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

%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}

\usepackage{amsmath}

% Taken from Fernando's slides.

\usepackage{ae,aecompl}

\usepackage{mathpazo,courier,euler}

\usepackage[scaled=.95]{helvet}

\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[Calculus]{Python for Science and Engg: Interpolation, Differentiation and Integration}

\author[FOSSEE] {FOSSEE}

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

\date[] {31, October 2009\\Day 1, Session 5}

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

%\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}

  \titlepage

\end{frame}

%% \begin{frame}

%%   \frametitle{Outline}

%%   \tableofcontents

%% %  \pausesections

%% \end{frame}

\section{\typ{loadtxt}}

\begin{frame}[fragile]

  \frametitle{Array slicing}

  \begin{lstlisting}

In []: A = array([[ 1,  1,  2, -1],

                  [ 2,  5, -1, -9],

                  [ 2,  1, -1,  3],

                  [ 1, -3,  2,  7]])

In []: A[:,0]

Out[]: array([ 1,  2,  2, 1])

In []: A[1:3,1:3]

Out[]: 

array([[ 5, -1],

       [ 1, -1]])

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

  \frametitle{\typ{loadtxt}}

  \begin{itemize}

  \item Load data from a text file.

  \item Each row must have same number of values.

  \end{itemize}

\begin{lstlisting}

In []: data = loadtxt('pendulum.txt')

In []: x = data[:, 0]

In []: y = data[:, 1]

\end{lstlisting}

\end{frame}

%% \begin{frame}[fragile]

%%   \frametitle{\typ{loadtxt}}

%% \end{frame}

\section{Interpolation}

\begin{frame}[fragile]

\frametitle{Loading data (revisited)}

\begin{itemize}

  \item Given data file \typ{points.txt}.

  \item It contains x,y position of particle.

  \item Plot the given points.

%%  \item Interpolate the missing region.

\end{itemize}

\begin{lstlisting}

In []: x, y = loadtxt('points.txt',

                       unpack = True)

In []: plot(x, y, '.')

\end{lstlisting}

\end{frame}

\begin{frame}

  \frametitle{Plot}

  \begin{center}

    \includegraphics[height=2in, interpolate=true]{data/missing_points}

  \end{center}

\end{frame}

%% \begin{frame}[fragile]

%% \frametitle{Interpolation \ldots}

%% \begin{small}

%%   \typ{In []: from scipy.interpolate import interp1d}

%% \end{small}

%% \begin{itemize}

%% \item The \typ{interp1d} function returns a function

%% \begin{lstlisting}

%%   In []: f = interp1d(L, T)

%% \end{lstlisting}

%% \item Functions can be assigned to variables 

%% \item This function interpolates between known data values to obtain unknown

%% \end{itemize}

%% \end{frame}

%% \begin{frame}[fragile]

%% \frametitle{Interpolation \ldots}

%% \begin{lstlisting}

%% In []: Ln = arange(0.1,0.99,0.005)

%% # Interpolating! 

%% # The new values in range of old data

%% In []: plot(L, T, 'o', Ln, f(Ln), '-')

%% In []: f = interp1d(L, T, kind='cubic')

%% # When kind not specified, it's linear

%% # Others are ...

%% # 'nearest', 'zero', 

%% # 'slinear', 'quadratic'

%% \end{lstlisting}

%% \end{frame}

\begin{frame}[fragile]

\frametitle{Spline Interpolation}

\begin{small}

\begin{lstlisting}

In []: from scipy.interpolate import splrep

In []: from scipy.interpolate import splev

\end{lstlisting}

\end{small}

\begin{itemize}

\item Involves two steps

  \begin{enumerate}

  \item Find out the spline curve, coefficients

  \item Evaluate the spline at new points

  \end{enumerate}

\end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{\typ{splrep}}

To find the spline curve

\begin{lstlisting}

In []: tck = splrep(x, y)

\end{lstlisting}

\typ{tck} contains parameters required for representing the spline curve!

\end{frame}

\begin{frame}[fragile]

\frametitle{\typ{splev}}

To Evaluate a spline and it's derivatives

\begin{lstlisting}

In []: Xnew = arange(0.01,3,0.02)

In []: Ynew = splev(Xnew, tck)

In []: y.shape

Out[]: (40,)

In []: Ynew.shape

Out[]: (150,)

In []: plot(Xnew, Ynew)

\end{lstlisting}

\end{frame}

%% \begin{frame}[fragile]

%% \frametitle{Interpolation \ldots}

%% \begin{itemize}

%% \item 

%% \end{itemize}

%% \end{frame}

\begin{frame}

  \frametitle{Plot}

  \begin{center}

    \includegraphics[height=2in, interpolate=true]{data/interpolate}

  \end{center}

\end{frame}

\section{Differentiation}

\begin{frame}[fragile]

\frametitle{Numerical Differentiation}

\begin{itemize}

\item Given function $f(x)$ or data points $y=f(x)$

\item We wish to calculate $f^{'}(x)$ at points $x$

\item Taylor series - finite difference approximations

\end{itemize}

\begin{center}

\begin{tabular}{l l}

$f(x+h)=f(x)+hf^{'}(x)$ &Forward \\

$f(x-h)=f(x)-hf^{'}(x)$ &Backward

\end{tabular}

\end{center}

\end{frame}

\begin{frame}[fragile]

\frametitle{Forward Difference}

\begin{lstlisting}

In []: x = linspace(0, 2*pi, 100)

In []: y = sin(x)

In []: deltax = x[1] - x[0]

\end{lstlisting}

Obtain the finite forward difference of y

\end{frame}

\begin{frame}[fragile]

\frametitle{Forward Difference \ldots}

\begin{lstlisting}

In []: fD = (y[1:] - y[:-1]) / deltax

In []: print len(fD)

Out[]: 99

In []: plot(x, y) 

In []: plot(x[:-1], fD)

\end{lstlisting}

\vspace{-.2in}

\begin{center}

  \includegraphics[height=1.8in, interpolate=true]{data/fwdDiff}

\end{center}

\end{frame}

\begin{frame}[fragile]

\frametitle{Example}

\begin{itemize}

\item Given x, y positions of a particle in \typ{pos.txt}

\item Find velocity \& acceleration in x, y directions

\end{itemize}

\small{

\begin{center}

\begin{tabular}{| c | c | c |}

\hline

$X$ & $Y$ \\ \hline

0.     &  0.\\ \hline

0.25   &  0.47775\\ \hline

0.5    &  0.931\\ \hline

0.75   &  1.35975\\ \hline

1.     &  1.764\\ \hline

1.25   &  2.14375\\ \hline

\vdots & \vdots\\ \hline

\end{tabular}

\end{center}}

\end{frame}

\begin{frame}[fragile]

\frametitle{Example \ldots}

\begin{itemize}

\item Read the file

\item Obtain an array of X, Y

\item Obtain velocity and acceleration

\item use \typ{deltaT = 0.05}

\end{itemize}

\begin{lstlisting}

In []: data = loadtxt('pos.txt')

In []: X,Y = data[:,0], data[:,1]

In []: S = array([X, Y])

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Example \ldots}

\begin{lstlisting}

In []: deltaT = 0.05

In []: v = (S[:,1:]-S[:,:-1])/deltaT

In []: a = (v[:,1:]-v[:,:-1])/deltaT

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Example \ldots}

Plotting Y, $v_y$, $a_y$

\begin{lstlisting}

In []: plot(Y)

In []: plot(v[1,:])

In []: plot(a[1,:])

\end{lstlisting}

\begin{center}

  \includegraphics[height=1.8in, interpolate=true]{data/pos_vel_accel}  

\end{center}

\end{frame}

\section{Quadrature}

\begin{frame}[fragile]

\frametitle{Quadrature}

\emphbar{$\int_0^1(sin(x) + x^2)$}

\typ{In []: from scipy.integrate import quad}

\begin{itemize}

\item Inputs - function to integrate, limits

\end{itemize}

\begin{lstlisting}

In []: quad(sin(x)+x**2, 0, 1)

NameError: name 'x' is not defined

In []: x = 0

In []: quad(sin(x)+x**2, 0, 1)

\end{lstlisting}

\begin{small}

\alert{\typ{error:}}

\typ{First argument must be a callable function.}

\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)+x**2

In []: quad(f, 0, 1)

\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[]: 0.0

In []: f(1)

Out[]: 1.8414709848078965

\end{lstlisting}

More on Functions later \ldots

\end{frame}

\begin{frame}[fragile]

\frametitle{Quadrature \ldots}

\begin{lstlisting}

In []: quad(f, 0, 1)

\end{lstlisting}

Returns the integral and an estimate of the absolute error in the result.

\begin{itemize}

\item \typ{dblquad}, \typ{tplquad} are available

\end{itemize}

\end{frame}

\begin{frame}

  \frametitle{Things we have learned}

  \begin{itemize}

  \item Interpolation

  \item Differentiation

  \item Functions

    \begin{itemize}

    \item Definition

    \item Calling

    \item Default Arguments

    \item Keyword Arguments

    \end{itemize}

  \item Quadrature

  \end{itemize}

\end{frame}

\end{document}