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+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%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}
+