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-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%Tutorial slides on Python.
-%
-% Author: FOSSEE
-% Copyright (c) 2009-2016, 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}
-\usepackage{pgf}
-
-% 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}
-}
-
-\newcommand{\myemph}[1]{\structure{\emph{#1}}}
-\newcommand{\PythonCode}[1]{\lstinline{#1}}
-
-\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[Basic SciPy]{Introductory Scientific Computing with
-Python}
-\subtitle{Basic SciPy}
-
-\author[Prabhu] {FOSSEE}
-
-\institute[FOSSEE -- IITB] {Department of Aerospace Engineering\\IIT Bombay}
-\date[] {
-Mumbai, India
-}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-%\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}
-
-
-\section{Solving linear systems}
-
-\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*}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Solution}
-Use \kwrd{solve()}
-\begin{align*}
-  x & = -5\\
-  y & = 2\\
-  z & = 3\\
-  w & = 0\\
-\end{align*}
-\inctime{5}
-\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}}
-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}}
-\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 - \alert{Function} and initial estimate
-\item Returns the solution
-\end{itemize}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{\typ{fsolve} \ldots}
-\begin{lstlisting}
-In []: def g(z):
- ....:     return sin(z)+cos(z)*cos(z)
-In []: fsolve(g, 0)
-Out[]: -0.66623943249251527
-\end{lstlisting}
-\begin{center}
-\includegraphics[height=2in, interpolate=true]{data/fsolve}
-\end{center}
-\inctime{10}
-\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}
-  \vspace*{-0.2in}
-  \begin{center}
-\includegraphics[height=2.5in, interpolate=true]{data/fsolve_tanx}
-\end{center}
-\vspace*{-0.5in}
-  \inctime{5}
-\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}
-\frametitle{Solving ODEs using SciPy}
-\begin{itemize}
-\item Consider the spread of an epidemic in a population
-  \vspace*{0.1in}
-\item $\frac{dy}{dt} = ky(L-y)$ gives the spread of the disease
-  \vspace*{0.1in}
-\item $L$ is the total population.
-\item Use $L = 2.5E5, k = 3E-5, y(0) = 250$
-\end{itemize}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Solving ODEs using SciPy}
-Define a function as below
-\small
-\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=3in, interpolate=true]{data/image}
-\end{center}
-\vspace*{-0.5in}
-\inctime{5}
-\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_rhs(state, t):
-  ....     theta = state[0]
-  ....     omega = state[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_rhs,
-                         initial,t)
-\end{lstlisting}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Result}
-\begin{center}
-\includegraphics[height=2in, interpolate=true]{data/ode}
-\end{center}
-  \inctime{10}
-\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.fft(y)
-In []: freq = fft.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 = fft.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.random(size=500)*0.2
-In []: yn = y +
-  ...:   random.normal(size=500)*0.2
-
-In []: plot(t, yr)
-In []: figure()
-In []: plot(freq[:250],
-  ...:      abs(fft.fft(yr))[: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.fft(yc))[:250])
-\end{lstlisting}
-Only scratched the surface here \dots
-
-\inctime{10}
-\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 FFTs and basic signal processing
-  \end{itemize}
-\end{frame}
-
-
-\begin{frame}
-    \frametitle{Further reading}
-    \begin{itemize}
-        \item \url{ipython.readthedocs.io}
-        \item \url{matplotlib.org/contents.html}
-        \item \url{docs.scipy.org/doc/numpy/user/quickstart.html}
-        \item \url{docs.scipy.org/doc/scipy/reference/tutorial}
-    \end{itemize}
-\end{frame}
-
-\end{document}