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% Tutorial slides on Python.

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

% Copyright (c) 2005-2009, Prabhu Ramachandran

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% Title page

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

\author[FOSSEE] {FOSSEE}

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

\date[] {31, October 2009}

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\frametitle{Least Squares Fit}

In this session - 

\begin{itemize}

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

\item Given a file containing L and T values

\end{itemize}

\end{frame}

\frametitle{Least Squares Fit \ldots}

Machinery Required -

\begin{itemize}

\item Reading files and parsing data

\item Plotting points, lines

\item Calculating the Coefficients of the Least Squares Fit curve

\begin{itemize}

  \item Arrays

\end{itemize}

\end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{Reading pendulum.txt}

\begin{itemize}

  \item The file has two columns

  \item Column1 - L; Column2 - T

\end{itemize}

\begin{lstlisting}

In []: L = []

In []: T = []

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

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

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

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

\end{lstlisting}

We now have two lists L and T

\end{frame}

\begin{frame}[fragile]

\begin{itemize}

\item Each element of the list T must be squared

\item Iterating over each element of the list works

\item But very slow \ldots

\item Instead, we use arrays

\end{itemize}

\begin{lstlisting}

In []: array(L)

In []: T = array(T)

In []: Tsq = T*T

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

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Arrays}

\begin{itemize}

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

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

\item Very easy to perform element-wise operations

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

\item More about arrays later

\end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{Least Square Polynomial}

\begin{enumerate}

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

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

\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. 

\end{enumerate}

\end{frame}

\begin{frame}[fragile]

\frametitle{Van der Monde Matrix}

Van der Monde matrix of order M

\begin{equation*}

  \begin{bmatrix}

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

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

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

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

  \end{bmatrix}

\end{equation*}

\begin{lstlisting}

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

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Least Square Fit Line}

\begin{itemize}

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

\item It returns the 

\begin{enumerate}

\item Least squares solution

\item Sum of residues

\item Rank of matrix A

\item Singular values of A

\end{enumerate}

\end{itemize}

\begin{lstlisting}

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

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Least Square Fit Line \ldots}

\begin{itemize}

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

\begin{lstlisting}

p=poly1d(coeffs)

\end{lstlisting}

\begin{lstlisting}

Tline = p(L)

\end{lstlisting}

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

\begin{lstlisting}

plot(L, Tline)

\end{lstlisting}

\end{frame}

\end{document}

Least squares: Smooth curve fit. 

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

Subject wise average. Introduce idea of dictionary. 

Session 3

import scipy

from scipy import linalg.

choose some meaningful plot. ??

Newton's law of cooling. 

u, v, f - optics

hooke's law

Least fit curves. 

Choose a named problem. 

ODE - first order. Whatever. 

arrays, etc etc. 

sum, average, mean. whatever. statistical

sslc data

numpy load text??