From 3fccee6f30b22fb8ac5f3c9e176d267896c14f00 Mon Sep 17 00:00:00 2001
From: Prabhu Ramachandran
Date: Mon, 2 Jan 2017 19:12:28 +0530
Subject: Add session on exercises.

---
 scipy/basic/exercises.tex | 476 ++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 476 insertions(+)
 create mode 100644 scipy/basic/exercises.tex

(limited to 'scipy/basic/exercises.tex')

diff --git a/scipy/basic/exercises.tex b/scipy/basic/exercises.tex
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+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%Tutorial slides on Python.
+%
+% Author: Prabhu Ramachandran <prabhu at aero.iitb.ac.in>
+% Copyright (c) 2005-2009, Prabhu Ramachandran
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\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}
+
+\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[Exercises]{Introductory Scientific Computing with Python}
+\subtitle{Exercises}
+
+\author[FOSSEE] {FOSSEE}
+
+\institute[FOSSEE -- IIT Bombay] {Department of Aerospace Engineering\\IIT Bombay}
+\date[] {Mumbai, India}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%\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}
+}
+
+
+% 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}[fragile]
+  \frametitle{Problem 1}
+  \begin{columns}
+    \column{0.5\textwidth}
+    \hspace*{-0.5in}
+    \includegraphics[height=2in, interpolate=true]{data/L-Tsq.png}
+    \column{0.45\textwidth}
+    \begin{block}{Example code}
+      \tiny
+\begin{lstlisting}
+In []: l, t = loadtxt('pendulum.txt',
+               unpack=True)
+In []: plot(l, t, '.')
+    \end{lstlisting}
+    \end{block}
+  \end{columns}
+  \begin{block}{Problem Statement}
+      Tweak above code to plot data in file \typ{pos.txt}.
+  \end{block}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Problem 1 cont...}
+  \begin{itemize}
+  \item Label both the axes.
+  \item What kind of motion is this?
+  \item Title the graph accordingly.
+  \item Annotate the position where vertical velocity is zero.
+  \end{itemize}
+\end{frame}
+
+\begin{frame}[fragile]
+  \frametitle{Solution}
+\begin{lstlisting}
+x, y = loadtxt('pos.txt', unpack=True)
+plot(x, y)
+xlabel('x')
+ylabel('y')
+title('Projectile motion')
+annotate('v = 0', xy=(5, 4.75))
+# Or
+annotate(r'$v_y = 0$', xy=(5, 4.75))
+\end{lstlisting}
+  \begin{itemize}
+  \item Note the \LaTeX\ syntax
+  \item Note the raw strings: \typ{r'$v_y = 0$'}
+  \end{itemize}
+\end{frame}
+
+\begin{frame}[fragile]
+  \frametitle{Problem 2}
+  \begin{columns}
+    \column{0.5\textwidth}
+    \hspace*{-0.5in}
+    \includegraphics[height=2in, interpolate=true]{data/exercises/triangle}
+    \column{0.45\textwidth}
+    \begin{block}{Plot points given x and y coordinates}
+    \tiny
+    \begin{lstlisting}
+In []: x = [3, 2, -2, 3]
+In []: y = [1, -3, 4, 1]
+In []: plot(x, y)
+    \end{lstlisting}
+    \end{block}
+  \end{columns}
+  Line can be plotted using arrays of coordinates.
+  \pause
+  \begin{block}{Problem statement}
+    Write a Program that plots a regular n-gon(Let n = 5).
+  \end{block}
+\end{frame}
+
+\begin{frame}[fragile]
+  \frametitle{Solution}
+\begin{lstlisting}
+n = 5
+t = linspace(0, 2*pi, n+1)
+x = cos(t)
+y = sin(t)
+plot(x, y)
+axis('equal')
+\end{lstlisting}
+\end{frame}
+
+\begin{frame}[fragile]
+  \frametitle{Better Solution}
+\begin{lstlisting}
+def plot_ngon(n):
+    t = linspace(0, 2*pi, n+1)
+    x = cos(t)
+    y = sin(t)
+    plot(x, y)
+    axis('equal')
+
+plot_ngon(5)
+clf()
+plot_ngon(10)
+\end{lstlisting}
+\end{frame}
+
+
+\begin{frame}[fragile]
+  \frametitle{Problem 3}
+  \begin{columns}
+    \column{0.5\textwidth}
+    \hspace*{-0.5in}
+    \includegraphics[height=2in, interpolate=true]{data/exercises/damp}
+    \column{0.45\textwidth}
+    \begin{block}{Damped Oscillation}
+    \tiny
+    \begin{lstlisting}
+In []: t = linspace(0, 4*pi)
+In []: plot(t, exp(-t/10)*sin(t))
+    \end{lstlisting}
+    \end{block}
+  \end{columns}
+\end{frame}
+
+\begin{frame}[fragile]
+  \frametitle{Problem 3 cont...}
+Create a sequence of images (say 10) in which the damped
+oscillator($e^{-t/10}sin(t)$) slowly evolves over time $t$.
+\begin{columns}
+\column{0.35\textwidth}
+\includegraphics[width=1.5in,height=1.5in, interpolate=true]{data/exercises/plot2}
+\column{0.35\textwidth}
+\includegraphics[width=1.5in,height=1.5in, interpolate=true]{data/exercises/plot4}
+\column{0.35\textwidth}
+\includegraphics[width=1.5in,height=1.5in, interpolate=true]{data/exercises/plot6}
+\end{columns}
+\begin{block}{Hint}
+\small
+  \begin{lstlisting}
+savefig('plot'+str(i)+'.png') #i is some int
+  \end{lstlisting}
+\end{block}
+\end{frame}
+
+\begin{frame}[fragile]
+  \frametitle{Naive Solution}
+\begin{lstlisting}
+for i in range(1, 11):
+    t = linspace(0, 0.5*pi*i, 50*i)
+    clf()
+    plot(t, exp(-t/10)*sin(t))
+    savefig('plot' + str(i) + '.png')
+\end{lstlisting}
+\end{frame}
+
+\begin{frame}[fragile]
+  \frametitle{Better Solution}
+\begin{lstlisting}
+for i in range(1, 11):
+    t = linspace(0, 0.5*pi*i, 50*i)
+    clf()
+    plot(t, exp(-t/10)*sin(t))
+    xlim(0, 5*pi)
+    ylim(-1, 1)
+    savefig('plot' + str(i) + '.png')
+\end{lstlisting}
+\end{frame}
+
+
+\begin{frame}[fragile]
+  \frametitle{Problem 4}
+  \begin{lstlisting}
+In []: x = imread('smoothing.gif')
+In []: x.shape
+Out[]: (256, 256)
+In []: imshow(x,cmap=cm.gray)
+In []: colorbar()
+\end{lstlisting}
+\emphbar{Replace each pixel with mean of neighboring pixels}
+  \begin{center}
+  \includegraphics[height=1in, interpolate=true]{data/exercises/neighbour}
+  \end{center}
+\end{frame}
+
+\begin{frame}
+  \begin{center}
+    \includegraphics[height=3in, interpolate=true]{data/exercises/smoothing}
+  \end{center}
+\end{frame}
+
+\begin{frame}[fragile]
+  \frametitle{Problem 4: Approach}
+  For \typ{y} being resultant image:
+  \begin{lstlisting}
+y[1, 1] = x[0, 1]/4 + x[1, 0]/4
+          + x[2, 1]/4 + x[1, 2]/4
+  \end{lstlisting}
+   \begin{columns}
+    \column{0.45\textwidth}
+    \hspace*{-0.5in}
+    \includegraphics[height=1.5in, interpolate=true]{data/exercises/smoothing}
+    \column{0.45\textwidth}
+    \hspace*{-0.5in}
+    \includegraphics[height=1.5in, interpolate=true]{data/exercises/after-filter}
+  \end{columns}
+   \begin{block}{Hint:}
+     Use array Slicing.
+   \end{block}
+\end{frame}
+
+\begin{frame}[fragile]
+  \frametitle{Solution}
+  \begin{lstlisting}
+In []: y = zeros_like(x)
+In []: y[1:-1,1:-1] = x[:-2,1:-1]/4 +
+                      x[2:,1:-1]/4 +
+                      x[1:-1,2:]/4 +
+                      x[1:-1,:-2]/4
+In []: imshow(y,cmap=cm.gray)
+  \end{lstlisting}
+\end{frame}
+
+\begin{frame}[fragile]
+  \frametitle{Problem 4 cont\ldots}
+  \begin{itemize}
+      \item Apply the smoothing operation repeatedly to the original
+          image
+
+      \item Subtract the smoothed image from the original to obtain
+          the edges
+  \end{itemize}
+\end{frame}
+
+\begin{frame}[fragile]
+  \frametitle{Problem 5}
+
+  What if you did the following in problem 4?
+\begin{lstlisting}
+In []: y1 = zeros_like(x)
+In []: y1[1:-1,1:-1] = (x[:-2,1:-1] +
+                        x[2:,1:-1] +
+                        x[1:-1,2:] +
+                        x[1:-1,:-2])/4
+  \end{lstlisting}
+
+    Are the answers different?
+
+\end{frame}
+
+\begin{frame}[fragile]
+  \frametitle{Problem 5 cont\ldots}
+  Why? The answer lies in the following:
+\begin{lstlisting}
+In []: x.dtype
+Out[]: dtype('uint8')
+
+In []: print(x.itemsize)
+1
+
+In []: z = x/4.0
+
+In []: print(z.dtype)
+float64
+
+\end{lstlisting}
+\end{frame}
+
+\begin{frame}[fragile]
+  \frametitle{Problem 5 cont\ldots}
+What if you did this?
+
+\begin{lstlisting}
+x = imread('smoothing.gif')
+y2 = zeros_like(x)
+y2[1:-1,1:-1] = x[:-2,1:-1]/4. + \
+                x[2:,1:-1]/4.  + \
+                x[1:-1,2:]/4.  + \
+                x[1:-1,:-2]/4.
+\end{lstlisting}
+\begin{itemize}
+    \item Will the answer be any different from \typ{y}?
+    \item What will the dtype of \typ{y2} be?
+    \item Discuss what is going on!
+\end{itemize}
+
+\end{frame}
+
+\begin{frame}[fragile]
+  \frametitle{Problem 5 cont\ldots}
+  \begin{itemize}
+  \item Did you do the right thing to find the edges earlier in problem 4?
+  \item If not, fix it!
+  \end{itemize}
+  Note that:
+  \small
+\begin{lstlisting}
+In []: print(x.dtype)
+uint8
+In []: x1 = x.astype('float64')
+In []: print(x1.dtype)
+float64
+In []: print(x.dtype.char)
+d
+In []: x.dtype.<TAB> # Explore!
+\end{lstlisting}
+\end{frame}
+
+\begin{frame}[fragile]
+  \frametitle{Problem 6}
+
+  Edge detection looks much nicer with \typ{lena.png}, try it!  The
+  caveat is that it is a 4 component RGBA image with elements in the
+  range $[0.0, 1.0]$.
+
+\begin{lstlisting}
+In []: x = imread('lena.png')
+
+In []: print(x.shape)
+(512, 512, 4)
+
+In []: print(x.min(), x.max())
+(0.0, 1.0)
+\end{lstlisting}
+
+ Repeat the edge detection with this image.
+
+\end{frame}
+
+
+\end{document}
+
+%% \begin{frame}
+%%   \frametitle{Problem 4}
+%%   Legendre polynomials $P_n(x)$ are defined by the following recurrence relation
+
+%% \center{$(n+1)P_{n+1}(x) - (2n+1)xP_n(x) + nP_{n-1}(x) = 0$}\\
+
+%% with $P_0(x) = 1$, $P_1(x) = x$ and $P_2(x) = (3x^2 - 1)/2$. Compute the next three
+%%    Legendre polynomials and plot all 6 over the interval [-1,1].
+%% \end{frame}
+
+%% \begin{frame}[fragile]
+%% \frametitle{Problem Set 5}
+%%   \begin{columns}
+%%     \column{0.6\textwidth}
+%%     \small{
+%%     \begin{itemize}
+%%       \item[3] Consider the iteration $x_{n+1} = f(x_n)$ where $f(x) = kx(1-x)$.  Plot the successive iterates of this process as explained below.
+%%     \end{itemize}}
+%%     \column{0.35\textwidth}
+%%     \hspace*{-0.5in}
+%%   \includegraphics[height=1.6in, interpolate=true]{data/cobweb}
+%% \end{columns}
+%% \end{frame}
+
+%% \begin{frame}
+%%   \frametitle{Problem Set 5.3}
+%%   Plot the cobweb plot as follows:
+%%   \begin{enumerate}
+%%     \item Start at $(x_0, 0)$ ($\implies$ i=0)
+%%     \item Draw a line to $(x_i, f(x_i))$
+%%     \item Set $x_{i+1} = f(x_i)$
+%%     \item Draw a line to $(x_{i+1}, x_{i+1})$
+%%     \item $(i\implies i+1)$
+%%     \item Repeat from 2 for as long as you want
+%%   \end{enumerate}
+%% \inctime{20}
+%% \end{frame}
-- 
cgit 

