1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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2 %Tutorial slides on Python. |
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3 % |
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4 % Author: FOSSEE |
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5 % Copyright (c) 2009, FOSSEE, IIT Bombay |
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6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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7 |
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8 \documentclass[14pt,compress]{beamer} |
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9 %\documentclass[draft]{beamer} |
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11 %\usepackage{pgfpages} |
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13 |
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14 % Modified from: generic-ornate-15min-45min.de.tex |
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15 \mode<presentation> |
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16 { |
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23 \usepackage[latin1]{inputenc} |
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24 %\usepackage{times} |
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25 \usepackage[T1]{fontenc} |
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26 |
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27 % Taken from Fernando's slides. |
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28 \usepackage{ae,aecompl} |
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29 \usepackage{mathpazo,courier,euler} |
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30 \usepackage[scaled=.95]{helvet} |
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31 \usepackage{amsmath} |
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32 |
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33 \definecolor{darkgreen}{rgb}{0,0.5,0} |
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34 |
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35 \usepackage{listings} |
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36 \lstset{language=Python, |
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37 basicstyle=\ttfamily\bfseries, |
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38 commentstyle=\color{red}\itshape, |
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40 showstringspaces=false, |
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41 keywordstyle=\color{blue}\bfseries} |
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42 |
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43 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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44 % Macros |
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45 \setbeamercolor{emphbar}{bg=blue!20, fg=black} |
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48 {#1} |
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50 } |
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53 \newcommand{\inctime}[1]{\addtocounter{time}{#1}{\tiny \thetime\ m}} |
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54 |
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55 \newcommand{\typ}[1]{\lstinline{#1}} |
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56 |
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57 \newcommand{\kwrd}[1]{ \texttt{\textbf{\color{blue}{#1}}} } |
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58 |
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59 %%% This is from Fernando's setup. |
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60 % \usepackage{color} |
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61 % \definecolor{orange}{cmyk}{0,0.4,0.8,0.2} |
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62 % % Use and configure listings package for nicely formatted code |
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63 % \usepackage{listings} |
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64 % \lstset{ |
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65 % language=Python, |
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66 % basicstyle=\small\ttfamily, |
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67 % commentstyle=\ttfamily\color{blue}, |
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68 % stringstyle=\ttfamily\color{orange}, |
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69 % showstringspaces=false, |
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70 % breaklines=true, |
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71 % postbreak = \space\dots |
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72 % } |
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73 |
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74 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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75 % Title page |
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76 \title[Solving Equations \& ODEs]{Python for Science and Engg:\\Solving Equations \& ODEs} |
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77 |
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78 \author[FOSSEE] {FOSSEE} |
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79 |
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80 \institute[IIT Bombay] {Department of Aerospace Engineering\\IIT Bombay} |
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81 \date[] {SciPy 2010, Introductory tutorials\\Day 1, Session 6} |
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82 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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83 |
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84 %\pgfdeclareimage[height=0.75cm]{iitmlogo}{iitmlogo} |
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85 %\logo{\pgfuseimage{iitmlogo}} |
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86 |
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87 |
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88 %% Delete this, if you do not want the table of contents to pop up at |
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89 %% the beginning of each subsection: |
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90 \AtBeginSubsection[] |
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91 { |
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92 \begin{frame}<beamer> |
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93 \frametitle{Outline} |
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94 \tableofcontents[currentsection,currentsubsection] |
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95 \end{frame} |
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96 } |
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97 |
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98 \AtBeginSection[] |
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99 { |
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100 \begin{frame}<beamer> |
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101 \frametitle{Outline} |
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102 \tableofcontents[currentsection,currentsubsection] |
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103 \end{frame} |
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104 } |
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105 |
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106 % If you wish to uncover everything in a step-wise fashion, uncomment |
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107 % the following command: |
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108 %\beamerdefaultoverlayspecification{<+->} |
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109 |
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110 %\includeonlyframes{current,current1,current2,current3,current4,current5,current6} |
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111 |
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112 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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113 % DOCUMENT STARTS |
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114 \begin{document} |
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115 |
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116 \begin{frame} |
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117 \maketitle |
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118 \end{frame} |
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119 |
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120 %% \begin{frame} |
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121 %% \frametitle{Outline} |
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122 %% \tableofcontents |
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123 %% % You might wish to add the option [pausesections] |
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124 %% \end{frame} |
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125 |
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126 \section{Solving linear equations} |
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127 |
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128 \begin{frame}[fragile] |
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129 \frametitle{Solution of equations} |
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130 Consider, |
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131 \begin{align*} |
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132 3x + 2y - z & = 1 \\ |
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133 2x - 2y + 4z & = -2 \\ |
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134 -x + \frac{1}{2}y -z & = 0 |
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135 \end{align*} |
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136 Solution: |
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137 \begin{align*} |
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138 x & = 1 \\ |
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139 y & = -2 \\ |
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140 z & = -2 |
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141 \end{align*} |
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142 \end{frame} |
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143 |
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144 \begin{frame}[fragile] |
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145 \frametitle{Solving using Matrices} |
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146 Let us now look at how to solve this using \kwrd{matrices} |
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147 \begin{lstlisting} |
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148 In []: A = array([[3,2,-1], |
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149 [2,-2,4], |
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150 [-1, 0.5, -1]]) |
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151 In []: b = array([1, -2, 0]) |
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152 In []: x = solve(A, b) |
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153 \end{lstlisting} |
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154 \end{frame} |
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155 |
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156 \begin{frame}[fragile] |
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157 \frametitle{Solution:} |
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158 \begin{lstlisting} |
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159 In []: x |
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160 Out[]: array([ 1., -2., -2.]) |
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161 \end{lstlisting} |
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162 \end{frame} |
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163 |
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164 \begin{frame}[fragile] |
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165 \frametitle{Let's check!} |
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166 \begin{small} |
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167 \begin{lstlisting} |
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168 In []: Ax = dot(A, x) |
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169 In []: Ax |
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170 Out[]: array([ 1.00000000e+00, -2.00000000e+00, -1.11022302e-16]) |
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171 \end{lstlisting} |
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172 \end{small} |
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173 \begin{block}{} |
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174 The last term in the matrix is actually \alert{0}!\\ |
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175 We can use \kwrd{allclose()} to check. |
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176 \end{block} |
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177 \begin{lstlisting} |
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178 In []: allclose(Ax, b) |
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179 Out[]: True |
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180 \end{lstlisting} |
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181 \inctime{10} |
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182 \end{frame} |
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183 |
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184 \begin{frame}[fragile] |
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185 \frametitle{Problem} |
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186 Solve the set of equations: |
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187 \begin{align*} |
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188 x + y + 2z -w & = 3\\ |
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189 2x + 5y - z - 9w & = -3\\ |
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190 2x + y -z + 3w & = -11 \\ |
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191 x - 3y + 2z + 7w & = -5\\ |
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192 \end{align*} |
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193 \inctime{5} |
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194 \end{frame} |
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195 |
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196 \begin{frame}[fragile] |
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197 \frametitle{Solution} |
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198 Use \kwrd{solve()} |
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199 \begin{align*} |
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200 x & = -5\\ |
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201 y & = 2\\ |
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202 z & = 3\\ |
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203 w & = 0\\ |
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204 \end{align*} |
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205 \end{frame} |
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206 |
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207 \section{Finding Roots} |
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208 |
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209 \begin{frame}[fragile] |
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210 \frametitle{SciPy: \typ{roots}} |
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211 \begin{itemize} |
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212 \item Calculates the roots of polynomials |
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213 \item To calculate the roots of $x^2-5x+6$ |
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214 \end{itemize} |
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215 \begin{lstlisting} |
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216 In []: coeffs = [1, -5, 6] |
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217 In []: roots(coeffs) |
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218 Out[]: array([3., 2.]) |
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219 \end{lstlisting} |
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220 \vspace*{-.2in} |
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221 \begin{center} |
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222 \includegraphics[height=1.6in, interpolate=true]{data/roots} |
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223 \end{center} |
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224 \end{frame} |
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225 |
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226 \begin{frame}[fragile] |
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227 \frametitle{SciPy: \typ{fsolve}} |
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228 \begin{small} |
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229 \begin{lstlisting} |
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230 In []: from scipy.optimize import fsolve |
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231 \end{lstlisting} |
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232 \end{small} |
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233 \begin{itemize} |
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234 \item Finds the roots of a system of non-linear equations |
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235 \item Input arguments - Function and initial estimate |
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236 \item Returns the solution |
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237 \end{itemize} |
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238 \end{frame} |
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239 |
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240 \begin{frame}[fragile] |
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241 \frametitle{\typ{fsolve}} |
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242 Find the root of $sin(z)+cos^2(z)$ nearest to $0$ |
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243 \vspace{-0.1in} |
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244 \begin{center} |
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245 \includegraphics[height=2.8in, interpolate=true]{data/fsolve} |
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246 \end{center} |
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247 \end{frame} |
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248 |
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249 \begin{frame}[fragile] |
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250 \frametitle{\typ{fsolve}} |
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251 Root of $sin(z)+cos^2(z)$ nearest to $0$ |
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252 \begin{lstlisting} |
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253 In []: fsolve(sin(z)+cos(z)*cos(z), 0) |
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254 NameError: name 'z' is not defined |
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255 \end{lstlisting} |
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256 \end{frame} |
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257 |
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258 \begin{frame}[fragile] |
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259 \frametitle{\typ{fsolve}} |
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260 \begin{lstlisting} |
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261 In []: z = linspace(-pi, pi) |
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262 In []: fsolve(sin(z)+cos(z)*cos(z), 0) |
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263 \end{lstlisting} |
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264 \begin{small} |
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265 \alert{\typ{TypeError:}} |
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266 \typ{'numpy.ndarray' object is not callable} |
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267 \end{small} |
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268 \end{frame} |
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269 |
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270 \begin{frame}[fragile] |
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271 \frametitle{Functions - Definition} |
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272 We have been using them all along. Now let's see how to define them. |
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273 \begin{lstlisting} |
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274 In []: def g(z): |
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275 ....: return sin(z)+cos(z)*cos(z) |
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276 \end{lstlisting} |
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277 \begin{itemize} |
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278 \item \typ{def} -- keyword |
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279 \item name: \typ{g} |
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280 \item arguments: \typ{z} |
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281 \item \typ{return} -- keyword |
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282 \end{itemize} |
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283 \end{frame} |
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284 |
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285 \begin{frame}[fragile] |
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286 \frametitle{Functions - Calling them} |
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287 \begin{lstlisting} |
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288 In []: g() |
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289 --------------------------------------- |
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290 \end{lstlisting} |
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291 \alert{\typ{TypeError:}}\typ{g() takes exactly 1 argument} |
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292 \typ{(0 given)} |
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293 \begin{lstlisting} |
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294 In []: g(0) |
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295 Out[]: 1.0 |
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296 In []: g(1) |
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297 Out[]: 1.1333975665343254 |
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298 \end{lstlisting} |
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299 More on Functions later \ldots |
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300 \end{frame} |
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301 |
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302 \begin{frame}[fragile] |
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303 \frametitle{\typ{fsolve} \ldots} |
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304 Find the root of $sin(z)+cos^2(z)$ nearest to $0$ |
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305 \begin{lstlisting} |
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306 In []: fsolve(g, 0) |
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307 Out[]: -0.66623943249251527 |
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308 \end{lstlisting} |
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309 \begin{center} |
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310 \includegraphics[height=2in, interpolate=true]{data/fsolve} |
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311 \end{center} |
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312 \end{frame} |
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313 |
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314 \begin{frame}[fragile] |
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315 \frametitle{Exercise Problem} |
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316 Find the root of the equation $x^2 - sin(x) + cos^2(x) = tan(x)$ nearest to $0$ |
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317 \end{frame} |
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318 |
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319 \begin{frame}[fragile] |
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320 \frametitle{Solution} |
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321 \begin{small} |
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322 \begin{lstlisting} |
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323 def g(x): |
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324 return x**2 - sin(x) + cos(x)*cos(x) - tan(x) |
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325 fsolve(g, 0) |
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326 \end{lstlisting} |
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327 \end{small} |
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328 \begin{center} |
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329 \includegraphics[height=2in, interpolate=true]{data/fsolve_tanx} |
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330 \end{center} |
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331 \end{frame} |
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332 |
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333 %% \begin{frame}[fragile] |
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334 %% \frametitle{Scipy Methods \dots} |
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335 %% \begin{small} |
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336 %% \begin{lstlisting} |
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337 %% In []: from scipy.optimize import fixed_point |
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338 |
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339 %% In []: from scipy.optimize import bisect |
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340 |
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341 %% In []: from scipy.optimize import newton |
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342 %% \end{lstlisting} |
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343 %% \end{small} |
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344 %% \end{frame} |
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345 |
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346 \section{ODEs} |
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347 |
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348 \begin{frame}[fragile] |
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349 \frametitle{Solving ODEs using SciPy} |
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350 \begin{itemize} |
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351 \item Consider the spread of an epidemic in a population |
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352 \item $\frac{dy}{dt} = ky(L-y)$ gives the spread of the disease |
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353 \item $L$ is the total population. |
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354 \item Use $L = 2.5E5, k = 3E-5, y(0) = 250$ |
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355 \item Define a function as below |
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356 \end{itemize} |
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357 \begin{lstlisting} |
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358 In []: from scipy.integrate import odeint |
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359 In []: def epid(y, t): |
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360 .... k = 3.0e-5 |
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361 .... L = 2.5e5 |
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362 .... return k*y*(L-y) |
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363 .... |
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364 \end{lstlisting} |
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365 \end{frame} |
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366 |
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367 \begin{frame}[fragile] |
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368 \frametitle{Solving ODEs using SciPy \ldots} |
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369 \begin{lstlisting} |
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370 In []: t = linspace(0, 12, 61) |
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371 |
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372 In []: y = odeint(epid, 250, t) |
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373 |
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374 In []: plot(t, y) |
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375 \end{lstlisting} |
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376 %Insert Plot |
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377 \end{frame} |
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378 |
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379 \begin{frame}[fragile] |
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380 \frametitle{Result} |
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381 \begin{center} |
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382 \includegraphics[height=2in, interpolate=true]{data/image} |
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383 \end{center} |
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384 \end{frame} |
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385 |
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386 |
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387 \begin{frame}[fragile] |
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388 \frametitle{ODEs - Simple Pendulum} |
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389 We shall use the simple ODE of a simple pendulum. |
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390 \begin{equation*} |
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391 \ddot{\theta} = -\frac{g}{L}sin(\theta) |
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392 \end{equation*} |
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393 \begin{itemize} |
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394 \item This equation can be written as a system of two first order ODEs |
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395 \end{itemize} |
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396 \begin{align} |
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397 \dot{\theta} &= \omega \\ |
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398 \dot{\omega} &= -\frac{g}{L}sin(\theta) \\ |
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399 \text{At}\ t &= 0 : \nonumber \\ |
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400 \theta = \theta_0(10^o)\quad & \&\quad \omega = 0\ (Initial\ values)\nonumber |
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401 \end{align} |
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402 \end{frame} |
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403 |
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404 \begin{frame}[fragile] |
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405 \frametitle{ODEs - Simple Pendulum \ldots} |
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406 \begin{itemize} |
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407 \item Use \typ{odeint} to do the integration |
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408 \end{itemize} |
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409 \begin{lstlisting} |
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410 In []: def pend_int(initial, t): |
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411 .... theta = initial[0] |
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412 .... omega = initial[1] |
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413 .... g = 9.81 |
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414 .... L = 0.2 |
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415 .... F=[omega, -(g/L)*sin(theta)] |
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416 .... return F |
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417 .... |
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418 \end{lstlisting} |
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419 \end{frame} |
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420 |
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421 \begin{frame}[fragile] |
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422 \frametitle{ODEs - Simple Pendulum \ldots} |
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423 \begin{itemize} |
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424 \item \typ{t} is the time variable \\ |
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425 \item \typ{initial} has the initial values |
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426 \end{itemize} |
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427 \begin{lstlisting} |
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428 In []: t = linspace(0, 20, 101) |
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429 In []: initial = [10*2*pi/360, 0] |
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430 \end{lstlisting} |
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431 \end{frame} |
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432 |
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433 \begin{frame}[fragile] |
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434 \frametitle{ODEs - Simple Pendulum \ldots} |
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435 %%\begin{small} |
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436 \typ{In []: from scipy.integrate import odeint} |
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437 %%\end{small} |
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438 \begin{lstlisting} |
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439 In []: pend_sol = odeint(pend_int, |
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440 initial,t) |
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441 \end{lstlisting} |
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442 \end{frame} |
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443 |
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444 \begin{frame}[fragile] |
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445 \frametitle{Result} |
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446 \begin{center} |
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447 \includegraphics[height=2in, interpolate=true]{data/ode} |
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448 \end{center} |
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449 \end{frame} |
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450 |
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451 \section{FFTs} |
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452 |
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453 \begin{frame}[fragile] |
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454 \frametitle{The FFT} |
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455 \begin{itemize} |
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456 \item We have a simple signal $y(t)$ |
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457 \item Find the FFT and plot it |
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458 \end{itemize} |
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459 \begin{lstlisting} |
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460 In []: t = linspace(0, 2*pi, 500) |
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461 In []: y = sin(4*pi*t) |
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462 |
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463 In []: f = fft(y) |
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464 In []: freq = fftfreq(500, t[1] - t[0]) |
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465 |
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466 In []: plot(freq[:250], abs(f)[:250]) |
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467 In []: grid() |
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468 \end{lstlisting} |
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469 \end{frame} |
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470 |
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471 \begin{frame}[fragile] |
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472 \frametitle{FFTs cont\dots} |
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473 \begin{lstlisting} |
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474 In []: y1 = ifft(f) # inverse FFT |
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475 In []: allclose(y, y1) |
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476 Out[]: True |
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477 \end{lstlisting} |
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478 \end{frame} |
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479 |
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480 \begin{frame}[fragile] |
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481 \frametitle{FFTs cont\dots} |
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482 Let us add some noise to the signal |
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483 \begin{lstlisting} |
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484 In []: yr = y + random(size=500)*0.2 |
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485 In []: yn = y + normal(size=500)*0.2 |
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486 |
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487 In []: plot(t, yr) |
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488 In []: figure() |
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489 In []: plot(freq[:250], |
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490 ...: abs(fft(yn))[:250]) |
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491 \end{lstlisting} |
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492 \begin{itemize} |
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493 \item \typ{random}: produces uniform deviates in $[0, 1)$ |
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494 \item \typ{normal}: draws random samples from a Gaussian |
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495 distribution |
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496 \item Useful to create a random matrix of any shape |
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497 \end{itemize} |
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498 \end{frame} |
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499 |
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500 \begin{frame}[fragile] |
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501 \frametitle{FFTs cont\dots} |
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502 Filter the noisy signal: |
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503 \begin{lstlisting} |
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504 In []: from scipy import signal |
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505 In []: yc = signal.wiener(yn, 5) |
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506 In []: clf() |
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507 In []: plot(t, yc) |
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508 In []: figure() |
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509 In []: plot(freq[:250], |
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510 ...: abs(fft(yc))[:250]) |
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511 \end{lstlisting} |
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512 Only scratched the surface here \dots |
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513 \end{frame} |
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514 |
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515 |
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516 \begin{frame} |
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517 \frametitle{Things we have learned} |
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518 \begin{itemize} |
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519 \item Solving Linear Equations |
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520 \item Defining Functions |
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521 \item Finding Roots |
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522 \item Solving ODEs |
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523 \item Random number generation |
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524 \item FFTs and basic signal processing |
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525 \end{itemize} |
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526 \end{frame} |
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527 |
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528 \end{document} |
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529 |
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530 %% Questions for Quiz %% |
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531 %% ------------------ %% |
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532 |
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533 \begin{frame} |
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534 \frametitle{\incqno } |
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535 Given a 4x4 matrix \texttt{A} and a 4-vector \texttt{b}, what command do |
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536 you use to solve for the equation \\ |
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537 \texttt{Ax = b}? |
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538 \end{frame} |
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539 |
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540 \begin{frame} |
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541 \frametitle{\incqno } |
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542 What command will you use if you wish to integrate a system of ODEs? |
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543 \end{frame} |
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544 |
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545 \begin{frame} |
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546 \frametitle{\incqno } |
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547 How do you calculate the roots of the polynomial, $y = 1 + 6x + 8x^2 + |
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548 x^3$? |
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549 \end{frame} |
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550 |
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551 \begin{frame} |
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552 \frametitle{\incqno } |
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553 Two arrays \lstinline+a+ and \lstinline+b+ are numerically almost equal, what command |
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554 do you use to check if this is true? |
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555 \end{frame} |
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556 |
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557 %% \begin{frame}[fragile] |
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558 %% \frametitle{\incqno } |
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559 %% \begin{lstlisting} |
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560 %% In []: x = arange(0, 1, 0.25) |
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561 %% In []: print x |
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562 %% \end{lstlisting} |
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563 %% What will be printed? |
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564 %% \end{frame} |
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565 |
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566 |
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567 %% \begin{frame}[fragile] |
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568 %% \frametitle{\incqno } |
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569 %% \begin{lstlisting} |
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570 %% from scipy.integrate import quad |
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571 %% def f(x): |
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572 %% res = x*cos(x) |
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573 %% quad(f, 0, 1) |
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574 %% \end{lstlisting} |
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575 %% What changes will you make to the above code to make it work? |
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576 %% \end{frame} |
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577 |
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578 %% \begin{frame} |
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579 %% \frametitle{\incqno } |
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580 %% What two commands will you use to create and evaluate a spline given |
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581 %% some data? |
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582 %% \end{frame} |
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583 |
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584 %% \begin{frame}[fragile] |
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585 %% \frametitle{\incqno } |
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586 %% What would be the result? |
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587 %% \begin{lstlisting} |
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588 %% In []: x |
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589 %% array([[0, 1, 2], |
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590 %% [3, 4, 5], |
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591 %% [6, 7, 8]]) |
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592 %% In []: x[::-1,:] |
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593 %% \end{lstlisting} |
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594 %% Hint: |
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595 %% \begin{lstlisting} |
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596 %% In []: x = arange(9) |
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597 %% In []: x[::-1] |
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598 %% array([8, 7, 6, 5, 4, 3, 2, 1, 0]) |
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599 %% \end{lstlisting} |
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600 %% \end{frame} |
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601 |
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602 %% \begin{frame}[fragile] |
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603 %% \frametitle{\incqno } |
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604 %% What would be the result? |
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605 %% \begin{lstlisting} |
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606 %% In []: y = arange(3) |
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607 %% In []: x = linspace(0,3,3) |
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608 %% In []: x-y |
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609 %% \end{lstlisting} |
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610 %% \end{frame} |
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611 |
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612 %% \begin{frame}[fragile] |
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613 %% \frametitle{\incqno } |
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614 %% \begin{lstlisting} |
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615 %% In []: x |
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616 %% array([[ 0, 1, 2, 3], |
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617 %% [ 4, 5, 6, 7], |
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618 %% [ 8, 9, 10, 11], |
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619 %% [12, 13, 14, 15]]) |
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620 %% \end{lstlisting} |
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621 %% How will you get the following \lstinline+x+? |
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622 %% \begin{lstlisting} |
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623 %% array([[ 5, 7], |
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624 %% [ 9, 11]]) |
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625 %% \end{lstlisting} |
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626 %% \end{frame} |
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627 |
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628 %% \begin{frame}[fragile] |
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629 %% \frametitle{\incqno } |
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630 %% What would be the output? |
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631 %% \begin{lstlisting} |
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632 %% In []: y = arange(4) |
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633 %% In []: x = array(([1,2,3,2],[1,3,6,0])) |
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634 %% In []: x + y |
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635 %% \end{lstlisting} |
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636 %% \end{frame} |
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637 |
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638 %% \begin{frame}[fragile] |
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639 %% \frametitle{\incqno } |
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640 %% \begin{lstlisting} |
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641 %% In []: line = plot(x, sin(x)) |
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642 %% \end{lstlisting} |
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643 %% Use the \lstinline+set_linewidth+ method to set width of \lstinline+line+ to 2. |
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644 %% \end{frame} |
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645 |
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646 %% \begin{frame}[fragile] |
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647 %% \frametitle{\incqno } |
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648 %% What would be the output? |
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649 %% \begin{lstlisting} |
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650 %% In []: x = arange(9) |
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651 %% In []: y = arange(9.) |
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652 %% In []: x == y |
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653 %% \end{lstlisting} |
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654 %% \end{frame} |
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655 |
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