--- a/presentations/least-square.tex	Fri Apr 16 16:16:13 2010 +0530
+++ b/presentations/least-square.tex	Sat Apr 17 12:50:42 2010 +0530
@@ -150,7 +150,9 @@
 
 \begin{frame}[fragile]
   \frametitle{Summary}
+  \begin{block}{}
     Obtaining the least fit curve from a data set
+  \end{block}    
 \end{frame}
 
 \begin{frame}

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/presentations/solving-equations.tex	Sat Apr 17 12:50:42 2010 +0530
@@ -0,0 +1,154 @@
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%Tutorial slides on Python.
+%
+% Author: FOSSEE 
+% Copyright (c) 2009, FOSSEE, IIT Bombay
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\documentclass[14pt,compress]{beamer}
+%\documentclass[draft]{beamer}
+%\documentclass[compress,handout]{beamer}
+%\usepackage{pgfpages} 
+%\pgfpagesuselayout{2 on 1}[a4paper,border shrink=5mm]
+
+% Modified from: generic-ornate-15min-45min.de.tex
+\mode<presentation>
+{
+  \usetheme{Warsaw}
+  \useoutertheme{infolines}
+  \setbeamercovered{transparent}
+}
+
+\usepackage[english]{babel}
+\usepackage[latin1]{inputenc}
+%\usepackage{times}
+\usepackage[T1]{fontenc}
+
+% Taken from Fernando's slides.
+\usepackage{ae,aecompl}
+\usepackage{mathpazo,courier,euler}
+\usepackage[scaled=.95]{helvet}
+
+\definecolor{darkgreen}{rgb}{0,0.5,0}
+
+\usepackage{listings}
+\lstset{language=Python,
+    basicstyle=\ttfamily\bfseries,
+    commentstyle=\color{red}\itshape,
+  stringstyle=\color{darkgreen},
+  showstringspaces=false,
+  keywordstyle=\color{blue}\bfseries}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Macros
+\setbeamercolor{emphbar}{bg=blue!20, fg=black}
+\newcommand{\emphbar}[1]
+{\begin{beamercolorbox}[rounded=true]{emphbar} 
+      {#1}
+ \end{beamercolorbox}
+}
+\newcounter{time}
+\setcounter{time}{0}
+\newcommand{\inctime}[1]{\addtocounter{time}{#1}{\tiny \thetime\ m}}
+
+\newcommand{\typ}[1]{\lstinline{#1}}
+
+\newcommand{\kwrd}[1]{ \texttt{\textbf{\color{blue}{#1}}}  }
+
+% Title page
+\title{Python for Scientific Computing : Least Square Fit}
+
+\author[FOSSEE] {FOSSEE}
+
+\institute[IIT Bombay] {Department of Aerospace Engineering\\IIT Bombay}
+\date{}
+
+% DOCUMENT STARTS
+\begin{document}
+
+\begin{frame}
+  \maketitle
+\end{frame}
+
+\begin{frame}
+  \frametitle{About the Session}
+  \begin{block}{Goal}
+Finding least square fit of given data-set
+  \end{block}
+  \begin{block}{Checklist}
+    \begin{itemize}
+    \item pendulum.txt
+  \end{itemize}
+  \end{block}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{Solution of equations}
+Consider,
+  \begin{align*}
+    3x + 2y - z  & = 1 \\
+    2x - 2y + 4z  & = -2 \\
+    -x + \frac{1}{2}y -z & = 0
+  \end{align*}
+Solution:
+  \begin{align*}
+    x & = 1 \\
+    y & = -2 \\
+    z & = -2
+  \end{align*}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{Scipy Methods - \typ{roots}}
+\begin{itemize}
+\item Calculates the roots of polynomials
+\item To calculate the roots of $x^2-5x+6$ 
+\end{itemize}
+\begin{lstlisting}
+  In []: coeffs = [1, -5, 6]
+  In []: roots(coeffs)
+  Out[]: array([3., 2.])
+\end{lstlisting}
+\vspace*{-.2in}
+\begin{center}
+\includegraphics[height=1.6in, interpolate=true]{data/roots}    
+\end{center}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{Scipy Methods - \typ{fsolve}}
+\begin{small}
+\begin{lstlisting}
+  In []: from scipy.optimize import fsolve
+\end{lstlisting}
+\end{small}
+\begin{itemize}
+\item Finds the roots of a system of non-linear equations
+\item Input arguments - Function and initial estimate
+\item Returns the solution
+\end{itemize}
+\end{frame}
+
+\begin{frame}[fragile]
+  \frametitle{Summary}
+  \begin{block}{}
+    \begin{itemize}
+    \item Solving linear equations
+    \item Finding roots
+    \item Roots of non linear equations
+    \item Basics of Functions
+    \end{itemize}
+  \end{block}    
+\end{frame}
+
+\begin{frame}
+  \frametitle{Thank you!}  
+  \begin{block}{}
+  This session is part of \textcolor{blue}{FOSSEE} project funded by:
+  \begin{center}
+    \textcolor{blue}{NME through ICT from MHRD, Govt. of India}.
+  \end{center}  
+  \end{block}
+\end{frame}
+
+\end{document}

--- a/solving-equations.org	Fri Apr 16 16:16:13 2010 +0530
+++ b/solving-equations.org	Sat Apr 17 12:50:42 2010 +0530
@@ -13,13 +13,17 @@
     of polynomials and other non-linear equations. In the process, we
     shall look at defining functions. 
 
+    We would be using concepts related to arrays which we have covered
+    in previous session
+
     Let's begin with solving linear equations. 
     {show a slide of the equations}
     We shall use the solve function, to solve this system of linear
-    equations.  Solve requires the coefficients and the constants to
+    equations. Solve requires the coefficients and the constants to
     be in the form of matrices to solve the system of linear equations. 
 
-    We begin by entering the coefficients and the constants as
+    Lets start ipython -pylab interpreter.    
+    Then we begin by entering the coefficients and the constants as
     matrices. 
 
     In []: A = array([[3,2,-1],
@@ -33,8 +37,9 @@
 
     Type x, to look at the solution obtained. 
 
-    Next, we verify the solution by obtaining a product of A and x,
-    and comparing it with b. Note that we should use the dot function
+    Equation is of the form Ax = b, so we verify the solution by 
+    obtaining a matrix product of A and x, and comparing it with b. 
+    As we have covered earlier that we should use the dot function 
     here, and not the * operator. 
 
     In []: Ax = dot(A, x)
@@ -60,22 +65,26 @@
     
     In []: coeffs = [1, -5, 6]
     In []: roots(coeffs)
-    As you can see, roots returns the coefficients in an array. 
+    As you can see, roots returns the result in an array. 
+    # It even works for polynomials with imaginary solutions.
+    # roots([1, 1, 1])
 
-    To find the roots of any arbitrary function, we use the fsolve
+    To find the roots of non linear equations, we use the fsolve
     function. We shall use the function sin(x)+cos^2(x) as our
-    function, in this tutorial. First, of all we import fsolve, since it
-    is not already available to us. 
+    function, in this tutorial. This function is not part of pylab
+    package which we import during starting, so we will have to 
+    import it. It is part of scipy package.
 
     In []: from scipy.optimize import fsolve
 
-    Now, let's look at the arguments of fsolve using fsolve?
+    Now, let's look at the documentation of fsolve using fsolve?    
     
     In []: fsolve?
 
-    The first argument, func, is a python function that takes atleast
-    one argument. So, we should now define a python function for the
-    given mathematical expression sin(x)+cos^2(x). 
+    As mentioned in docs the first argument, func, is a python 
+    function that takes atleast one argument. So, we should now 
+    define a python function for the given mathematical expression
+    sin(x)+cos^2(x). 
 
     The second argument, x0, is the initial estimate of the roots of
     the function. Based on this initial guess, fsolve returns a root. 
@@ -86,16 +95,21 @@
     given mathematical expression (sin(x)+cos^2(x)) for a each input. 
 
     In []: def f(x):
-               return sin(x)+cos(x)*cos(x)
+    ...        return sin(x)+cos(x)*cos(x)
+    ...
+    ...
+    hit return thrice for coming out of function definition. 
    
     def, is a key word in python that tells the interpreter that a
     function definition is beginning. f, here, is the name of the
     function and x is the lone argument of the function. The whole
-    definition of the function is done with in an indented block. Our
-    function f has just one line in it's definition. 
+    definition of the function is done with in an indented block same
+    as for loops and conditional statements we have used in our 
+    earlier sessions. Our function f has just one line in it's 
+    definition. 
 
     You can test your function, by calling it with an argument for
-    which the output value is know, say x = 0. We can see that
+    which the output value is known, say x = 0. We can see that
     sin(x) + cos^2(x) has a value of 1, when x = 0. 
 
     Let's check our function definition, by calling it with 0 as an
@@ -111,8 +125,9 @@
     In []: fsolve(f, 0)
     fsolve has returned a root of sin(x)+cos^2(x) that is close to 0. 
 
-    That brings us to the end of this tutorial on solving linear
-    equations, finding roots of polynomials and other non-linear
+    That brings us to the end of this tutorial. We have covered solution
+    of linear equations, finding roots of polynomials and other 
+    non-linear
     equations. We have also learnt how to define functions and call
     them.