+
−\documentclass[12pt]{article}
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−\usepackage{amsmath}
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−\title{Python Workshop\\Problems and Exercises}
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−\author{Asokan Pichai\\Prabhu Ramachandran}
+
−\begin{document}
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−\maketitle
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−
+
−\section{Matrices and Arrays \& 2D Plotting}
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−\subsection{Matrices and Arrays}
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−\subsubsection{Basic Numpy}
+
−\begin{verbatim}
+
−# Simple array math example
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−>>> import numpy as np
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−>>> a = np.array([1,2,3,4])
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−>>> b = np.arange(2,6)
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−>>> b
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−array([2,3,4,5])
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−>>> a*2 + b + 1 # Basic math!
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−array([5, 8, 11, 14])
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−
+
−# Pi and e are defined.
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−>>> x = np.linspace(0.0, 10.0, 1000)
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−>>> x *= 2*np.pi/10 # inplace.
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−# apply functions to array.
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−>>> y = np.sin(x)
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−>>> z = np.exp(y)
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−
+
−>>> x = np.array([1., 2, 3, 4])
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−>>> np.size(x)
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−4
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−>>> x.dtype # What is a.dtype?
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−dtype('float64')
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−>>> x.shape
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−(4,)
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−>>> print np.rank(x), x.itemsize
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−1 8
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−>>> x[0] = 10
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−>>> print x[0], x[-1]
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−10.0 4.0
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−
+
−>>> a = np.array([[ 0, 1, 2, 3],
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−...            [10,11,12,13]])
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−>>> a.shape # (rows, columns)
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−(2, 4)
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−# Accessing and setting values
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−>>> a[1,3] 
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−13
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−>>> a[1,3] = -1
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−>>> a[1] # The second row
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−array([10,11,12,-1])
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−
+
−>>> b=np.array([[0,2,4,2],[1,2,3,4]])
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−>>> np.add(a,b,a)
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−>>> np.sum(x,axis=1)
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−
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−>>> np.greater(a,4)
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−>>> np.sqrt(a)
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−\end{verbatim}
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−
+
−\subsubsection{Array Creation}
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−\begin{verbatim}
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−>>> np.array([2,3,4])  
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−array([2, 3, 4])
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−
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−>>> np.linspace(0, 2, 4)   
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−array([0.,0.6666667,1.3333333,2.])
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−
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−>>>np.ones([2,2])
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−array([[ 1.,  1.],
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−     [ 1.,  1.]])
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−
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−>>>a = np.array([[1,2,3],[4,5,6]])
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−>>>np.ones_like(a)
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−array([[1, 1, 1],
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−       [1, 1, 1]])
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−\end{verbatim}
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−\subsubsection{Slicing, Striding Arrays}
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−\begin{verbatim}
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−>>> a = np.array([[1,2,3], [4,5,6], 
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−               [7,8,9]])
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−>>> a[0,1:3]
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−array([2, 3])
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−>>> a[1:,1:]
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−array([[5, 6],
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−       [8, 9]])
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−>>> a[:,2]
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−array([3, 6, 9])
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−>>> a[...,2]
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−array([3, 6, 9])
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−
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−>>> a[0::2,0::2]
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−array([[1, 3],
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−       [7, 9]])
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−# Slices are references to the 
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−# same memory!
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−\end{verbatim}
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−\subsubsection{Random Numbers}
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−\begin{verbatim}
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−>>> np.random.rand(3,2)
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−array([[ 0.96276665,  0.77174861],
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−       [ 0.35138557,  0.61462271],
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−       [ 0.16789255,  0.43848811]])
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−>>> np.random.randint(1,100)
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−42
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−\end{verbatim}
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−
+
−\subsubsection{Problem Set}
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−\begin{verbatim}
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−    >>> from scipy import misc
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−    >>> A=misc.imread(name)
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−    >>> misc.imshow(A)
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−\end{verbatim}
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−\begin{enumerate}
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−  \item Convert an RGB image to Grayscale. $ Y = 0.5R + 0.25G + 0.25B $.
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−  \item Scale the image to 50\%.
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−  \item Introduce some random noise.
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−  \item Smooth the image using a mean filter.
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−  \\\small{Each element in the array is replaced by mean of all the neighbouring elements}
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−  \\\small{How fast does your code run?}
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−\end{enumerate}
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−
+
−\subsection{2D Plotting}
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−\subsubsection{Basic 2D Plotting}
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−\begin{verbatim}
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−$ ipython -pylab
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−>>> x = linspace(0, 2*pi, 1000)
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−>>> plot(x, sin(x)) 
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−>>> plot(x, sin(x), 'ro')
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−>>> xlabel(r'$\chi$', color='g')
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−# LaTeX markup!
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−>>> ylabel(r'sin($\chi$)', color='r')
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−>>> title('Simple figure', fontsize=20)
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−>>> savefig('/tmp/test.eps')
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−\end{verbatim}
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−\subsubsection{Tweaking plots}
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−\begin{verbatim}
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−# Set properties of objects:
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−>>> l, = plot(x, sin(x))
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−# Why "l,"?
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−>>> setp(l, linewidth=2.0, color='r')
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−>>> l.set_linewidth(2.0)
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−>>> draw() # Redraw.
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−>>> setp(l) # Print properties.
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−>>> clf() # Clear figure.
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−>>> close() # Close figure.
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−\end{verbatim}
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−
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−\subsubsection{Working with text}
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−\begin{verbatim}
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−>>> w = arange(-2,2,.1)
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−>>> plot(w,exp(-(w*w))*cos)
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−>>> ylabel('$f(\omega)$')
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−>>> xlabel('$\omega$')
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−>>> title(r"$f(\omega)=e^{-\omega^2}
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−            cos({\omega^2})$")
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−>>> annotate('maxima',xy=(0, 1), 
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−             xytext=(1, 0.8), 
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−             arrowprops=dict(
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−             facecolor='black', 
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−             shrink=0.05))
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−\end{verbatim}
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−
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−\subsubsection{Legends}
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−\begin{verbatim}
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−>>> x = linspace(0, 2*np.pi, 1000)
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−>>> plot(x, cos(5*x), 'r--', 
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−         label='cosine')
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−>>> plot(x, sin(5*x), 'g--', 
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−         label='sine')
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−>>> legend() 
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−# Or use:
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−>>> legend(['cosine', 'sine'])
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−\end{verbatim}
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−
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−\subsubsection{Multiple figures}
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−\begin{verbatim}
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−>>> figure(1)
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−>>> plot(x, sin(x))
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−>>> figure(2)
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−>>> plot(x, tanh(x))
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−>>> figure(1)
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−>>> title('Easy as 1,2,3')
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−
+
−\end{verbatim}
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−
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−\subsubsection{Problem Set}
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−  \begin{enumerate}
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−    \item Write a function that plots any regular n-gon given n.
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−    \item Consider the logistic map, $f(x) = kx(1-x)$, plot it for
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−          $k=2.5, 3.5$ and $4$ in the same plot.
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−
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−  \item Consider the iteration $x_{n+1} = f(x_n)$ where $f(x) =
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−        kx(1-x)$.  Plot the successive iterates of this process.
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−  \item Plot this using a cobweb plot as follows:
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−  \begin{enumerate}
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−    \item Start at $(x_0, 0)$
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−    \item Draw line to $(x_i, f(x_i))$; 
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−    \item Set $x_{i+1} = f(x_i)$
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−    \item Draw line to $(x_i, x_i)$
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−    \item Repeat from 2 for as long as you want 
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−  \end{enumerate}
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−\end{enumerate}
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−
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−\section{Advanced Numpy}
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−
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−\subsection{Broadcasting}
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−\begin{verbatim}
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−>>> a = np.arange(4)
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−>>> b = np.arange(5)
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−>>> a+b #Does this work?
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−>>> a+3
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−>>> c=np.array([3])
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−>>> a+c #Works!
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−>>> b+c #But how?
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−>>> a.shape, b.shape, c.shape
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−
+
−>>> a = np.arange(4)
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−>>> a+3
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−array([3, 4, 5, 6])
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−
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−>>> x = np.ones((3, 5))
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−>>> y = np.ones(8)
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−>>> (x[..., None] + y).shape
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−(3, 5, 8)
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−
+
−\end{verbatim}
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−
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−\subsection{Copies \& Views}
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−\begin{verbatim}
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−>>> a = np.array([[1,2,3],[4,5,6]])
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−>>> b = a
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−>>> b is a
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−>>> b[0,0]=0; print a
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−>>> c = a.view()
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−>>> c is a
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−>>> c.base is a
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−>>> c.flags.owndata
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−>>> d = a.copy()
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−>>> d.base is a
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−>>> d.flags.owndata
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−
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−>>> a = np.arange(1,9)
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−>>> a.shape=3,3
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−>>> b = a[0,1:3]
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−>>> c = a[0::2,0::2]
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−>>> a.flags.owndata
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−>>> b.flags.owndata
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−>>> b.base
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−>>> c.base is a
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−
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−>>> b = a[np.array([0,1,2])]
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−array([[1, 2, 3],
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−       [4, 5, 6],
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−       [7, 8, 9]])
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−>>> b.flags.owndata
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−>>> abool=np.greater(a,2)
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−>>> c = a[abool]
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−>>> c.flags.owndata
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−
+
−\end{verbatim}
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−
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−\section{Scipy}
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−\subsection{Linear Algebra}
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−\begin{verbatim}
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−>>> import scipy as sp
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−>>> from scipy import linalg
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−>>> A=sp.mat(np.arange(1,10))
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−>>> A.shape=3,3
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−>>> linalg.inv(A)
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−>>> linalg.det(A)
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−>>> linalg.norm(A)
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−>>> linalg.expm(A) #logm
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−>>> linalg.sinm(A) #cosm, tanm, ...
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−
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−>>> A = sp.mat(np.arange(1,10))
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−>>> A.shape=3,3
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−>>> linalg.lu(A)
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−>>> linalg.eig(A)
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−>>> linalg.eigvals(A)
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−\end{verbatim}
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−
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−\subsection{Solving Linear Equations}
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−
+
−\begin{align*}
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−  3x + 2y - z  & = 1 \\
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−  2x - 2y + 4z  & = -2 \\
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−  -x + \frac{1}{2}y -z & = 0
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−\end{align*}
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−To Solve this, 
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−\begin{verbatim}
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−>>> A = sp.mat([[3,2,-1],[2,-2,4]
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−              ,[-1,1/2,-1]])
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−>>> B = sp.mat([[1],[-2],[0]])
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−>>> linalg.solve(A,B)
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−\end{verbatim}
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−
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−\subsection{Integrate}
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−\subsubsection{Quadrature}
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−Calculate the area under $(sin(x) + x^2)$ in the range $(0,1)$
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−\begin{verbatim}
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−>>> def f(x):
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−        return np.sin(x)+x**2
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−>>> integrate.quad(f, 0, 1)
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−\end{verbatim}
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−
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−\subsubsection{ODE Integration}
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−Numerically solve ODEs\\
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−\begin{align*}
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−\frac{dx}{dt} &=-e^{-t}x^2\\ 
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−         x(0) &=2    
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−\end{align*}
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−\begin{verbatim}
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−>>> def dx_dt(x,t):
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−...     return -np.exp(-t)*x**2
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−>>> x=integrate.odeint(dx_dt, 2, t)
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−>>> plt.plot(x,t)
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−\end{verbatim}
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−
+
−\subsection{Interpolation}
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−\subsubsection{1D Interpolation}
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−\begin{verbatim}
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−>>> from scipy import interpolate
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−>>> interpolate.interp1d?
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−>>> x = np.arange(0,2*np.pi,np.pi/4)
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−>>> y = np.sin(x)
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−>>> fl = interpolate.interp1d(
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−            x,y,kind='linear')
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−>>> fc = interpolate.interp1d(
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−             x,y,kind='cubic')
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−>>> fl(np.pi/3)
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−>>> fc(np.pi/3)
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−\end{verbatim}
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−
+
−\subsubsection{Splines}
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−Plot the Cubic Spline of $sin(x)$
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−\begin{verbatim}
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−>>> x = np.arange(0,2*np.pi,np.pi/4)
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−>>> y = np.sin(x)
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−>>> tck = interpolate.splrep(x,y)
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−>>> X = np.arange(0,2*np.pi,np.pi/50)
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−>>> Y = interpolate.splev(X,tck,der=0)
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−>>> plt.plot(x,y,'o',x,y,X,Y)
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−>>> plt.show()
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−\end{verbatim}
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−
+
−\subsection{Signal \& Image Processing}
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−Applying a simple median filter
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−\begin{verbatim}
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−>>> from scipy import signal, ndimage
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−>>> from scipy import lena
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−>>> A=lena().astype('float32')
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−>>> B=signal.medfilt2d(A)
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−>>> imshow(B)
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−\end{verbatim}
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−Zooming an array - uses spline interpolation
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−\begin{verbatim}
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−>>> b=ndimage.zoom(A,0.5)
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−>>> imshow(b)
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−\end{verbatim}
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−
+
−\section{3D Data Visualization}
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−\subsection{Using mlab}
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−\begin{verbatim}
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−>>> from enthought.mayavi import mlab
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−
+
−>>> mlab.test_<TAB>
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−>>> mlab.test_contour3d()
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−>>> mlab.test_contour3d??
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−\end{verbatim}
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−
+
−\subsubsection{Plotting Functions}
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−\begin{verbatim}
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−>>> from numpy import *
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−>>> t = linspace(0, 2*pi, 50)
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−>>> u = cos(t)*pi
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−>>> x, y, z = sin(u), cos(u), sin(t)
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−
+
−>>> x = mgrid[-3:3:100j,-3:3:100j]
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−>>> z = sin(x*x + y*y)
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−>>> mlab.surf(x, y, z)
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−\end{verbatim}
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−
+
−\subsubsection{Large 2D Data}
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−\begin{verbatim}
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−>>> mlab.mesh(x, y, z)
+
−
+
−>>> phi, theta = numpy.mgrid[0:pi:20j, 
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−...                         0:2*pi:20j]
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−>>> x = sin(phi)*cos(theta)
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−>>> y = sin(phi)*sin(theta)
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−>>> z = cos(phi)
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−>>> mlab.mesh(x, y, z, 
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−...           representation=
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−...           'wireframe')
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−\end{verbatim}
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−
+
−\subsubsection{Large 3D Data}
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−\begin{verbatim}
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−>>> x, y, z = ogrid[-5:5:64j, 
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−...                -5:5:64j, 
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−...                -5:5:64j]
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−>>> mlab.contour3d(x*x*0.5 + y*y + 
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−                   z*z*2)
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−
+
−>>> mlab.test_quiver3d()
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−\end{verbatim}
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−
+
−\subsection{Motivational Problem}
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−Atmospheric data of temperature over the surface of the earth. Let temperature ($T$) vary linearly with height ($z$)\\
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−$T = 288.15 - 6.5z$
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−
+
−\begin{verbatim}
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−lat = linspace(-89, 89, 37)
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−lon = linspace(0, 360, 37)
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−z = linspace(0, 100, 11)
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−\end{verbatim}
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−
+
−\begin{verbatim}
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−x, y, z = mgrid[0:360:37j,-89:89:37j,
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−                0:100:11j]
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−t = 288.15 - 6.5*z
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−mlab.contour3d(x, y, z, t)
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−mlab.outline()
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−mlab.colorbar()
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−\end{verbatim}
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−
+
−\subsection{Lorenz equation}
+
−\begin{eqnarray*}
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−\frac{d x}{dt} &=& s (y-x)\\
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−\frac{d y}{d t} &=& rx -y -xz\\
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−\frac{d z}{d t} &=& xy - bz\\
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−\end{eqnarray*}
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−
+
−Let $s=10,$
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−$r=28,$ 
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−$b=8./3.$
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−
+
−\begin{verbatim}
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−x, y, z = mgrid[-50:50:20j,-50:50:20j,
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−                -10:60:20j]
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−
+
−\end{verbatim}
+
−
+
−\end{document}