--- a/day1/session4.tex Thu Dec 09 18:42:33 2010 +0530
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,733 +0,0 @@
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%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[Matrices \& Curve Fitting]{Python for Science and Engg: Matrices
-\& Least Squares Fit}
-
-\author[FOSSEE] {FOSSEE}
-
-\institute[IIT Bombay] {Department of Aerospace Engineering\\IIT Bombay}
-\date[] {SciPy 2010, Introductory tutorials\\Day 1, Session 4}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-%\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{Matrices}
-
-\begin{frame}
-\frametitle{Matrices: Introduction}
-\alert{All matrix operations are done using \kwrd{arrays}}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Matrices: Initializing}
-\begin{lstlisting}
-In []: c = array([[11,12,13],
- [21,22,23],
- [31,32,33]])
-
-In []: c
-Out[]:
-array([[11, 12, 13],
- [21, 22, 23],
- [31, 32, 33]])
-\end{lstlisting}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Initializing some special matrices}
-\begin{small}
- \begin{lstlisting}
-In []: ones((3,5))
-Out[]:
-array([[ 1., 1., 1., 1., 1.],
- [ 1., 1., 1., 1., 1.],
- [ 1., 1., 1., 1., 1.]])
-
-In []: ones_like([1, 2, 3, 4])
-Out[]: array([1, 1, 1, 1])
-
-In []: identity(2)
-Out[]:
-array([[ 1., 0.],
- [ 0., 1.]])
- \end{lstlisting}
-Also available \alert{\typ{zeros, zeros_like, empty, empty_like}}
-\end{small}
-\end{frame}
-
-
-\begin{frame}[fragile]
- \frametitle{Accessing elements}
- \begin{small}
- \begin{lstlisting}
-In []: c
-Out[]:
-array([[11, 12, 13],
- [21, 22, 23],
- [31, 32, 33]])
-
-In []: c[1][2]
-Out[]: 23
-In []: c[1,2]
-Out[]: 23
-
-In []: c[1]
-Out[]: array([21, 22, 23])
- \end{lstlisting}
- \end{small}
-\end{frame}
-
-\begin{frame}[fragile]
- \frametitle{Changing elements}
- \begin{small}
- \begin{lstlisting}
-In []: c[1,1] = -22
-In []: c
-Out[]:
-array([[ 11, 12, 13],
- [ 21, -22, 23],
- [ 31, 32, 33]])
-
-In []: c[1] = 0
-In []: c
-Out[]:
-array([[11, 12, 13],
- [ 0, 0, 0],
- [31, 32, 33]])
- \end{lstlisting}
- \end{small}
-How do you access one \alert{column}?
-\end{frame}
-
-\begin{frame}[fragile]
- \frametitle{Slicing}
-\begin{small}
- \begin{lstlisting}
-In []: c[:,1]
-Out[]: array([12, 0, 32])
-
-In []: c[1,:]
-Out[]: array([0, 0, 0])
-
-In []: c[0:2,:]
-Out[]:
-array([[11, 12, 13],
- [ 0, 0, 0]])
-
-In []: c[1:3,:]
-Out[]:
-array([[ 0, 0, 0],
- [31, 32, 33]])
- \end{lstlisting}
-\end{small}
-\end{frame}
-
-\begin{frame}[fragile]
- \frametitle{Slicing \ldots}
-\begin{small}
- \begin{lstlisting}
-In []: c[:2,:]
-Out[]:
-array([[11, 12, 13],
- [ 0, 0, 0]])
-
-In []: c[1:,:]
-Out[]:
-array([[ 0, 0, 0],
- [31, 32, 33]])
-
-In []: c[1:,:2]
-Out[]:
-array([[ 0, 0],
- [31, 32]])
- \end{lstlisting}
-
-\end{small}
-\end{frame}
-
-\begin{frame}[fragile]
- \frametitle{Striding}
- \begin{small}
- \begin{lstlisting}
-In []: c[::2,:]
-Out[]:
-array([[11, 12, 13],
- [31, 32, 33]])
-
-In []: c[:,::2]
-Out[]:
-array([[11, 13],
- [ 0, 0],
- [31, 33]])
-
-In []: c[::2,::2]
-Out[]:
-array([[11, 13],
- [31, 33]])
- \end{lstlisting}
- \end{small}
-\end{frame}
-
-\begin{frame}[fragile]
- \frametitle{Shape of a matrix}
- \begin{lstlisting}
-In []: c
-Out[]:
-array([[11, 12, 13],
- [ 0, 0, 0],
- [31, 32, 33]])
-
-In []: c.shape
-Out[]: (3, 3)
- \end{lstlisting}
-\emphbar{Shape specifies shape or dimensions of a matrix}
-\end{frame}
-
-\begin{frame}[fragile]
- \frametitle{Elementary image processing}
-\begin{small}
- \begin{lstlisting}
-In []: a = imread('lena.png')
-
-In []: imshow(a)
-Out[]: <matplotlib.image.AxesImage object at 0xa0384cc>
- \end{lstlisting}
- \end{small}
-\typ{imread} returns an array of shape (512, 512, 4) which represents an image of 512x512 pixels and 4 shades.\\
-\typ{imshow} renders the array as an image.
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Slicing \& Striding Exercises}
- \begin{itemize}
- \item Crop the image to get the top-left quarter
- \item Crop the image to get only the face
- \item Resize image to half by dropping alternate pixels
- \end{itemize}
-
-\end{frame}
-\begin{frame}[fragile]
- \frametitle{Solutions}
-\begin{small}
- \begin{lstlisting}
-In []: imshow(a[:256,:256])
-Out[]: <matplotlib.image.AxesImage object at 0xb6f658c>
-
-In []: imshow(a[200:400,200:400])
-Out[]: <matplotlib.image.AxesImage object at 0xb757c2c>
-
-In []: imshow(a[::2,::2])
-Out[]: <matplotlib.image.AxesImage object at 0xb765c8c>
- \end{lstlisting}
-\end{small}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Transpose of a Matrix}
-\begin{lstlisting}
-In []: a = array([[ 1, 1, 2, -1],
- ...: [ 2, 5, -1, -9],
- ...: [ 2, 1, -1, 3],
- ...: [ 1, -3, 2, 7]])
-
-In []: a.T
-Out[]:
-array([[ 1, 2, 2, 1],
- [ 1, 5, 1, -3],
- [ 2, -1, -1, 2],
- [-1, -9, 3, 7]])
-\end{lstlisting}
-\end{frame}
-
-\begin{frame}[fragile]
- \frametitle{Matrix Addition}
- \begin{lstlisting}
-In []: b = array([[3,2,-1,5],
- [2,-2,4,9],
- [-1,0.5,-1,-7],
- [9,-5,7,3]])
-In []: a + b
-Out[]:
-array([[ 4. , 3. , 1. , 4. ],
- [ 4. , 3. , 3. , 0. ],
- [ 1. , 1.5, -2. , -4. ],
- [ 10. , -8. , 9. , 10. ]])
- \end{lstlisting}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Elementwise Multiplication}
-\begin{lstlisting}
-In []: a*b
-Out[]:
-array([[ 3. , 2. , -2. , -5. ],
- [ 4. , -10. , -4. , -81. ],
- [ -2. , 0.5, 1. , -21. ],
- [ 9. , 15. , 14. , 21. ]])
-
-\end{lstlisting}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Matrix Multiplication}
-\begin{lstlisting}
-In []: dot(a, b)
-Out[]:
-array([[ -6. , 6. , -6. , -3. ],
- [-64. , 38.5, -44. , 35. ],
- [ 36. , -13.5, 24. , 35. ],
- [ 58. , -26. , 34. , -15. ]])
-\end{lstlisting}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Inverse of a Matrix}
-\begin{lstlisting}
-
-\end{lstlisting}
-\begin{small}
-\begin{lstlisting}
-In []: inv(a)
-Out[]:
-array([[-0.5 , 0.55, -0.15, 0.7 ],
- [ 0.75, -0.5 , 0.5 , -0.75],
- [ 0.5 , -0.15, -0.05, -0.1 ],
- [ 0.25, -0.25, 0.25, -0.25]])
-\end{lstlisting}
-\end{small}
-\emphbar{Try this: \typ{I = dot(a, inv(a))}}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Determinant and sum of all elements}
-\begin{lstlisting}
-In []: det(a)
-Out[]: 80.0
-\end{lstlisting}
- \begin{lstlisting}
-In []: sum(a)
-Out[]: 12
- \end{lstlisting}
-
-\end{frame}
-
-%%use S=array(X,Y)
-\begin{frame}[fragile]
-\frametitle{Eigenvalues and Eigen Vectors}
-\begin{small}
-\begin{lstlisting}
-In []: e = array([[3,2,4],[2,0,2],[4,2,3]])
-
-In []: eig(e)
-Out[]:
-(array([-1., 8., -1.]),
- array([[-0.74535599, 0.66666667, -0.1931126 ],
- [ 0.2981424 , 0.33333333, -0.78664085],
- [ 0.59628479, 0.66666667, 0.58643303]]))
-
-In []: eigvals(e)
-Out[]: array([-1., 8., -1.])
-\end{lstlisting}
-\end{small}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Computing Norms}
-\begin{lstlisting}
-In []: norm(e)
-Out[]: 8.1240384046359608
-\end{lstlisting}
-\end{frame}
-
-\begin{frame}[fragile]
- \frametitle{Singular Value Decomposition}
- \begin{small}
- \begin{lstlisting}
-In []: svd(e)
-Out[]:
-(array(
-[[ -6.66666667e-01, -1.23702565e-16, 7.45355992e-01],
- [ -3.33333333e-01, -8.94427191e-01, -2.98142397e-01],
- [ -6.66666667e-01, 4.47213595e-01, -5.96284794e-01]]),
- array([ 8., 1., 1.]),
- array([[-0.66666667, -0.33333333, -0.66666667],
- [-0. , 0.89442719, -0.4472136 ],
- [-0.74535599, 0.2981424 , 0.59628479]]))
- \end{lstlisting}
- \end{small}
-\end{frame}
-
-\section{Least Squares Fit}
-\begin{frame}[fragile]
-\frametitle{$L$ vs. $T^2$ - Scatter}
-Linear trend visible.
-\vspace{-0.1in}
-\begin{figure}
-\includegraphics[width=4in]{data/L-Tsq-points}
-\end{figure}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{$L$ vs. $T^2$ - Line}
-This line does not make any mathematical sense.
-\vspace{-0.1in}
-\begin{figure}
-\includegraphics[width=4in]{data/L-Tsq-Line}
-\end{figure}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{$L$ vs. $T^2$ - Least Square Fit}
-This is what our intention is.
-\vspace{-0.1in}
-\begin{figure}
-\includegraphics[width=4in]{data/least-sq-fit}
-\end{figure}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Matrix Formulation}
-\begin{itemize}
-\item We need to fit a line through points for the equation $T^2 = m \cdot L+c$
-\item In matrix form, the equation can be represented as $T_{sq} = A \cdot p$, where $T_{sq}$ is
- $\begin{bmatrix}
- T^2_1 \\
- T^2_2 \\
- \vdots\\
- T^2_N \\
- \end{bmatrix}$
-, A is
- $\begin{bmatrix}
- L_1 & 1 \\
- L_2 & 1 \\
- \vdots & \vdots\\
- L_N & 1 \\
- \end{bmatrix}$
- and p is
- $\begin{bmatrix}
- m\\
- c\\
- \end{bmatrix}$
-\item We need to find $p$ to plot the line
-\end{itemize}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Getting $L$ and $T^2$}
-%If you \alert{closed} IPython after session 2
-\begin{lstlisting}
-In []: L = []
-In []: t = []
-In []: for line in open('pendulum.txt'):
- .... point = line.split()
- .... L.append(float(point[0]))
- .... t.append(float(point[1]))
- ....
- ....
-\end{lstlisting}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Getting $L$ and $T^2$ \dots}
-\begin{lstlisting}
-In []: L = array(L)
-In []: t = array(t)
-\end{lstlisting}
-\alert{\typ{In []: tsq = t*t}}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Generating $A$}
-\begin{lstlisting}
-In []: A = array([L, ones_like(L)])
-In []: A = A.T
-\end{lstlisting}
-%% \begin{itemize}
-%% \item A is also called a Van der Monde matrix
-%% \item It can also be generated using \typ{vander}
-%% \end{itemize}
-%% \begin{lstlisting}
-%% In []: A = vander(L, 2)
-%% \end{lstlisting}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{\typ{lstsq} \ldots}
-\begin{itemize}
-\item Now use the \typ{lstsq} function
-\item Along with a lot of things, it returns the least squares solution
-\end{itemize}
-\begin{lstlisting}
-In []: result = lstsq(A,tsq)
-In []: coef = result[0]
-\end{lstlisting}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Least Square Fit Line \ldots}
-We get the points of the line from \typ{coef}
-\begin{lstlisting}
-In []: Tline = coef[0]*L + coef[1]
-
-In []: Tline.shape
-\end{lstlisting}
-\begin{itemize}
-\item Now plot \typ{Tline} vs. \typ{L}, to get the Least squares fit line.
-\end{itemize}
-\begin{lstlisting}
-In []: plot(L, Tline, 'r')
-
-In []: plot(L, tsq, 'o')
-\end{lstlisting}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Least Squares Fit}
-\vspace{-0.15in}
-\begin{figure}
-\includegraphics[width=4in]{data/least-sq-fit}
-\end{figure}
-\end{frame}
-
-\section{Summary}
-\begin{frame}
- \frametitle{What did we learn?}
- \begin{itemize}
- \item Matrices
- \begin{itemize}
- \item Initializing
- \item Accessing elements
- \item Slicing and Striding
- \item Transpose
- \item Addition
- \item Multiplication
- \item Inverse of a matrix
- \item Determinant
- \item Eigenvalues and Eigen vector
- \item Singular Value Decomposition
- \end{itemize}
- \item Least Square Curve fitting
- \end{itemize}
-\end{frame}
-
-\end{document}
-
-%% Questions for Quiz %%
-%% ------------------ %%
-
-\begin{frame}[fragile]
-\frametitle{\incqno }
-\begin{lstlisting}
-In []: a = array([[1, 2],
- [3, 4]])
-In []: a[1,0] = 0
-\end{lstlisting}
-What is the resulting array?
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{\incqno }
-\begin{lstlisting}
- In []: x = array(([1,2,3,4],
- [2,3,4,5]))
- In []: x[-2][-3] = 4
- In []: print x
-\end{lstlisting}
-What will be printed?
-\end{frame}
-
-%% \begin{frame}[fragile]
-%% \frametitle{\incqno }
-%% \begin{lstlisting}
-%% In []: x = array([[1,2,3,4],
-%% [3,4,2,5]])
-%% \end{lstlisting}
-%% What is the \lstinline+shape+ of this array?
-%% \end{frame}
-
-\begin{frame}[fragile]
-\frametitle{\incqno }
-\begin{lstlisting}
- In []: x = array([[1,2,3,4]])
-\end{lstlisting}
-How to change \lstinline+x+ to \lstinline+array([[1,2,0,4]])+?
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{\incqno }
-\begin{lstlisting}
- In []: x = array([[1,2,3,4],
- [3,4,2,5]])
-\end{lstlisting}
-How do you get the following slice of \lstinline+x+?
-\begin{lstlisting}
-array([[2,3],
- [4,2]])
-\end{lstlisting}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{\incqno }
-\begin{lstlisting}
- In []: x = array([[9,18,27],
- [30,60,90],
- [14,7,1]])
-\end{lstlisting}
-What is the output of \lstinline+x[::3,::3]+
-\end{frame}
-
-
-\begin{frame}[fragile]
-\frametitle{\incqno }
-\begin{lstlisting}
-In []: a = array([[1, 2],
- [3, 4]])
-\end{lstlisting}
-How do you get the transpose of this array?
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{\incqno }
-\begin{lstlisting}
-In []: a = array([[1, 2],
- [3, 4]])
-In []: b = array([[1, 1],
- [2, 2]])
-In []: a*b
-\end{lstlisting}
-What does this produce?
-\end{frame}
-
-\begin{frame}
-\frametitle{\incqno }
-What command do you use to find the inverse of a matrix and its
-eigenvalues?
-\end{frame}
-
-%% \begin{frame}
-%% \frametitle{\incqno }
-%% The file \lstinline+datafile.txt+ contains 3 columns of data. What
-%% command will you use to read the entire data file into an array?
-%% \end{frame}
-
-%% \begin{frame}
-%% \frametitle{\incqno }
-%% If the contents of the file \lstinline+datafile.txt+ is read into an
-%% $N\times3$ array called \lstinline+data+, how would you obtain the third
-%% column of this data?
-%% \end{frame}
-