%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%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[ODEs \& Root Finding]{Python for Science and Engg:\\ODEs \& Finding Roots}

\author[FOSSEE] {FOSSEE}

\institute[IIT Bombay] {Department of Aerospace Engineering\\IIT Bombay}

\date[] {31, October 2009\\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{ODEs}

\begin{frame}[fragile]

\frametitle{ODE Integration}

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{Solving ODEs using SciPy}

\begin{itemize}

\item We use the \typ{odeint} function from scipy to do the integration

\item Define a function as below

\end{itemize}

\begin{lstlisting}

In []: def pend_int(initial, t):

  ....     theta, omega = initial

  ....     g, L = -9.81, 0.2

  ....     f=[omega, -(g/L)*sin(theta)]

  ....     return f

  ....

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Solving ODEs using SciPy \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, 10, 101)

In []: initial = [10*2*pi/360, 0]

\end{lstlisting} 

\end{frame}

\begin{frame}[fragile]

\frametitle{Solving ODEs using SciPy \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}

\section{Finding Roots}

\begin{frame}[fragile]

\frametitle{Roots of $f(x)=0$}

\begin{itemize}

\item Roots --- values of $x$ satisfying $f(x)=0$

\item $f(x)=0$ may have real or complex roots

\item Presently, let's look at real roots

\end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{Roots of $f(x)=0$}

\begin{itemize}

\item Given function $cosx-x^2$ 

\item First we find \alert{a} root in $(-\pi/2, \pi/2)$

\item Then we find \alert{all} roots in $(-\pi/2, \pi/2)$

\end{itemize}

\end{frame}

%% \begin{frame}[fragile]

%% \frametitle{Fixed Point Method}

%% \begin{enumerate}

%% \item Convert $f(x)=0$ to the form $x=g(x)$

%% \item Start with an initial value of $x$ and iterate successively

%% \item $x_{n+1}=g(x_n)$ and $x_0$ is our initial guess

%% \item Iterate until $x_{n+1}-x_n \le tolerance$

%% \end{enumerate}

%% \end{frame}

%% \begin{frame}[fragile]

%% \frametitle{Fixed Point \dots}

%% \begin{lstlisting}

%% In []: def our_g(x):

%%  ....:     return sqrt(cos(x))

%%  ....: 

%% In []: tolerance = 1e-5

%% In []: while abs(x1-x0)>tolerance:

%%  ....:     x0 = x1

%%  ....:     x1 = our_g(x1)

%%  ....:   

%% In []: x0

%% \end{lstlisting}

%% \end{frame}

\begin{frame}[fragile]

\frametitle{Bisection Method}

\begin{enumerate}

\item Start with the given interval $(-\pi/2, \pi/2)$ ($(a, b)$)

\item $f(a)\cdot f(b) < 0$

\item Bisect the interval. $c = \frac{a+b}{2}$

\item Change the interval to $(a, c)$ if $f(a)\cdot f(c) < 0$

\item Else, change the interval to $(c, b)$

\item Go back to 3 until $(b-a) \le$ tolerance

\end{enumerate}

\alert{\typ{tolerance = 1e-5}}

\end{frame}

%% \begin{frame}[fragile]

%% \frametitle{Bisection \dots}

%% \begin{lstlisting}

%% In []: tolerance = 1e-5

%% In []: a = -pi/2

%% In []: b = 0

%% In []: while b-a > tolerance:

%%  ....:     c = (a+b)/2

%%  ....:     if our_f(a)*our_f(c) < 0:

%%  ....:         b = c

%%  ....:     else:

%%  ....:         a = c

%%  ....:         

%% \end{lstlisting}

%% \end{frame}

\begin{frame}[fragile]

\frametitle{Newton-Raphson Method}

\begin{enumerate}

\item Start with an initial guess of x for the root

\item $\Delta x = -f(x)/f^{'}(x)$

\item $ x \leftarrow x + \Delta x$

\item Go back to 2 until $|\Delta x| \le$ tolerance

\end{enumerate}

\end{frame}

%% \begin{frame}[fragile]

%% \frametitle{Newton-Raphson \dots}

%% \begin{lstlisting}

%% In []: def our_df(x):

%%  ....:     return -sin(x)-2*x

%%  ....: 

%% In []: delx = 10*tolerance

%% In []: while delx > tolerance:

%%  ....:     delx = -our_f(x)/our_df(x)

%%  ....:     x = x + delx

%%  ....:     

%%  ....:     

%% \end{lstlisting}

%% \end{frame}

%% \begin{frame}[fragile]

%% \frametitle{Newton-Raphson \ldots}

%% \begin{itemize}

%% \item What if $f^{'}(x) = 0$?

%% \item Can you write a better version of the Newton-Raphson?

%% \item What if the differential is not easy to calculate?

%% \item Look at Secant Method

%% \end{itemize}

%% \end{frame}

\begin{frame}[fragile]

\frametitle{Initial Estimates}

\begin{itemize}

\item Given an interval

\item How to find \alert{all} the roots?

\end{itemize}

\begin{enumerate}

\item Check for change of signs of $f(x)$ in the given interval

\item A root lies in the interval where the sign change occurs

\end{enumerate}

\end{frame}

\begin{frame}[fragile]

\frametitle{Initial Estimates \ldots}

\begin{lstlisting}

In []: def our_f(x):

 ....:     return cos(x) - x*x

 ....: 

In []: x = linspace(-pi/2, pi/2, 11)

In []: y = our_f(x)

\end{lstlisting}

Get the intervals of x, where sign changes occur

\end{frame}

\begin{frame}[fragile]

\frametitle{Initial Estimates \ldots}

\begin{lstlisting}

In []: pos = y[:-1]*y[1:] < 0

In []: rpos = zeros(x.shape, dtype=bool)

In []: rpos[:-1] = pos

In []: rpos[1:] += pos

In []: rts = x[rpos]

\end{lstlisting}

Now use Newton-Raphson?

\end{frame}

\begin{frame}[fragile]

\frametitle{Scipy Methods - \typ{roots}}

\begin{itemize}

\item Calculates the roots of polynomials

\end{itemize}

\begin{lstlisting}

  In []: coeffs = [1, 6, 13]

  In []: roots(coeffs)

\end{lstlisting}

\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}

\begin{lstlisting}

  In []: fsolve(our_f, -pi/2)

\end{lstlisting}

\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}

\begin{frame}

  \frametitle{Things we have learned}

  \begin{itemize}

  \item Solving ODEs

  \item Finding Roots

    \begin{itemize}

    \item Estimating Interval

    \item Newton-Raphson

    \item Scipy methods

    \end{itemize}

  \end{itemize}

\end{frame}

\end{document}