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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Tutorial slides on Python.
%
% Author: FOSSEE
% Copyright (c) 2009, FOSSEE, IIT Bombay
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Taken from Fernando's slides.
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\usepackage[scaled=.95]{helvet}
\usepackage{amsmath}
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basicstyle=\ttfamily\bfseries,
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stringstyle=\color{darkgreen},
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keywordstyle=\color{blue}\bfseries}
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%%% 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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% 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]
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}
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\frametitle{Outline}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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{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
\end{itemize}
\end{frame}
\end{document}
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