1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

     2 % Tutorial slides on Python.

     3 %

     4 % Author: Prabhu Ramachandran <prabhu at aero.iitb.ac.in>

     5 % Copyright (c) 2005-2009, Prabhu Ramachandran

     6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

     7 

     8 \documentclass[14pt,compress]{beamer}

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    22 \usepackage[english]{babel}

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    24 %\usepackage{times}

    25 \usepackage[T1]{fontenc}

    26 

    27 % Taken from Fernando's slides.

    28 \usepackage{ae,aecompl}

    29 \usepackage{mathpazo,courier,euler}

    30 \usepackage[scaled=.95]{helvet}

    31 \usepackage{amsmath}

    32 

    33 \definecolor{darkgreen}{rgb}{0,0.5,0}

    34 

    35 \usepackage{listings}

    36 \lstset{language=Python,

    37     basicstyle=\ttfamily\bfseries,

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    58 

    59 %%% This is from Fernando's setup.

    60 % \usepackage{color}

    61 % \definecolor{orange}{cmyk}{0,0.4,0.8,0.2}

    62 % % Use and configure listings package for nicely formatted code

    63 % \usepackage{listings}

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    73 

    74 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    75 % Title page

    76 \title[]{Arrays \& Least Squares Fit}

    77 

    78 \author[FOSSEE] {FOSSEE}

    79 

    80 \institute[IIT Bombay] {Department of Aerospace Engineering\\IIT Bombay}

    81 \date[] {31, October 2009}

    82 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    83 

    84 %\pgfdeclareimage[height=0.75cm]{iitmlogo}{iitmlogo}

    85 %\logo{\pgfuseimage{iitmlogo}}

    86 

    87 

    88 %% Delete this, if you do not want the table of contents to pop up at

    89 %% the beginning of each subsection:

    90 \AtBeginSubsection[]

    91 {

    92   \begin{frame}<beamer>

    93     \frametitle{Outline}

    94     \tableofcontents[currentsection,currentsubsection]

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    96 }

    97 

    98 \AtBeginSection[]

    99 {

   100   \begin{frame}<beamer>

   101     \frametitle{Outline}

   102     \tableofcontents[currentsection,currentsubsection]

   103   \end{frame}

   104 }

   105 

   106 \newcommand{\num}{\texttt{numpy}}

   107 

   108 

   109 % If you wish to uncover everything in a step-wise fashion, uncomment

   110 % the following command: 

   111 %\beamerdefaultoverlayspecification{<+->}

   112 

   113 %\includeonlyframes{current,current1,current2,current3,current4,current5,current6}

   114 

   115 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

   116 % DOCUMENT STARTS

   117 \begin{document}

   118 

   119 \begin{frame}

   120   \maketitle

   121 \end{frame}

   122 

   123 %% \begin{frame}

   124 %%   \frametitle{Outline}

   125 %%   \tableofcontents

   126 %%   % You might wish to add the option [pausesections]

   127 %% \end{frame}

   128 

   129 \begin{frame}

   130 \frametitle{Least Squares Fit}

   131 In this session - 

   132 \begin{itemize}

   133 \item We shall plot a least squares fit curve for time-period(T) squared vs. length(L) plot of a Simple Pendulum. 

   134 \item Given a file containing L and T values

   135 \end{itemize}

   136 \end{frame}

   137 

   138 \begin{frame}[fragile]

   139 \frametitle{Least Squares Fit \ldots}

   140 Machinery Required -

   141 \begin{itemize}

   142 \item Reading files and parsing data

   143 \item Plotting points, lines

   144 \item Calculating the Coefficients of the Least Squares Fit curve

   145 \begin{itemize}

   146   \item Arrays

   147 \end{itemize}

   148 \end{itemize}

   149 \end{frame}

   150 

   151 \begin{frame}[fragile]

   152 \frametitle{Reading pendulum.txt}

   153 \begin{itemize}

   154   \item The file has two columns

   155   \item Column1 - L; Column2 - T

   156 \end{itemize}

   157 \begin{lstlisting}

   158 In []: L = []

   159 In []: T = []

   160 In []: for line in open('pendulum.txt'):

   161   ....     len, t = line.split()

   162   ....     L.append(float(len))

   163   ....     T.append(float(t))

   164 \end{lstlisting}

   165 We now have two lists L and T

   166 \end{frame}

   167 

   168 \begin{frame}[fragile]

   169 \frametitle{Calculating $T^2$}

   170 \begin{itemize}

   171 \item Each element of the list T must be squared

   172 \item Iterating over each element of the list works

   173 \item But very slow \ldots

   174 \item Instead, we use arrays

   175 \end{itemize}

   176 \begin{lstlisting}

   177 In []: array(L)

   178 In []: T = array(T)

   179 In []: Tsq = T*T

   180 In []: plot(L, Tsq, 'o')

   181 \end{lstlisting}

   182 \end{frame}

   183 

   184 \begin{frame}[fragile]

   185 \frametitle{Arrays}

   186 \begin{itemize}

   187 \item T is now a \typ{numpy array}

   188 \item \typ{numpy} arrays are very efficient and powerful 

   189 \item Very easy to perform element-wise operations

   190 \item \typ{+, -, *, /, \%}

   191 \item More about arrays later

   192 \end{itemize}

   193 \end{frame}

   194 

   195 \begin{frame}[fragile]

   196 \frametitle{Least Square Polynomial}

   197 \begin{enumerate}

   198 \item $T^2 = \frac{4\pi^2}{g}L$

   199 \item $T^2$ and $L$ have a linear relationship

   200 \item We find an approximate solution to $Ax = y$, where A is the Van der Monde matrix to get coefficients of the least squares fit line. 

   201 \end{enumerate}

   202 \end{frame}

   203 

   204 \begin{frame}[fragile]

   205 \frametitle{Van der Monde Matrix}

   206 Van der Monde matrix of order M

   207 \begin{equation*}

   208   \begin{bmatrix}

   209   l_1^{M-1} & \ldots & l_1 & 1 \\

   210   l_2^{M-1} & \ldots &l_2 & 1 \\

   211   \vdots & \ldots & \vdots & \vdots\\

   212   l_N^{M-1} & \ldots & l_N & 1 \\

   213   \end{bmatrix}

   214 \end{equation*}

   215 \begin{lstlisting}

   216 In []: A=vander(L,2)

   217 \end{lstlisting}

   218 \end{frame}

   219 

   220 \begin{frame}[fragile]

   221 \frametitle{Least Square Fit Line}

   222 \begin{itemize}

   223 \item We use the \typ{lstsq} function of pylab

   224 \item It returns the 

   225 \begin{enumerate}

   226 \item Least squares solution

   227 \item Sum of residues

   228 \item Rank of matrix A

   229 \item Singular values of A

   230 \end{enumerate}

   231 \end{itemize}

   232 \begin{lstlisting}

   233 coeffs, res, rank, sing = lstsq(A,Tsq)

   234 \end{lstlisting}

   235 \end{frame}

   236 

   237 \begin{frame}[fragile]

   238 \frametitle{Least Square Fit Line \ldots}

   239 \begin{itemize}

   240 \item Use the poly1d function of pylab, to create a function for the line equation using the coefficients obtained

   241 \begin{lstlisting}

   242 p=poly1d(coeffs)

   243 \end{lstlisting}

   244 \item Get new $T^2$ values using the function \typ{p} obtained

   245 \begin{lstlisting}

   246 Tline = p(L)

   247 \end{lstlisting}

   248 \item Now plot Tline vs. L, to get the Least squares fit line. 

   249 \begin{lstlisting}

   250 plot(L, Tline)

   251 \end{lstlisting}

   252 \end{itemize}

   253 \end{frame}

   254 

   255 \end{document}

   256 

   257 Least squares: Smooth curve fit. 

   258 Array Operations: Mean, average (etc region wise like district wise and state wise from SSLC.txt) 

   259 Subject wise average. Introduce idea of dictionary. 

   260 

   261 Session 3

   262 

   263 import scipy

   264 from scipy import linalg.

   265 

   266 choose some meaningful plot. ??

   267 Newton's law of cooling. 

   268 u, v, f - optics

   269 hooke's law

   270 Least fit curves. 

   271 

   272 

   273 Choose a named problem. 

   274 ODE - first order. Whatever. 

   275 

   276 

   277 arrays, etc etc. 

   278 sum, average, mean. whatever. statistical

   279 sslc data

   280 numpy load text??