author | Puneeth Chaganti <punchagan@fossee.in> |
Fri, 06 Nov 2009 14:28:51 +0530 | |
changeset 285 | e072cadee389 |
parent 282 | b2dafce03b8d |
child 286 | ac457f7d1702 |
permissions | -rw-r--r-- |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%Tutorial slides on Python. |
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% |
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% Author: FOSSEE |
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% Copyright (c) 2009, FOSSEE, IIT Bombay |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\documentclass[14pt,compress]{beamer} |
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%\documentclass[draft]{beamer} |
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%\documentclass[compress,handout]{beamer} |
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%\usepackage{pgfpages} |
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%\pgfpagesuselayout{2 on 1}[a4paper,border shrink=5mm] |
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|
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% Modified from: generic-ornate-15min-45min.de.tex |
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\mode<presentation> |
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{ |
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\usetheme{Warsaw} |
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\useoutertheme{infolines} |
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\setbeamercovered{transparent} |
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} |
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\usepackage[english]{babel} |
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\usepackage[latin1]{inputenc} |
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%\usepackage{times} |
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\usepackage[T1]{fontenc} |
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\usepackage{amsmath} |
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% Taken from Fernando's slides. |
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\usepackage{ae,aecompl} |
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\usepackage{mathpazo,courier,euler} |
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\usepackage[scaled=.95]{helvet} |
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\definecolor{darkgreen}{rgb}{0,0.5,0} |
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\usepackage{listings} |
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\lstset{language=Python, |
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basicstyle=\ttfamily\bfseries, |
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commentstyle=\color{red}\itshape, |
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stringstyle=\color{darkgreen}, |
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showstringspaces=false, |
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keywordstyle=\color{blue}\bfseries} |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% Macros |
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\setbeamercolor{emphbar}{bg=blue!20, fg=black} |
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\newcommand{\emphbar}[1] |
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{\begin{beamercolorbox}[rounded=true]{emphbar} |
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{#1} |
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\end{beamercolorbox} |
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} |
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\newcounter{time} |
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\setcounter{time}{0} |
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\newcommand{\inctime}[1]{\addtocounter{time}{#1}{\tiny \thetime\ m}} |
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\newcommand{\typ}[1]{\lstinline{#1}} |
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\newcommand{\kwrd}[1]{ \texttt{\textbf{\color{blue}{#1}}} } |
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%%% This is from Fernando's setup. |
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% \usepackage{color} |
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% \definecolor{orange}{cmyk}{0,0.4,0.8,0.2} |
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% % Use and configure listings package for nicely formatted code |
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% \usepackage{listings} |
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% \lstset{ |
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% language=Python, |
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% basicstyle=\small\ttfamily, |
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% commentstyle=\ttfamily\color{blue}, |
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% stringstyle=\ttfamily\color{orange}, |
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% showstringspaces=false, |
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% breaklines=true, |
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% postbreak = \space\dots |
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% } |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% Title page |
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\title[Matrices \& Curve Fitting]{Python for Science and Engg: Matrices \& Least Square Fit} |
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\author[FOSSEE] {FOSSEE} |
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\institute[IIT Bombay] {Department of Aerospace Engineering\\IIT Bombay} |
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\date[] {31 October, 2009\\Day 1, Session 4} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%\pgfdeclareimage[height=0.75cm]{iitmlogo}{iitmlogo} |
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%\logo{\pgfuseimage{iitmlogo}} |
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%% Delete this, if you do not want the table of contents to pop up at |
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%% the beginning of each subsection: |
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\AtBeginSubsection[] |
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{ |
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\begin{frame}<beamer> |
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\frametitle{Outline} |
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\tableofcontents[currentsection,currentsubsection] |
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\end{frame} |
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} |
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\AtBeginSection[] |
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{ |
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\begin{frame}<beamer> |
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\frametitle{Outline} |
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\tableofcontents[currentsection,currentsubsection] |
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\end{frame} |
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} |
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% If you wish to uncover everything in a step-wise fashion, uncomment |
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% the following command: |
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%\beamerdefaultoverlayspecification{<+->} |
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%\includeonlyframes{current,current1,current2,current3,current4,current5,current6} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% DOCUMENT STARTS |
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\begin{document} |
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\begin{frame} |
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\titlepage |
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\end{frame} |
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\begin{frame} |
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\frametitle{Outline} |
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\tableofcontents |
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% \pausesections |
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\end{frame} |
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\section{Matrices} |
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\begin{frame} |
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\frametitle{Matrices: Introduction} |
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\alert{All matrix operations are done using \kwrd{arrays}} |
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\end{frame} |
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\begin{frame}[fragile] |
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\frametitle{Matrices: Initializing} |
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\begin{lstlisting} |
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In []: A = array([[ 1, 1, 2, -1], |
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[ 2, 5, -1, -9], |
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[ 2, 1, -1, 3], |
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[ 1, -3, 2, 7]]) |
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In []: A |
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Out[]: |
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array([[ 1, 1, 2, -1], |
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[ 2, 5, -1, -9], |
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[ 2, 1, -1, 3], |
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[ 1, -3, 2, 7]]) |
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\end{lstlisting} |
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\end{frame} |
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\begin{frame}[fragile] |
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\frametitle{Accessing elements} |
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\begin{lstlisting} |
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In []: C = array([[1,1,2], |
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[2,4,1], |
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[-1,3,7]]) |
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In []: C[1][2] |
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Out[]: 1 |
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In []: C[1,2] |
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Out[]: 1 |
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|
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In []: C[1] |
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Out[]: array([2, 4, 1]) |
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\end{lstlisting} |
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\end{frame} |
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\begin{frame}[fragile] |
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\frametitle{Changing elements} |
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\begin{small} |
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\begin{lstlisting} |
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In []: C[1,1] = -2 |
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In []: C |
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Out[]: |
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array([[ 1, 1, 2], |
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[ 2, -2, 1], |
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[-1, 3, 7]]) |
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In []: C[1] = [0,0,0] |
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In []: C |
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Out[]: |
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array([[ 1, 1, 2], |
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[ 0, 0, 0], |
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[-1, 3, 7]]) |
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\end{lstlisting} |
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\end{small} |
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How to change one \alert{column}? |
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\end{frame} |
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\begin{frame}[fragile] |
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\frametitle{Slicing} |
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\begin{small} |
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\begin{lstlisting} |
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In []: C[:,1] |
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Out[]: array([1, 0, 3]) |
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In []: C[1,:] |
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Out[]: array([0, 0, 0]) |
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In []: C[0:2,:] |
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Out[]: |
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array([[1, 1, 2], |
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[0, 0, 0]]) |
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In []: C[1:3,:] |
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array([[ 0, 0, 0], |
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[-1, 3, 7]]) |
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\end{lstlisting} |
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\end{small} |
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\end{frame} |
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\begin{frame}[fragile] |
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\frametitle{Slicing \ldots} |
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\begin{small} |
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\begin{lstlisting} |
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In []: C[:2,:] |
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Out[]: |
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array([[1, 1, 2], |
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[0, 0, 0]]) |
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In []: C[1:,:] |
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Out[]: |
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array([[ 0, 0, 0], |
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[-1, 3, 7]]) |
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In []: C[1:,:2] |
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array([[ 0, 0], |
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[-1, 3]]) |
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\end{lstlisting} |
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|
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\end{small} |
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\end{frame} |
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|
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\begin{frame}[fragile] |
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\frametitle{Striding} |
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\begin{small} |
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\begin{lstlisting} |
279
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In []: C[::2,:] |
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Out[]: |
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array([[ 1, 1, 2], |
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[-1, 3, 7]]) |
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|
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In []: C[:,::2] |
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Out[]: |
281 | 247 |
xarray([[ 1, 2], |
279
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[ 0, 0], |
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[-1, 7]]) |
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|
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In []: C[::2,::2] |
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Out[]: |
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array([[ 1, 2], |
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[-1, 7]]) |
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\end{lstlisting} |
279
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\end{small} |
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\end{frame} |
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|
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\begin{frame}[fragile] |
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\frametitle{Slicing \& Striding Exercises} |
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\begin{small} |
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\begin{lstlisting} |
282
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In []: A = imread('lena.png') |
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|
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In []: imshow(A) |
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Out[]: <matplotlib.image.AxesImage object at 0xa0384cc> |
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|
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In []: A.shape |
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Out[]: (512, 512, 4) |
272
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\end{lstlisting} |
282
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\end{small} |
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\begin{itemize} |
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\item Crop the image to get the top-left quarter |
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\item Crop the image to get only the face |
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\item Resize image to half by dropping alternate pixels |
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\end{itemize} |
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\end{frame} |
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278 |
|
282
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279 |
\begin{frame}[fragile] |
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\frametitle{Solutions} |
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281 |
\begin{small} |
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\begin{lstlisting} |
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In []: imshow(A[:256,:256]) |
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Out[]: <matplotlib.image.AxesImage object at 0xb6f658c> |
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285 |
|
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286 |
In []: imshow(A[200:400,200:400]) |
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Out[]: <matplotlib.image.AxesImage object at 0xb757c2c> |
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288 |
|
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In []: imshow(A[::2,::2]) |
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Out[]: <matplotlib.image.AxesImage object at 0xb765c8c> |
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291 |
\end{lstlisting} |
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292 |
\end{small} |
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293 |
\end{frame} |
214
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|
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\begin{frame}[fragile] |
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\frametitle{Transpose of a Matrix} |
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297 |
\begin{lstlisting} |
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298 |
In []: A.T |
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Out[]: |
274
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array([[ 1, 2, 2, 1], |
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[ 1, 5, 1, -3], |
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[ 2, -1, -1, 2], |
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303 |
[-1, -9, 3, 7]]) |
214
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|
304 |
\end{lstlisting} |
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|
305 |
\end{frame} |
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|
306 |
|
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|
307 |
\begin{frame}[fragile] |
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|
308 |
\frametitle{Sum of all elements} |
19592f802dde
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|
309 |
\begin{lstlisting} |
274
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|
310 |
In []: sum(A) |
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311 |
Out[]: 12 |
214
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|
312 |
\end{lstlisting} |
19592f802dde
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|
313 |
\end{frame} |
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|
314 |
|
19592f802dde
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|
315 |
\begin{frame}[fragile] |
19592f802dde
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|
316 |
\frametitle{Matrix Addition} |
19592f802dde
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|
317 |
\begin{lstlisting} |
274
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|
318 |
In []: B = array([[3,2,-1,5], |
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|
319 |
[2,-2,4,9], |
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|
320 |
[-1,0.5,-1,-7], |
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|
321 |
[9,-5,7,3]]) |
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|
322 |
In []: A + B |
214
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|
323 |
Out[]: |
274
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|
324 |
array([[ 4. , 3. , 1. , 4. ], |
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|
325 |
[ 4. , 3. , 3. , 0. ], |
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|
326 |
[ 1. , 1.5, -2. , -4. ], |
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|
327 |
[ 10. , -8. , 9. , 10. ]]) |
214
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|
328 |
\end{lstlisting} |
19592f802dde
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|
329 |
\end{frame} |
19592f802dde
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|
330 |
|
19592f802dde
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|
331 |
\begin{frame}[fragile] |
274
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|
332 |
\frametitle{Elementwise Multiplication} |
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|
333 |
\begin{lstlisting} |
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|
334 |
In []: A*B |
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|
335 |
Out[]: |
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|
336 |
array([[ 3. , 2. , -2. , -5. ], |
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|
337 |
[ 4. , -10. , -4. , -81. ], |
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|
338 |
[ -2. , 0.5, 1. , -21. ], |
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|
339 |
[ 9. , 15. , 14. , 21. ]]) |
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340 |
|
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|
341 |
\end{lstlisting} |
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342 |
\end{frame} |
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343 |
|
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344 |
\begin{frame}[fragile] |
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\frametitle{Matrix Multiplication} |
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346 |
\begin{lstlisting} |
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347 |
In []: dot(A,B) |
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348 |
Out[]: |
274
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349 |
array([[ -6. , 6. , -6. , -3. ], |
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350 |
[-64. , 38.5, -44. , 35. ], |
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351 |
[ 36. , -13.5, 24. , 35. ], |
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352 |
[ 58. , -26. , 34. , -15. ]]) |
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353 |
\end{lstlisting} |
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354 |
\end{frame} |
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355 |
|
118
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|
356 |
\begin{frame}[fragile] |
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357 |
\frametitle{Inverse of a Matrix} |
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358 |
\begin{lstlisting} |
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359 |
In []: inv(A) |
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360 |
\end{lstlisting} |
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361 |
\begin{small} |
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362 |
\begin{lstlisting} |
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363 |
Out[]: |
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364 |
array([[-0.5 , 0.55, -0.15, 0.7 ], |
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365 |
[ 0.75, -0.5 , 0.5 , -0.75], |
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366 |
[ 0.5 , -0.15, -0.05, -0.1 ], |
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367 |
[ 0.25, -0.25, 0.25, -0.25]]) |
118
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368 |
\end{lstlisting} |
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369 |
\end{small} |
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370 |
\end{frame} |
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|
371 |
|
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|
372 |
\begin{frame}[fragile] |
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|
373 |
\frametitle{Determinant} |
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374 |
\begin{lstlisting} |
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375 |
In []: det(A) |
274
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376 |
Out[]: 80.0 |
118
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377 |
\end{lstlisting} |
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|
378 |
\end{frame} |
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379 |
|
263
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380 |
%%use S=array(X,Y) |
118
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381 |
\begin{frame}[fragile] |
274
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382 |
\frametitle{Eigenvalues and Eigen Vectors} |
131
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|
383 |
\begin{small} |
118
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|
384 |
\begin{lstlisting} |
274
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385 |
In []: E = array([[3,2,4],[2,0,2],[4,2,3]]) |
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|
386 |
|
274
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387 |
In []: eig(E) |
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|
388 |
Out[]: |
253
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389 |
(array([-1., 8., -1.]), |
274
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|
390 |
array([[-0.74535599, 0.66666667, -0.1931126 ], |
253
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|
391 |
[ 0.2981424 , 0.33333333, -0.78664085], |
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|
392 |
[ 0.59628479, 0.66666667, 0.58643303]])) |
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|
393 |
|
274
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394 |
In []: eigvals(E) |
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395 |
Out[]: array([-1., 8., -1.]) |
118
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|
396 |
\end{lstlisting} |
131
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|
397 |
\end{small} |
118
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|
398 |
\end{frame} |
127
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|
399 |
|
272
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400 |
%% \begin{frame}[fragile] |
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|
401 |
%% \frametitle{Computing Norms} |
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402 |
%% \begin{lstlisting} |
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|
403 |
%% In []: norm(E) |
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404 |
%% Out[]: 8.1240384046359608 |
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405 |
%% \end{lstlisting} |
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406 |
%% \end{frame} |
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|
407 |
|
272
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408 |
%% \begin{frame}[fragile] |
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409 |
%% \frametitle{Singular Value Decomposition} |
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|
410 |
%% \begin{small} |
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|
411 |
%% \begin{lstlisting} |
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|
412 |
%% In []: svd(E) |
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|
413 |
%% Out[]: |
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|
414 |
%% (array( |
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|
415 |
%% [[ -6.66666667e-01, -1.23702565e-16, 7.45355992e-01], |
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|
416 |
%% [ -3.33333333e-01, -8.94427191e-01, -2.98142397e-01], |
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|
417 |
%% [ -6.66666667e-01, 4.47213595e-01, -5.96284794e-01]]), |
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|
418 |
%% array([ 8., 1., 1.]), |
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|
419 |
%% array([[-0.66666667, -0.33333333, -0.66666667], |
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|
420 |
%% [-0. , 0.89442719, -0.4472136 ], |
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|
421 |
%% [-0.74535599, 0.2981424 , 0.59628479]])) |
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|
422 |
%% \end{lstlisting} |
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423 |
%% \end{small} |
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424 |
%% \inctime{15} |
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|
425 |
%% \end{frame} |
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|
426 |
|
268
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|
427 |
\section{Least Squares Fit} |
214
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|
428 |
\begin{frame}[fragile] |
272
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|
429 |
\frametitle{$L$ vs. $T^2$} |
268
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|
430 |
\vspace{-0.15in} |
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|
431 |
\begin{figure} |
282
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|
432 |
\includegraphics[width=4in]{data/L-Tsq-points} |
268
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|
433 |
\end{figure} |
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434 |
\end{frame} |
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435 |
|
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|
436 |
\begin{frame}[fragile] |
272
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|
437 |
\frametitle{$L$ vs. $T^2$} |
268
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|
438 |
\vspace{-0.15in} |
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|
439 |
\begin{figure} |
282
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440 |
\includegraphics[width=4in]{data/L-Tsq-Line} |
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441 |
\end{figure} |
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442 |
\end{frame} |
127
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|
443 |
|
232
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444 |
\begin{frame}[fragile] |
268
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|
445 |
\frametitle{Least Squares Fit} |
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|
446 |
\vspace{-0.15in} |
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|
447 |
\begin{figure} |
282
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|
448 |
\includegraphics[width=4in]{data/least-sq-fit} |
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\end{figure} |
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\end{frame} |
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451 |
|
268
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452 |
\begin{frame} |
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453 |
\frametitle{Least Square Fit Curve} |
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454 |
\begin{itemize} |
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455 |
\item $T^2$ and $L$ have a linear relationship |
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456 |
\item Hence, Least Square Fit Curve is a line |
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457 |
\item we shall use the \typ{lstsq} function |
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458 |
\end{itemize} |
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459 |
\end{frame} |
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460 |
|
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461 |
\begin{frame}[fragile] |
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462 |
\frametitle{\typ{lstsq}} |
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\begin{itemize} |
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464 |
\item We need to fit a line through points for the equation $T^2 = m \cdot L+c$ |
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465 |
\item In matrix form, the equation can be represented as $T^2 = A \cdot p$, where A is |
268
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466 |
$\begin{bmatrix} |
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467 |
L_1 & 1 \\ |
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468 |
L_2 & 1 \\ |
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469 |
\vdots & \vdots\\ |
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470 |
L_N & 1 \\ |
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471 |
\end{bmatrix}$ |
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472 |
and p is |
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473 |
$\begin{bmatrix} |
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474 |
m\\ |
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475 |
c\\ |
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476 |
\end{bmatrix}$ |
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477 |
\item We need to find $p$ to plot the line |
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478 |
\end{itemize} |
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479 |
\end{frame} |
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480 |
|
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481 |
\begin{frame}[fragile] |
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482 |
\frametitle{Generating $A$} |
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483 |
\begin{lstlisting} |
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484 |
In []: A = array([L, ones_like(L)]) |
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485 |
In []: A = A.T |
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486 |
\end{lstlisting} |
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487 |
\begin{small} |
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488 |
\begin{block}{} |
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489 |
\begin{lstlisting} |
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490 |
In []: ones((3,5)) |
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491 |
Out[]: |
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492 |
array([[ 1., 1., 1., 1., 1.], |
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[ 1., 1., 1., 1., 1.], |
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494 |
[ 1., 1., 1., 1., 1.]]) |
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495 |
|
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496 |
In []: ones_like([1, 2, 3, 4, 5]) |
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497 |
Out[]: array([1, 1, 1, 1, 1]) |
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|
498 |
\end{lstlisting} |
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499 |
Also available \alert{\typ{zeros, zeros_like, empty, empty_like}} |
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500 |
\end{block} |
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501 |
\end{small} |
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|
502 |
|
268
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503 |
%% \begin{itemize} |
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|
504 |
%% \item A is also called a Van der Monde matrix |
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|
505 |
%% \item It can also be generated using \typ{vander} |
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|
506 |
%% \end{itemize} |
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|
507 |
%% \begin{lstlisting} |
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|
508 |
%% In []: A = vander(L, 2) |
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|
509 |
%% \end{lstlisting} |
274
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|
510 |
\end{frame} |
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|
511 |
|
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|
512 |
\begin{frame}[fragile] |
268
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|
513 |
\frametitle{\typ{lstsq} \ldots} |
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|
514 |
\begin{itemize} |
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changeset
|
515 |
\item Now use the \typ{lstsq} function |
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|
516 |
\item Along with a lot of things, it returns the least squares solution |
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|
517 |
\end{itemize} |
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|
518 |
\begin{lstlisting} |
272
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|
519 |
In []: result = lstsq(A,TSq) |
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|
520 |
In []: coef = result[0] |
268
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|
521 |
\end{lstlisting} |
253
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|
522 |
\end{frame} |
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|
523 |
|
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|
524 |
\begin{frame}[fragile] |
268
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|
525 |
\frametitle{Least Square Fit Line \ldots} |
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|
526 |
We get the points of the line from \typ{coef} |
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|
527 |
\begin{lstlisting} |
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changeset
|
528 |
In []: Tline = coef[0]*L + coef[1] |
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|
529 |
\end{lstlisting} |
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|
530 |
\begin{itemize} |
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|
531 |
\item Now plot Tline vs. L, to get the Least squares fit line. |
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|
532 |
\end{itemize} |
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|
533 |
\begin{lstlisting} |
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|
534 |
In []: plot(L, Tline) |
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|
535 |
\end{lstlisting} |
127
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Updated Integration section in Session 4.
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|
536 |
\end{frame} |
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|
537 |
|
285
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|
538 |
\begin{frame}[fragile] |
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|
539 |
\frametitle{Least Squares Fit} |
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|
540 |
\vspace{-0.15in} |
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|
541 |
\begin{figure} |
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542 |
\includegraphics[width=4in]{data/least-sq-fit} |
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|
543 |
\end{figure} |
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|
544 |
\end{frame} |
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|
545 |
|
213
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|
546 |
\section{Summary} |
185
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547 |
\begin{frame} |
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548 |
\frametitle{What did we learn?} |
185
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|
549 |
\begin{itemize} |
213
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|
550 |
\item Matrices |
214
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|
551 |
\begin{itemize} |
263
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|
552 |
\item Accessing elements |
285
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553 |
\item Slicing and Striding |
214
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|
554 |
\item Transpose |
19592f802dde
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|
555 |
\item Addition |
19592f802dde
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changeset
|
556 |
\item Multiplication |
19592f802dde
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|
557 |
\item Inverse of a matrix |
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|
558 |
\item Determinant |
274
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\item Eigenvalues and Eigen vector |
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%% \item Norms |
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%% \item Singular Value Decomposition |
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\end{itemize} |
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\item Least Square Curve fitting |
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\end{itemize} |
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\end{frame} |
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|
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\end{document} |