f0fd3317 Made usage of FOSSEE consistent, in comments also.
Puneeth Chaganti <punchagan@fossee.in>
parents: 141
diff changeset
     4
% Author: FOSSEE
57e0f0fd3317 Made usage of FOSSEE consistent, in comments also.
Puneeth Chaganti <punchagan@fossee.in>
parents: 141
diff changeset
     5
% Copyright (c) 2009, FOSSEE, IIT Bombay
133
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
     6
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
     7
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
     8
\documentclass[14pt,compress]{beamer}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
     9
%\documentclass[draft]{beamer}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    10
%\documentclass[compress,handout]{beamer}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    11
%\usepackage{pgfpages} 
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    12
%\pgfpagesuselayout{2 on 1}[a4paper,border shrink=5mm]
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    13
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    14
% Modified from: generic-ornate-15min-45min.de.tex
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    15
\mode<presentation>
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    16
{
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    17
  \usetheme{Warsaw}
239
8953675dc056 Added slide numbers.
Puneeth Chaganti <punchagan@fossee.in>
parents: 223
diff changeset
    18
  \useoutertheme{infolines}
133
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    19
  \setbeamercovered{transparent}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    20
}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    21
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    22
\usepackage[english]{babel}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    23
\usepackage[latin1]{inputenc}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    24
%\usepackage{times}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    25
\usepackage[T1]{fontenc}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    26
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    27
% Taken from Fernando's slides.
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    28
\usepackage{ae,aecompl}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    29
\usepackage{mathpazo,courier,euler}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    30
\usepackage[scaled=.95]{helvet}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    31
\usepackage{amsmath}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    32
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    33
\definecolor{darkgreen}{rgb}{0,0.5,0}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    34
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    35
\usepackage{listings}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    36
\lstset{language=Python,
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    37
    basicstyle=\ttfamily\bfseries,
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    38
    commentstyle=\color{red}\itshape,
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    39
  stringstyle=\color{darkgreen},
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    40
  showstringspaces=false,
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    41
  keywordstyle=\color{blue}\bfseries}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    42
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    43
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    44
% Macros
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    45
\setbeamercolor{emphbar}{bg=blue!20, fg=black}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    46
\newcommand{\emphbar}[1]
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    47
{\begin{beamercolorbox}[rounded=true]{emphbar} 
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    48
      {#1}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    49
 \end{beamercolorbox}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    50
}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    51
\newcounter{time}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    52
\setcounter{time}{0}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    53
\newcommand{\inctime}[1]{\addtocounter{time}{#1}{\tiny \thetime\ m}}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    54
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    55
\newcommand{\typ}[1]{\lstinline{#1}}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    56
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    57
\newcommand{\kwrd}[1]{ \texttt{\textbf{\color{blue}{#1}}}  }
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    58
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    59
%%% This is from Fernando's setup.
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    60
% \usepackage{color}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    61
% \definecolor{orange}{cmyk}{0,0.4,0.8,0.2}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    62
% % Use and configure listings package for nicely formatted code
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    63
% \usepackage{listings}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    64
% \lstset{
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    65
%    language=Python,
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    66
%    basicstyle=\small\ttfamily,
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    67
%    commentstyle=\ttfamily\color{blue},
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    68
%    stringstyle=\ttfamily\color{orange},
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    69
%    showstringspaces=false,
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    70
%    breaklines=true,
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    71
%    postbreak = \space\dots
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    72
% }
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    73
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    74
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    75
% Title page
273
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
    76
\title[Solving Equations \& ODEs]{Python for Science and Engg:\\Solving Equations \& ODEs}
133
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    77
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    78
\author[FOSSEE] {FOSSEE}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    79
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    80
\institute[IIT Bombay] {Department of Aerospace Engineering\\IIT Bombay}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    81
\date[] {31, October 2009\\Day 1, Session 6}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    82
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    83
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    84
%\pgfdeclareimage[height=0.75cm]{iitmlogo}{iitmlogo}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    85
%\logo{\pgfuseimage{iitmlogo}}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    86
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    87
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    88
%% Delete this, if you do not want the table of contents to pop up at
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    89
%% the beginning of each subsection:
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    90
\AtBeginSubsection[]
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    91
{
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    92
  \begin{frame}<beamer>
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    93
    \frametitle{Outline}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    94
    \tableofcontents[currentsection,currentsubsection]
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    95
  \end{frame}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    96
}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    97
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    98
\AtBeginSection[]
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
    99
{
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   100
  \begin{frame}<beamer>
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   101
    \frametitle{Outline}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   102
    \tableofcontents[currentsection,currentsubsection]
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   103
  \end{frame}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   104
}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   105
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   106
% If you wish to uncover everything in a step-wise fashion, uncomment
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   107
% the following command: 
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   108
%\beamerdefaultoverlayspecification{<+->}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   109
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   110
%\includeonlyframes{current,current1,current2,current3,current4,current5,current6}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   111
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   112
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   113
% DOCUMENT STARTS
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   114
\begin{document}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   115
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   116
\begin{frame}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   117
  \maketitle
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   118
\end{frame}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   119
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   120
%% \begin{frame}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   121
%%   \frametitle{Outline}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   122
%%   \tableofcontents
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   123
%%   % You might wish to add the option [pausesections]
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   124
%% \end{frame}
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   125
273
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   126
\section{Solving linear equations}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   127
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   128
\begin{frame}[fragile]
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   129
\frametitle{Solution of equations}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   130
Consider,
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   131
  \begin{align*}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   132
    3x + 2y - z  & = 1 \\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   133
    2x - 2y + 4z  & = -2 \\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   134
    -x + \frac{1}{2}y -z & = 0
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   135
  \end{align*}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   136
Solution:
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   137
  \begin{align*}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   138
    x & = 1 \\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   139
    y & = -2 \\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   140
    z & = -2
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   141
  \end{align*}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   142
\end{frame}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   143
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   144
\begin{frame}[fragile]
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   145
\frametitle{Solving using Matrices}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   146
Let us now look at how to solve this using \kwrd{matrices}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   147
  \begin{lstlisting}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   148
    In []: A = array([[3,2,-1],
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   149
                      [2,-2,4],                   
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   150
                      [-1, 0.5, -1]])
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   151
    In []: b = array([[1], [-2], [0]])
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   152
    In []: x = solve(A, b)
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   153
    In []: Ax = dot(A,x)
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   154
  \end{lstlisting}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   155
\end{frame}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   156
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   157
\begin{frame}[fragile]
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   158
\frametitle{Solution:}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   159
\begin{lstlisting}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   160
In []: x
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   161
Out[]: 
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   162
array([[ 1.],
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   163
       [-2.],
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   164
       [-2.]])
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   165
\end{lstlisting}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   166
\end{frame}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   167
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   168
\begin{frame}[fragile]
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   169
\frametitle{Let's check!}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   170
\begin{lstlisting}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   171
In []: Ax
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   172
Out[]: 
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   173
array([[  1.00000000e+00],
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   174
       [ -2.00000000e+00],
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   175
       [  2.22044605e-16]])
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   176
\end{lstlisting}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   177
\begin{block}{}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   178
The last term in the matrix is actually \alert{0}!\\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   179
We can use \kwrd{allclose()} to check.
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   180
\end{block}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   181
\begin{lstlisting}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   182
In []: allclose(Ax, b)
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   183
Out[]: True
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   184
\end{lstlisting}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   185
\inctime{15}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   186
\end{frame}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   187
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   188
\subsection{Exercises}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   189
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   190
\begin{frame}[fragile]
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   191
\frametitle{Problem 1}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   192
Given the matrix:\\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   193
\begin{center}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   194
$\begin{bmatrix}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   195
-2 & 2 & 3\\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   196
 2 & 1 & 6\\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   197
-1 &-2 & 0\\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   198
\end{bmatrix}$
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   199
\end{center}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   200
Find:
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   201
\begin{itemize}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   202
  \item[i] Transpose
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   203
  \item[ii]Inverse
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   204
  \item[iii]Determinant
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   205
  \item[iv] Eigenvalues and Eigen vectors
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   206
  \item[v] Singular Value decomposition
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   207
\end{itemize}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   208
\end{frame}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   209
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   210
\begin{frame}[fragile]
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   211
\frametitle{Problem 2}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   212
Given 
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   213
\begin{center}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   214
A = 
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   215
$\begin{bmatrix}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   216
-3 & 1 & 5 \\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   217
1 & 0 & -2 \\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   218
5 & -2 & 4 \\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   219
\end{bmatrix}$
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   220
, B = 
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   221
$\begin{bmatrix}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   222
0 & 9 & -12 \\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   223
-9 & 0 & 20 \\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   224
12 & -20 & 0 \\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   225
\end{bmatrix}$
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   226
\end{center}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   227
Find:
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   228
\begin{itemize}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   229
  \item[i] Sum of A and B
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   230
  \item[ii]Elementwise Product of A and B
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   231
  \item[iii] Matrix product of A and B
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   232
\end{itemize}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   233
\end{frame}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   234
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   235
\begin{frame}[fragile]
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   236
\frametitle{Solution}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   237
Sum: 
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   238
$\begin{bmatrix}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   239
-3 & 10 & 7 \\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   240
-8 & 0 & 18 \\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   241
17 & -22 & 4 \\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   242
\end{bmatrix}$
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   243
,\\ Elementwise Product:
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   244
$\begin{bmatrix}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   245
0 & 9 & -60 \\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   246
-9 & 0 & -40 \\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   247
60 & 40 & 0 \\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   248
\end{bmatrix}$
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   249
,\\ Matrix product:
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   250
$\begin{bmatrix}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   251
51 & -127 & 56 \\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   252
-24 & 49 & -12 \\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   253
66 & -35 & -100 \\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   254
\end{bmatrix}$
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   255
\end{frame}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   256
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   257
\begin{frame}[fragile]
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   258
\frametitle{Problem 3}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   259
Solve the set of equations:
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   260
\begin{align*}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   261
  x + y + 2z -w & = 3\\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   262
  2x + 5y - z - 9w & = -3\\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   263
  2x + y -z + 3w & = -11 \\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   264
  x - 3y + 2z + 7w & = -5\\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   265
\end{align*}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   266
\inctime{10}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   267
\end{frame}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   268
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   269
\begin{frame}[fragile]
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   270
\frametitle{Solution}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   271
Use \kwrd{solve()}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   272
\begin{align*}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   273
  x & = -5\\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   274
  y & = 2\\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   275
  z & = 3\\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   276
  w & = 0\\
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   277
\end{align*}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   278
\end{frame}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   279
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   280
\section{Finding Roots}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   281
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   282
\begin{frame}[fragile]
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   283
\frametitle{Scipy Methods - \typ{roots}}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   284
\begin{itemize}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   285
\item Calculates the roots of polynomials
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   286
\end{itemize}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   287
\begin{lstlisting}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   288
  In []: coeffs = [1, 6, 13]
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   289
  In []: roots(coeffs)
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   290
\end{lstlisting}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   291
\end{frame}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   292
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   293
\begin{frame}[fragile]
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   294
\frametitle{Scipy Methods - \typ{fsolve}}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   295
\begin{small}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   296
\begin{lstlisting}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   297
  In []: from scipy.optimize import fsolve
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   298
\end{lstlisting}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   299
\end{small}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   300
\begin{itemize}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   301
\item Finds the roots of a system of non-linear equations
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   302
\item Input arguments - Function and initial estimate
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   303
\item Returns the solution
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   304
\end{itemize}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   305
\begin{lstlisting}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   306
  In []: fsolve(our_f, -pi/2)
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   307
\end{lstlisting}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   308
\end{frame}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   309
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   310
%% \begin{frame}[fragile]
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   311
%% \frametitle{Scipy Methods \dots}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   312
%% \begin{small}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   313
%% \begin{lstlisting}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   314
%% In []: from scipy.optimize import fixed_point
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   315
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   316
%% In []: from scipy.optimize import bisect
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   317
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   318
%% In []: from scipy.optimize import newton
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   319
%% \end{lstlisting}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   320
%% \end{small}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   321
%% \end{frame}
c378d1ffb1d1 Moved Solving Linear Equations to session 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 271
diff changeset
   322
221
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   323
\section{ODEs}
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   324
\begin{frame}[fragile]
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   325
\frametitle{ODE Integration}
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   326
We shall use the simple ODE of a simple pendulum. 
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   327
\begin{equation*}
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   328
\ddot{\theta} = -\frac{g}{L}sin(\theta)
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   329
\end{equation*}
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   330
\begin{itemize}
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   331
\item This equation can be written as a system of two first order ODEs
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   332
\end{itemize}
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   333
\begin{align}
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   334
\dot{\theta} &= \omega \\
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   335
\dot{\omega} &= -\frac{g}{L}sin(\theta) \\
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   336
 \text{At}\ t &= 0 : \nonumber \\
263
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 258
diff changeset
   337
 \theta = \theta_0(10^o)\quad & \&\quad  \omega = 0\ (Initial\ values)\nonumber 
221
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   338
\end{align}
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   339
\end{frame}
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   340
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   341
\begin{frame}[fragile]
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   342
\frametitle{Solving ODEs using SciPy}
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   343
\begin{itemize}
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   344
\item We use the \typ{odeint} function from scipy to do the integration
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   345
\item Define a function as below
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   346
\end{itemize}
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   347
\begin{lstlisting}
263
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 258
diff changeset
   348
In []: def pend_int(initial, t):
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 258
diff changeset
   349
  ....     theta, omega = initial
280
9bed85f05eb8 Added save command in session1.
Puneeth Chaganti <punchagan@fossee.in>
parents: 273
diff changeset
   350
  ....     g, L = 9.81, 0.2
221
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   351
  ....     f=[omega, -(g/L)*sin(theta)]
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   352
  ....     return f
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   353
  ....
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   354
\end{lstlisting}
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   355
\end{frame}
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   356
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   357
\begin{frame}[fragile]
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   358
\frametitle{Solving ODEs using SciPy \ldots}
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   359
\begin{itemize}
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   360
\item \typ{t} is the time variable \\ 
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   361
\item \typ{initial} has the initial values
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   362
\end{itemize}
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   363
\begin{lstlisting}
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   364
In []: t = linspace(0, 10, 101)
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   365
In []: initial = [10*2*pi/360, 0]
263
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 258
diff changeset
   366
\end{lstlisting} 
221
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   367
\end{frame}
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   368
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   369
\begin{frame}[fragile]
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   370
\frametitle{Solving ODEs using SciPy \ldots}
263
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 258
diff changeset
   371
%%\begin{small}
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 258
diff changeset
   372
\typ{In []: from scipy.integrate import odeint}
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 258
diff changeset
   373
%%\end{small}
221
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   374
\begin{lstlisting}
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   375
In []: pend_sol = odeint(pend_int, 
263
8a4a1e5aec85 Changes made for Goa workshop .
Shantanu <shantanu@fossee.in>
parents: 258
diff changeset
   376
                         initial,t)
221
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   377
\end{lstlisting}
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   378
\end{frame}
9ed9539446bc Updated day1 session5 and session6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 142
diff changeset
   379
133
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   380
223
081600805dde Minor edits to sessions 5 and 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 221
diff changeset
   381
\begin{frame}
081600805dde Minor edits to sessions 5 and 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 221
diff changeset
   382
  \frametitle{Things we have learned}
081600805dde Minor edits to sessions 5 and 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 221
diff changeset
   383
  \begin{itemize}
081600805dde Minor edits to sessions 5 and 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 221
diff changeset
   384
  \item Solving ODEs
081600805dde Minor edits to sessions 5 and 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 221
diff changeset
   385
  \item Finding Roots
081600805dde Minor edits to sessions 5 and 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 221
diff changeset
   386
  \end{itemize}
081600805dde Minor edits to sessions 5 and 6.
Puneeth Chaganti <punchagan@fossee.in>
parents: 221
diff changeset
   387
\end{frame}
133
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   388
578db74dfea0 Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff changeset
   389
\end{document}