# HG changeset patch
# User Puneeth Chaganti <punchagan@fossee.in>
# Date 1291914442 -19800
# Node ID 956b486ffe6fd71c2ca4787d939ff01f96666d10
# Parent  a1117e03f98a3afc886db269afb87ec4e8cf09b3
Renamed day1/session6 to day1/session4.

diff -r a1117e03f98a -r 956b486ffe6f day1/session4.tex
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/day1/session4.tex	Thu Dec 09 22:37:22 2010 +0530
@@ -0,0 +1,655 @@
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%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}
+\usepackage{amsmath}
+
+\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}}}  }
+
+%%% This is from Fernando's setup.
+% \usepackage{color}
+% \definecolor{orange}{cmyk}{0,0.4,0.8,0.2}
+% % Use and configure listings package for nicely formatted code
+% \usepackage{listings}
+% \lstset{
+%    language=Python,
+%    basicstyle=\small\ttfamily,
+%    commentstyle=\ttfamily\color{blue},
+%    stringstyle=\ttfamily\color{orange},
+%    showstringspaces=false,
+%    breaklines=true,
+%    postbreak = \space\dots
+% }
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Title page
+\title[Solving Equations \& ODEs]{Python for Science and Engg:\\Solving Equations \& ODEs}
+
+\author[FOSSEE] {FOSSEE}
+
+\institute[IIT Bombay] {Department of Aerospace Engineering\\IIT Bombay}
+\date[] {SciPy 2010, Introductory tutorials\\Day 1, Session 6}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%\pgfdeclareimage[height=0.75cm]{iitmlogo}{iitmlogo}
+%\logo{\pgfuseimage{iitmlogo}}
+
+
+%% Delete this, if you do not want the table of contents to pop up at
+%% the beginning of each subsection:
+\AtBeginSubsection[]
+{
+  \begin{frame}<beamer>
+    \frametitle{Outline}
+    \tableofcontents[currentsection,currentsubsection]
+  \end{frame}
+}
+
+\AtBeginSection[]
+{
+  \begin{frame}<beamer>
+    \frametitle{Outline}
+    \tableofcontents[currentsection,currentsubsection]
+  \end{frame}
+}
+
+% If you wish to uncover everything in a step-wise fashion, uncomment
+% the following command: 
+%\beamerdefaultoverlayspecification{<+->}
+
+%\includeonlyframes{current,current1,current2,current3,current4,current5,current6}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% DOCUMENT STARTS
+\begin{document}
+
+\begin{frame}
+  \maketitle
+\end{frame}
+
+%% \begin{frame}
+%%   \frametitle{Outline}
+%%   \tableofcontents
+%%   % You might wish to add the option [pausesections]
+%% \end{frame}
+
+\section{Solving linear equations}
+
+\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{Solving using Matrices}
+Let us now look at how to solve this using \kwrd{matrices}
+  \begin{lstlisting}
+In []: A = array([[3,2,-1],
+                  [2,-2,4],                   
+                  [-1, 0.5, -1]])
+In []: b = array([1, -2, 0])
+In []: x = solve(A, b)
+  \end{lstlisting}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{Solution:}
+\begin{lstlisting}
+In []: x
+Out[]: array([ 1., -2., -2.])
+\end{lstlisting}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{Let's check!}
+\begin{small}
+\begin{lstlisting}
+In []: Ax = dot(A, x)
+In []: Ax
+Out[]: array([  1.00000000e+00,  -2.00000000e+00,  -1.11022302e-16])
+\end{lstlisting}
+\end{small}
+\begin{block}{}
+The last term in the matrix is actually \alert{0}!\\
+We can use \kwrd{allclose()} to check.
+\end{block}
+\begin{lstlisting}
+In []: allclose(Ax, b)
+Out[]: True
+\end{lstlisting}
+\inctime{10}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{Problem}
+Solve the set of equations:
+\begin{align*}
+  x + y + 2z -w & = 3\\
+  2x + 5y - z - 9w & = -3\\
+  2x + y -z + 3w & = -11 \\
+  x - 3y + 2z + 7w & = -5\\
+\end{align*}
+\inctime{5}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{Solution}
+Use \kwrd{solve()}
+\begin{align*}
+  x & = -5\\
+  y & = 2\\
+  z & = 3\\
+  w & = 0\\
+\end{align*}
+\end{frame}
+
+\section{Finding Roots}
+
+\begin{frame}[fragile]
+\frametitle{SciPy: \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: \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{\typ{fsolve}}
+Find the root of $sin(z)+cos^2(z)$ nearest to $0$
+\vspace{-0.1in}
+\begin{center}
+\includegraphics[height=2.8in, interpolate=true]{data/fsolve}    
+\end{center}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{\typ{fsolve}}
+Root of $sin(z)+cos^2(z)$ nearest to $0$
+\begin{lstlisting}
+In []: fsolve(sin(z)+cos(z)*cos(z), 0)
+NameError: name 'z' is not defined
+\end{lstlisting}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{\typ{fsolve}}
+\begin{lstlisting}
+In []: z = linspace(-pi, pi)
+In []: fsolve(sin(z)+cos(z)*cos(z), 0)
+\end{lstlisting}
+\begin{small}
+\alert{\typ{TypeError:}}
+\typ{'numpy.ndarray' object is not callable}
+\end{small}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{Functions - Definition}
+We have been using them all along. Now let's see how to define them.
+\begin{lstlisting}
+In []: def g(z):
+ ....:     return sin(z)+cos(z)*cos(z)
+\end{lstlisting}
+\begin{itemize}
+\item \typ{def} -- keyword
+\item name: \typ{g}
+\item arguments: \typ{z}
+\item \typ{return} -- keyword
+\end{itemize}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{Functions - Calling them}
+\begin{lstlisting}
+In []: g()
+---------------------------------------
+\end{lstlisting}
+\alert{\typ{TypeError:}}\typ{g() takes exactly 1 argument}
+\typ{(0 given)}
+\begin{lstlisting}
+In []: g(0)
+Out[]: 1.0
+In []: g(1)
+Out[]: 1.1333975665343254
+\end{lstlisting}
+More on Functions later \ldots
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{\typ{fsolve} \ldots}
+Find the root of $sin(z)+cos^2(z)$ nearest to $0$
+\begin{lstlisting}
+In []: fsolve(g, 0)
+Out[]: -0.66623943249251527
+\end{lstlisting}
+\begin{center}
+\includegraphics[height=2in, interpolate=true]{data/fsolve}    
+\end{center}
+\end{frame}
+
+\begin{frame}[fragile]
+  \frametitle{Exercise Problem}
+  Find the root of the equation $x^2 - sin(x) + cos^2(x) = tan(x)$ nearest to $0$
+\end{frame}
+
+\begin{frame}[fragile]
+  \frametitle{Solution}
+  \begin{small}
+  \begin{lstlisting}
+def g(x):
+    return x**2 - sin(x) + cos(x)*cos(x) - tan(x)
+fsolve(g, 0)
+  \end{lstlisting}
+  \end{small}
+  \begin{center}
+\includegraphics[height=2in, interpolate=true]{data/fsolve_tanx}
+  \end{center}
+\end{frame}
+
+%% \begin{frame}[fragile]
+%% \frametitle{Scipy Methods \dots}
+%% \begin{small}
+%% \begin{lstlisting}
+%% In []: from scipy.optimize import fixed_point
+
+%% In []: from scipy.optimize import bisect
+
+%% In []: from scipy.optimize import newton
+%% \end{lstlisting}
+%% \end{small}
+%% \end{frame}
+
+\section{ODEs}
+
+\begin{frame}[fragile]
+\frametitle{Solving ODEs using SciPy}
+\begin{itemize}
+\item Consider the spread of an epidemic in a population
+\item $\frac{dy}{dt} = ky(L-y)$ gives the spread of the disease
+\item $L$ is the total population.
+\item Use $L = 2.5E5, k = 3E-5, y(0) = 250$
+\item Define a function as below
+\end{itemize}
+\begin{lstlisting}
+In []: from scipy.integrate import odeint
+In []: def epid(y, t):
+  ....     k = 3.0e-5
+  ....     L = 2.5e5
+  ....     return k*y*(L-y)
+  ....
+\end{lstlisting}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{Solving ODEs using SciPy \ldots}
+\begin{lstlisting}
+In []: t = linspace(0, 12, 61)
+
+In []: y = odeint(epid, 250, t)
+
+In []: plot(t, y)
+\end{lstlisting}
+%Insert Plot
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{Result}
+\begin{center}
+\includegraphics[height=2in, interpolate=true]{data/image}  
+\end{center}
+\end{frame}
+
+
+\begin{frame}[fragile]
+\frametitle{ODEs - Simple Pendulum}
+We shall use the simple ODE of a simple pendulum. 
+\begin{equation*}
+\ddot{\theta} = -\frac{g}{L}sin(\theta)
+\end{equation*}
+\begin{itemize}
+\item This equation can be written as a system of two first order ODEs
+\end{itemize}
+\begin{align}
+\dot{\theta} &= \omega \\
+\dot{\omega} &= -\frac{g}{L}sin(\theta) \\
+ \text{At}\ t &= 0 : \nonumber \\
+ \theta = \theta_0(10^o)\quad & \&\quad  \omega = 0\ (Initial\ values)\nonumber 
+\end{align}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{ODEs - Simple Pendulum \ldots}
+\begin{itemize}
+\item Use \typ{odeint} to do the integration
+\end{itemize}
+\begin{lstlisting}
+In []: def pend_int(initial, t):
+  ....     theta = initial[0]
+  ....     omega = initial[1]
+  ....     g = 9.81
+  ....     L = 0.2
+  ....     F=[omega, -(g/L)*sin(theta)]
+  ....     return F
+  ....
+\end{lstlisting}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{ODEs - Simple Pendulum \ldots}
+\begin{itemize}
+\item \typ{t} is the time variable \\ 
+\item \typ{initial} has the initial values
+\end{itemize}
+\begin{lstlisting}
+In []: t = linspace(0, 20, 101)
+In []: initial = [10*2*pi/360, 0]
+\end{lstlisting} 
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{ODEs - Simple Pendulum \ldots}
+%%\begin{small}
+\typ{In []: from scipy.integrate import odeint}
+%%\end{small}
+\begin{lstlisting}
+In []: pend_sol = odeint(pend_int, 
+                         initial,t)
+\end{lstlisting}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{Result}
+\begin{center}
+\includegraphics[height=2in, interpolate=true]{data/ode}  
+\end{center}
+\end{frame}
+
+\section{FFTs}
+
+\begin{frame}[fragile]
+\frametitle{The FFT}
+\begin{itemize}
+    \item We have a simple signal $y(t)$
+    \item Find the FFT and plot it
+\end{itemize}
+\begin{lstlisting}
+In []: t = linspace(0, 2*pi, 500)
+In []: y = sin(4*pi*t)
+
+In []: f = fft(y)
+In []: freq = fftfreq(500, t[1] - t[0])
+
+In []: plot(freq[:250], abs(f)[:250])
+In []: grid()
+\end{lstlisting} 
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{FFTs cont\dots}
+\begin{lstlisting}
+In []: y1 = ifft(f) # inverse FFT
+In []: allclose(y, y1)
+Out[]: True
+\end{lstlisting} 
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{FFTs cont\dots}
+Let us add some noise to the signal
+\begin{lstlisting}
+In []: yr = y + random(size=500)*0.2
+In []: yn = y + normal(size=500)*0.2
+
+In []: plot(t, yr)
+In []: figure()
+In []: plot(freq[:250],
+  ...:      abs(fft(yn))[:250])
+\end{lstlisting}
+\begin{itemize}
+    \item \typ{random}: produces uniform deviates in $[0, 1)$
+    \item \typ{normal}: draws random samples from a Gaussian
+        distribution
+    \item Useful to create a random matrix of any shape
+\end{itemize}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{FFTs cont\dots}
+Filter the noisy signal:
+\begin{lstlisting}
+In []: from scipy import signal
+In []: yc = signal.wiener(yn, 5)
+In []: clf()
+In []: plot(t, yc)
+In []: figure()
+In []: plot(freq[:250], 
+  ...:      abs(fft(yc))[:250])
+\end{lstlisting}
+Only scratched the surface here \dots
+\end{frame}
+
+
+\begin{frame}
+  \frametitle{Things we have learned}
+  \begin{itemize}
+  \item Solving Linear Equations
+  \item Defining Functions
+  \item Finding Roots
+  \item Solving ODEs
+  \item Random number generation
+  \item FFTs and basic signal processing
+  \end{itemize}
+\end{frame}
+
+\end{document}
+
+%% Questions for Quiz %%
+%% ------------------ %%
+
+\begin{frame}
+\frametitle{\incqno }
+Given a 4x4 matrix \texttt{A} and a 4-vector \texttt{b}, what command do
+you use to solve for the equation \\
+\texttt{Ax = b}?
+\end{frame}
+
+\begin{frame}
+\frametitle{\incqno }
+What command will you use if you wish to integrate a system of ODEs?
+\end{frame}
+
+\begin{frame}
+\frametitle{\incqno }
+How do you calculate the roots of the polynomial, $y = 1 + 6x + 8x^2 +
+x^3$?
+\end{frame}
+
+\begin{frame}
+\frametitle{\incqno }
+Two arrays \lstinline+a+ and \lstinline+b+ are numerically almost equal, what command
+do you use to check if this is true?
+\end{frame}
+
+%% \begin{frame}[fragile]
+%% \frametitle{\incqno }
+%% \begin{lstlisting}
+%%   In []: x = arange(0, 1, 0.25)
+%%   In []: print x
+%% \end{lstlisting}
+%% What will be printed?
+%% \end{frame}
+
+
+%% \begin{frame}[fragile]
+%% \frametitle{\incqno }
+%% \begin{lstlisting}
+%% from scipy.integrate import quad
+%% def f(x):
+%%     res = x*cos(x)
+%% quad(f, 0, 1)
+%% \end{lstlisting}
+%% What changes will you make to the above code to make it work?
+%% \end{frame}
+
+%% \begin{frame}
+%% \frametitle{\incqno }
+%% What two commands will you use to create and evaluate a spline given
+%% some data?
+%% \end{frame}
+
+%% \begin{frame}[fragile]
+%% \frametitle{\incqno }
+%% What would be the result?
+%% \begin{lstlisting}
+%%   In []: x
+%%   array([[0, 1, 2],
+%%          [3, 4, 5],
+%%          [6, 7, 8]])
+%%   In []: x[::-1,:]
+%% \end{lstlisting}
+%% Hint:
+%% \begin{lstlisting}
+%%   In []: x = arange(9)
+%%   In []: x[::-1]
+%%   array([8, 7, 6, 5, 4, 3, 2, 1, 0])
+%% \end{lstlisting}
+%% \end{frame}
+
+%% \begin{frame}[fragile]
+%% \frametitle{\incqno }
+%% What would be the result?
+%% \begin{lstlisting}
+%%   In []: y = arange(3)
+%%   In []: x = linspace(0,3,3)
+%%   In []: x-y
+%% \end{lstlisting}
+%% \end{frame}
+
+%% \begin{frame}[fragile]
+%% \frametitle{\incqno }
+%% \begin{lstlisting}
+%%   In []: x
+%%   array([[ 0, 1, 2, 3],
+%%          [ 4, 5, 6, 7],
+%%          [ 8, 9, 10, 11],
+%%          [12, 13, 14, 15]])
+%% \end{lstlisting}
+%% How will you get the following \lstinline+x+?
+%% \begin{lstlisting}
+%%   array([[ 5, 7],
+%%          [ 9, 11]])
+%% \end{lstlisting}
+%% \end{frame}
+
+%% \begin{frame}[fragile]
+%% \frametitle{\incqno }
+%% What would be the output?
+%% \begin{lstlisting}
+%%   In []: y = arange(4)
+%%   In []: x = array(([1,2,3,2],[1,3,6,0]))
+%%   In []: x + y
+%% \end{lstlisting}
+%% \end{frame}
+
+%% \begin{frame}[fragile]
+%% \frametitle{\incqno }
+%% \begin{lstlisting}
+%%   In []: line = plot(x, sin(x))
+%% \end{lstlisting}
+%% Use the \lstinline+set_linewidth+ method to set width of \lstinline+line+ to 2.
+%% \end{frame}
+
+%% \begin{frame}[fragile]
+%% \frametitle{\incqno }
+%% What would be the output?
+%% \begin{lstlisting}
+%%   In []: x = arange(9)
+%%   In []: y = arange(9.)
+%%   In []: x == y
+%% \end{lstlisting}
+%% \end{frame}
+
diff -r a1117e03f98a -r 956b486ffe6f day1/session6.tex
--- a/day1/session6.tex	Thu Dec 09 22:35:05 2010 +0530
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,655 +0,0 @@
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%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}
-\usepackage{amsmath}
-
-\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}}}  }
-
-%%% This is from Fernando's setup.
-% \usepackage{color}
-% \definecolor{orange}{cmyk}{0,0.4,0.8,0.2}
-% % Use and configure listings package for nicely formatted code
-% \usepackage{listings}
-% \lstset{
-%    language=Python,
-%    basicstyle=\small\ttfamily,
-%    commentstyle=\ttfamily\color{blue},
-%    stringstyle=\ttfamily\color{orange},
-%    showstringspaces=false,
-%    breaklines=true,
-%    postbreak = \space\dots
-% }
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% Title page
-\title[Solving Equations \& ODEs]{Python for Science and Engg:\\Solving Equations \& ODEs}
-
-\author[FOSSEE] {FOSSEE}
-
-\institute[IIT Bombay] {Department of Aerospace Engineering\\IIT Bombay}
-\date[] {SciPy 2010, Introductory tutorials\\Day 1, Session 6}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
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-
-%% Delete this, if you do not want the table of contents to pop up at
-%% the beginning of each subsection:
-\AtBeginSubsection[]
-{
-  \begin{frame}<beamer>
-    \frametitle{Outline}
-    \tableofcontents[currentsection,currentsubsection]
-  \end{frame}
-}
-
-\AtBeginSection[]
-{
-  \begin{frame}<beamer>
-    \frametitle{Outline}
-    \tableofcontents[currentsection,currentsubsection]
-  \end{frame}
-}
-
-% If you wish to uncover everything in a step-wise fashion, uncomment
-% the following command: 
-%\beamerdefaultoverlayspecification{<+->}
-
-%\includeonlyframes{current,current1,current2,current3,current4,current5,current6}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% DOCUMENT STARTS
-\begin{document}
-
-\begin{frame}
-  \maketitle
-\end{frame}
-
-%% \begin{frame}
-%%   \frametitle{Outline}
-%%   \tableofcontents
-%%   % You might wish to add the option [pausesections]
-%% \end{frame}
-
-\section{Solving linear equations}
-
-\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{Solving using Matrices}
-Let us now look at how to solve this using \kwrd{matrices}
-  \begin{lstlisting}
-In []: A = array([[3,2,-1],
-                  [2,-2,4],                   
-                  [-1, 0.5, -1]])
-In []: b = array([1, -2, 0])
-In []: x = solve(A, b)
-  \end{lstlisting}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Solution:}
-\begin{lstlisting}
-In []: x
-Out[]: array([ 1., -2., -2.])
-\end{lstlisting}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Let's check!}
-\begin{small}
-\begin{lstlisting}
-In []: Ax = dot(A, x)
-In []: Ax
-Out[]: array([  1.00000000e+00,  -2.00000000e+00,  -1.11022302e-16])
-\end{lstlisting}
-\end{small}
-\begin{block}{}
-The last term in the matrix is actually \alert{0}!\\
-We can use \kwrd{allclose()} to check.
-\end{block}
-\begin{lstlisting}
-In []: allclose(Ax, b)
-Out[]: True
-\end{lstlisting}
-\inctime{10}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Problem}
-Solve the set of equations:
-\begin{align*}
-  x + y + 2z -w & = 3\\
-  2x + 5y - z - 9w & = -3\\
-  2x + y -z + 3w & = -11 \\
-  x - 3y + 2z + 7w & = -5\\
-\end{align*}
-\inctime{5}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Solution}
-Use \kwrd{solve()}
-\begin{align*}
-  x & = -5\\
-  y & = 2\\
-  z & = 3\\
-  w & = 0\\
-\end{align*}
-\end{frame}
-
-\section{Finding Roots}
-
-\begin{frame}[fragile]
-\frametitle{SciPy: \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: \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{\typ{fsolve}}
-Find the root of $sin(z)+cos^2(z)$ nearest to $0$
-\vspace{-0.1in}
-\begin{center}
-\includegraphics[height=2.8in, interpolate=true]{data/fsolve}    
-\end{center}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{\typ{fsolve}}
-Root of $sin(z)+cos^2(z)$ nearest to $0$
-\begin{lstlisting}
-In []: fsolve(sin(z)+cos(z)*cos(z), 0)
-NameError: name 'z' is not defined
-\end{lstlisting}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{\typ{fsolve}}
-\begin{lstlisting}
-In []: z = linspace(-pi, pi)
-In []: fsolve(sin(z)+cos(z)*cos(z), 0)
-\end{lstlisting}
-\begin{small}
-\alert{\typ{TypeError:}}
-\typ{'numpy.ndarray' object is not callable}
-\end{small}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Functions - Definition}
-We have been using them all along. Now let's see how to define them.
-\begin{lstlisting}
-In []: def g(z):
- ....:     return sin(z)+cos(z)*cos(z)
-\end{lstlisting}
-\begin{itemize}
-\item \typ{def} -- keyword
-\item name: \typ{g}
-\item arguments: \typ{z}
-\item \typ{return} -- keyword
-\end{itemize}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Functions - Calling them}
-\begin{lstlisting}
-In []: g()
----------------------------------------
-\end{lstlisting}
-\alert{\typ{TypeError:}}\typ{g() takes exactly 1 argument}
-\typ{(0 given)}
-\begin{lstlisting}
-In []: g(0)
-Out[]: 1.0
-In []: g(1)
-Out[]: 1.1333975665343254
-\end{lstlisting}
-More on Functions later \ldots
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{\typ{fsolve} \ldots}
-Find the root of $sin(z)+cos^2(z)$ nearest to $0$
-\begin{lstlisting}
-In []: fsolve(g, 0)
-Out[]: -0.66623943249251527
-\end{lstlisting}
-\begin{center}
-\includegraphics[height=2in, interpolate=true]{data/fsolve}    
-\end{center}
-\end{frame}
-
-\begin{frame}[fragile]
-  \frametitle{Exercise Problem}
-  Find the root of the equation $x^2 - sin(x) + cos^2(x) = tan(x)$ nearest to $0$
-\end{frame}
-
-\begin{frame}[fragile]
-  \frametitle{Solution}
-  \begin{small}
-  \begin{lstlisting}
-def g(x):
-    return x**2 - sin(x) + cos(x)*cos(x) - tan(x)
-fsolve(g, 0)
-  \end{lstlisting}
-  \end{small}
-  \begin{center}
-\includegraphics[height=2in, interpolate=true]{data/fsolve_tanx}
-  \end{center}
-\end{frame}
-
-%% \begin{frame}[fragile]
-%% \frametitle{Scipy Methods \dots}
-%% \begin{small}
-%% \begin{lstlisting}
-%% In []: from scipy.optimize import fixed_point
-
-%% In []: from scipy.optimize import bisect
-
-%% In []: from scipy.optimize import newton
-%% \end{lstlisting}
-%% \end{small}
-%% \end{frame}
-
-\section{ODEs}
-
-\begin{frame}[fragile]
-\frametitle{Solving ODEs using SciPy}
-\begin{itemize}
-\item Consider the spread of an epidemic in a population
-\item $\frac{dy}{dt} = ky(L-y)$ gives the spread of the disease
-\item $L$ is the total population.
-\item Use $L = 2.5E5, k = 3E-5, y(0) = 250$
-\item Define a function as below
-\end{itemize}
-\begin{lstlisting}
-In []: from scipy.integrate import odeint
-In []: def epid(y, t):
-  ....     k = 3.0e-5
-  ....     L = 2.5e5
-  ....     return k*y*(L-y)
-  ....
-\end{lstlisting}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Solving ODEs using SciPy \ldots}
-\begin{lstlisting}
-In []: t = linspace(0, 12, 61)
-
-In []: y = odeint(epid, 250, t)
-
-In []: plot(t, y)
-\end{lstlisting}
-%Insert Plot
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Result}
-\begin{center}
-\includegraphics[height=2in, interpolate=true]{data/image}  
-\end{center}
-\end{frame}
-
-
-\begin{frame}[fragile]
-\frametitle{ODEs - Simple Pendulum}
-We shall use the simple ODE of a simple pendulum. 
-\begin{equation*}
-\ddot{\theta} = -\frac{g}{L}sin(\theta)
-\end{equation*}
-\begin{itemize}
-\item This equation can be written as a system of two first order ODEs
-\end{itemize}
-\begin{align}
-\dot{\theta} &= \omega \\
-\dot{\omega} &= -\frac{g}{L}sin(\theta) \\
- \text{At}\ t &= 0 : \nonumber \\
- \theta = \theta_0(10^o)\quad & \&\quad  \omega = 0\ (Initial\ values)\nonumber 
-\end{align}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{ODEs - Simple Pendulum \ldots}
-\begin{itemize}
-\item Use \typ{odeint} to do the integration
-\end{itemize}
-\begin{lstlisting}
-In []: def pend_int(initial, t):
-  ....     theta = initial[0]
-  ....     omega = initial[1]
-  ....     g = 9.81
-  ....     L = 0.2
-  ....     F=[omega, -(g/L)*sin(theta)]
-  ....     return F
-  ....
-\end{lstlisting}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{ODEs - Simple Pendulum \ldots}
-\begin{itemize}
-\item \typ{t} is the time variable \\ 
-\item \typ{initial} has the initial values
-\end{itemize}
-\begin{lstlisting}
-In []: t = linspace(0, 20, 101)
-In []: initial = [10*2*pi/360, 0]
-\end{lstlisting} 
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{ODEs - Simple Pendulum \ldots}
-%%\begin{small}
-\typ{In []: from scipy.integrate import odeint}
-%%\end{small}
-\begin{lstlisting}
-In []: pend_sol = odeint(pend_int, 
-                         initial,t)
-\end{lstlisting}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Result}
-\begin{center}
-\includegraphics[height=2in, interpolate=true]{data/ode}  
-\end{center}
-\end{frame}
-
-\section{FFTs}
-
-\begin{frame}[fragile]
-\frametitle{The FFT}
-\begin{itemize}
-    \item We have a simple signal $y(t)$
-    \item Find the FFT and plot it
-\end{itemize}
-\begin{lstlisting}
-In []: t = linspace(0, 2*pi, 500)
-In []: y = sin(4*pi*t)
-
-In []: f = fft(y)
-In []: freq = fftfreq(500, t[1] - t[0])
-
-In []: plot(freq[:250], abs(f)[:250])
-In []: grid()
-\end{lstlisting} 
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{FFTs cont\dots}
-\begin{lstlisting}
-In []: y1 = ifft(f) # inverse FFT
-In []: allclose(y, y1)
-Out[]: True
-\end{lstlisting} 
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{FFTs cont\dots}
-Let us add some noise to the signal
-\begin{lstlisting}
-In []: yr = y + random(size=500)*0.2
-In []: yn = y + normal(size=500)*0.2
-
-In []: plot(t, yr)
-In []: figure()
-In []: plot(freq[:250],
-  ...:      abs(fft(yn))[:250])
-\end{lstlisting}
-\begin{itemize}
-    \item \typ{random}: produces uniform deviates in $[0, 1)$
-    \item \typ{normal}: draws random samples from a Gaussian
-        distribution
-    \item Useful to create a random matrix of any shape
-\end{itemize}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{FFTs cont\dots}
-Filter the noisy signal:
-\begin{lstlisting}
-In []: from scipy import signal
-In []: yc = signal.wiener(yn, 5)
-In []: clf()
-In []: plot(t, yc)
-In []: figure()
-In []: plot(freq[:250], 
-  ...:      abs(fft(yc))[:250])
-\end{lstlisting}
-Only scratched the surface here \dots
-\end{frame}
-
-
-\begin{frame}
-  \frametitle{Things we have learned}
-  \begin{itemize}
-  \item Solving Linear Equations
-  \item Defining Functions
-  \item Finding Roots
-  \item Solving ODEs
-  \item Random number generation
-  \item FFTs and basic signal processing
-  \end{itemize}
-\end{frame}
-
-\end{document}
-
-%% Questions for Quiz %%
-%% ------------------ %%
-
-\begin{frame}
-\frametitle{\incqno }
-Given a 4x4 matrix \texttt{A} and a 4-vector \texttt{b}, what command do
-you use to solve for the equation \\
-\texttt{Ax = b}?
-\end{frame}
-
-\begin{frame}
-\frametitle{\incqno }
-What command will you use if you wish to integrate a system of ODEs?
-\end{frame}
-
-\begin{frame}
-\frametitle{\incqno }
-How do you calculate the roots of the polynomial, $y = 1 + 6x + 8x^2 +
-x^3$?
-\end{frame}
-
-\begin{frame}
-\frametitle{\incqno }
-Two arrays \lstinline+a+ and \lstinline+b+ are numerically almost equal, what command
-do you use to check if this is true?
-\end{frame}
-
-%% \begin{frame}[fragile]
-%% \frametitle{\incqno }
-%% \begin{lstlisting}
-%%   In []: x = arange(0, 1, 0.25)
-%%   In []: print x
-%% \end{lstlisting}
-%% What will be printed?
-%% \end{frame}
-
-
-%% \begin{frame}[fragile]
-%% \frametitle{\incqno }
-%% \begin{lstlisting}
-%% from scipy.integrate import quad
-%% def f(x):
-%%     res = x*cos(x)
-%% quad(f, 0, 1)
-%% \end{lstlisting}
-%% What changes will you make to the above code to make it work?
-%% \end{frame}
-
-%% \begin{frame}
-%% \frametitle{\incqno }
-%% What two commands will you use to create and evaluate a spline given
-%% some data?
-%% \end{frame}
-
-%% \begin{frame}[fragile]
-%% \frametitle{\incqno }
-%% What would be the result?
-%% \begin{lstlisting}
-%%   In []: x
-%%   array([[0, 1, 2],
-%%          [3, 4, 5],
-%%          [6, 7, 8]])
-%%   In []: x[::-1,:]
-%% \end{lstlisting}
-%% Hint:
-%% \begin{lstlisting}
-%%   In []: x = arange(9)
-%%   In []: x[::-1]
-%%   array([8, 7, 6, 5, 4, 3, 2, 1, 0])
-%% \end{lstlisting}
-%% \end{frame}
-
-%% \begin{frame}[fragile]
-%% \frametitle{\incqno }
-%% What would be the result?
-%% \begin{lstlisting}
-%%   In []: y = arange(3)
-%%   In []: x = linspace(0,3,3)
-%%   In []: x-y
-%% \end{lstlisting}
-%% \end{frame}
-
-%% \begin{frame}[fragile]
-%% \frametitle{\incqno }
-%% \begin{lstlisting}
-%%   In []: x
-%%   array([[ 0, 1, 2, 3],
-%%          [ 4, 5, 6, 7],
-%%          [ 8, 9, 10, 11],
-%%          [12, 13, 14, 15]])
-%% \end{lstlisting}
-%% How will you get the following \lstinline+x+?
-%% \begin{lstlisting}
-%%   array([[ 5, 7],
-%%          [ 9, 11]])
-%% \end{lstlisting}
-%% \end{frame}
-
-%% \begin{frame}[fragile]
-%% \frametitle{\incqno }
-%% What would be the output?
-%% \begin{lstlisting}
-%%   In []: y = arange(4)
-%%   In []: x = array(([1,2,3,2],[1,3,6,0]))
-%%   In []: x + y
-%% \end{lstlisting}
-%% \end{frame}
-
-%% \begin{frame}[fragile]
-%% \frametitle{\incqno }
-%% \begin{lstlisting}
-%%   In []: line = plot(x, sin(x))
-%% \end{lstlisting}
-%% Use the \lstinline+set_linewidth+ method to set width of \lstinline+line+ to 2.
-%% \end{frame}
-
-%% \begin{frame}[fragile]
-%% \frametitle{\incqno }
-%% What would be the output?
-%% \begin{lstlisting}
-%%   In []: x = arange(9)
-%%   In []: y = arange(9.)
-%%   In []: x == y
-%% \end{lstlisting}
-%% \end{frame}
-