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-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%Tutorial slides on Python.
-%
-% Author: FOSSEE
-% Copyright (c) 2009, FOSSEE, IIT Bombay
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
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-%\usepackage{times}
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-% Taken from Fernando's slides.
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-\usepackage[scaled=.95]{helvet}
-\usepackage{amsmath}
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-\usepackage{listings}
-\lstset{language=Python,
-    basicstyle=\ttfamily\bfseries,
-    commentstyle=\color{red}\itshape,
-  stringstyle=\color{darkgreen},
-  showstringspaces=false,
-  keywordstyle=\color{blue}\bfseries}
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-\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{
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-%    showstringspaces=false,
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-%    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}
-




