+
−%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
−%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[] {7 November, 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{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)
+
−    In []: Ax = dot(A,x)
+
−  \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{lstlisting}
+
−In []: Ax
+
−Out[]: 
+
−array([[  1.00000000e+00],
+
−       [ -2.00000000e+00],
+
−       [  2.22044605e-16]])
+
−\end{lstlisting}
+
−\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{15}
+
−\end{frame}
+
−
+
−\subsection{Exercises}
+
−
+
−\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{10}
+
−\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 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}
+
−\end{frame}
+
−
+
−\begin{frame}[fragile]
+
−\frametitle{\typ{fsolve}}
+
−Find the root of $sin(x)+cos^2(x)$ nearest to $0$
+
−\begin{lstlisting}
+
−In []: fsolve(sin(x)+cos(x)**2, 0)
+
−NameError: name 'x' is not defined
+
−In []: x = linspace(-pi, pi)
+
−In []: fsolve(sin(x)+cos(x)**2, 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 f(x):
+
−           return sin(x)+cos(x)**2
+
−\end{lstlisting}
+
−\begin{itemize}
+
−\item \typ{def}
+
−\item name
+
−\item arguments
+
−\item \typ{return}
+
−\end{itemize}
+
−\end{frame}
+
−
+
−\begin{frame}[fragile]
+
−\frametitle{Functions - Calling them}
+
−\begin{lstlisting}
+
−In [15]: f()
+
−---------------------------------------
+
−\end{lstlisting}
+
−\alert{\typ{TypeError:}}\typ{f() takes exactly 1 argument}
+
−\typ{(0 given)}
+
−\begin{lstlisting}
+
−In []: f(0)
+
−Out[]: 1.0
+
−In []: f(1)
+
−Out[]: 1.1333975665343254
+
−\end{lstlisting}
+
−More on Functions later \ldots
+
−\end{frame}
+
−
+
−\begin{frame}[fragile]
+
−\frametitle{\typ{fsolve} \ldots}
+
−Find the root of $sin(x)+cos^2(x)$ nearest to $0$
+
−\begin{lstlisting}
+
−In []: fsolve(f, 0)
+
−Out[]: -0.66623943249251527
+
−\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}
+
−
+
−\section{ODEs}
+
−
+
−\begin{frame}[fragile]
+
−\frametitle{Solving ODEs using SciPy}
+
−\begin{itemize}
+
−\item Let's 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 = 25000, k = 0.00003, y(0) = 250
+
−\item Define a function as below
+
−\end{itemize}
+
−\begin{lstlisting}
+
−In []: def epid(y, t):
+
−  ....     k, L = 0.00003, 25000
+
−  ....     return k*y*(L-y)
+
−  ....
+
−\end{lstlisting}
+
−\end{frame}
+
−
+
−\begin{frame}[fragile]
+
−\frametitle{Solving ODEs using SciPy \ldots}
+
−\begin{lstlisting}
+
−In []: t = arange(0, 12, 0.2)
+
−
+
−In []: y = odeint(epid, 250, t)
+
−
+
−In []: plot(t, y)
+
−\end{lstlisting}
+
−%Insert Plot
+
−\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, omega = initial
+
−  ....     g, L = 9.81, 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, 10, 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}
+
−  \frametitle{Things we have learned}
+
−  \begin{itemize}
+
−  \item Solving Linear Equations
+
−  \item Defining Functions
+
−  \item Finding Roots
+
−  \item Solving ODEs
+
−  \end{itemize}
+
−\end{frame}
+
−
+
−\end{document}