26 |
26 |
27 % Taken from Fernando's slides. |
27 % Taken from Fernando's slides. |
28 \usepackage{ae,aecompl} |
28 \usepackage{ae,aecompl} |
29 \usepackage{mathpazo,courier,euler} |
29 \usepackage{mathpazo,courier,euler} |
30 \usepackage[scaled=.95]{helvet} |
30 \usepackage[scaled=.95]{helvet} |
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31 \usepackage{amsmath} |
31 |
32 |
32 \definecolor{darkgreen}{rgb}{0,0.5,0} |
33 \definecolor{darkgreen}{rgb}{0,0.5,0} |
33 |
34 |
34 \usepackage{listings} |
35 \usepackage{listings} |
35 \lstset{language=Python, |
36 \lstset{language=Python, |
145 \item Arrays |
146 \item Arrays |
146 \end{itemize} |
147 \end{itemize} |
147 \end{itemize} |
148 \end{itemize} |
148 \end{frame} |
149 \end{frame} |
149 |
150 |
150 \begin{frame} |
151 \begin{frame}[fragile] |
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152 \frametitle{Reading pendulum.txt} |
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153 \begin{itemize} |
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154 \item The file has two columns |
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155 \item Column1 - L; Column2 - T |
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156 \end{itemize} |
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157 \begin{lstlisting} |
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158 In []: L = [] |
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159 In []: T = [] |
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160 In []: for line in open('pendulum.txt'): |
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161 .... len, t = line.split() |
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162 .... L.append(float(len)) |
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163 .... T.append(float(t)) |
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164 \end{lstlisting} |
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165 We now have two lists L and T |
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166 \end{frame} |
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167 |
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168 \begin{frame}[fragile] |
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169 \frametitle{Calculating T^2} |
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170 \begin{itemize} |
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171 \item Each element of the list T must be squared |
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172 \item Iterating over each element of the list works |
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173 \item But very slow \ldots |
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174 \item Instead, we use arrays |
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175 \end{itemize} |
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176 \begin{lstlisting} |
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177 In []: array(L) |
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178 In []: T = array(T) |
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179 In []: Tsq = T*T |
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180 In []: plot(L, Tsq, 'o') |
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181 \end{lstlisting} |
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182 \end{frame} |
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183 |
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184 \begin{frame}[fragile] |
151 \frametitle{Arrays} |
185 \frametitle{Arrays} |
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186 \begin{itemize} |
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187 \item T is now a \typ{numpy array} |
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188 \item \typ{numpy} arrays are very efficient and powerful |
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189 \item Very easy to perform element-wise operations |
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190 \item \typ{+, -, *, /, \%} |
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191 \item More about arrays later |
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192 \end{itemize} |
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193 \end{frame} |
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194 |
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195 \begin{frame}[fragile] |
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196 \frametitle{Least Square Polynomial} |
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197 \begin{enumerate} |
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198 \item $T^2 = \frac{4\pi^2}{g}L$ |
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199 \item $T^2$ and $L$ have a linear relationship |
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200 \item We find an approximate solution to $Ax = y$, where A is the Van der Monde matrix to get coefficients of the least squares fit line. |
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201 \end{enumerate} |
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202 \end{frame} |
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203 |
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204 \begin{frame}[fragile] |
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205 \frametitle{Van der Monde Matrix} |
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206 Van der Monde matrix of order M |
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207 \begin{equation*} |
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208 \begin{bmatrix} |
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209 l_1^{M-1} & \ldots & l_1 & 1 \\ |
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210 l_2^{M-1} & \ldots &l_2 & 1 \\ |
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211 \vdots & \ldots & \vdots & \vdots\\ |
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212 l_N^{M-1} & \ldots & l_N & 1 \\ |
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213 \end{bmatrix} |
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214 \end{equation*} |
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215 \begin{lstlisting} |
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216 In []: A=vander(L,2) |
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217 \end{lstlisting} |
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218 \end{frame} |
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219 |
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220 \begin{frame}[fragile] |
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221 \frametitle{Least Square Fit Line} |
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222 \begin{itemize} |
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223 \item We use the \typ{lstsq} function of pylab |
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224 \item It returns the |
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225 \begin{enumerate} |
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226 \item Least squares solution |
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227 \item Sum of residues |
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228 \item Rank of matrix A |
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229 \item Singular values of A |
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230 \end{enumerate} |
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231 \end{itemize} |
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232 \begin{lstlisting} |
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233 coeffs, res, rank, sing = lstsq(A,Tsq) |
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234 \end{lstlisting} |
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235 \end{frame} |
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236 |
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237 \begin{frame}[fragile] |
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238 \frametitle{Least Square Fit Line \ldots} |
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239 \begin{itemize} |
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240 \item Use the poly1d function of pylab, to create a function for the line equation using the coefficients obtained |
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241 \begin{lstlisting} |
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242 p=poly1d(coeffs) |
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243 \end{lstlisting} |
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244 \item Get new T^2 values using the function \typ{p} obtained |
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245 \begin{lstlisting} |
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246 Tline = p(L) |
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247 \end{lstlisting} |
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248 \item Now plot Tline vs. L, to get the Least squares fit line. |
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249 \begin{lstlisting} |
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250 plot(L, Tline) |
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251 \end{lstlisting} |
152 \end{frame} |
252 \end{frame} |
153 |
253 |
154 \end{document} |
254 \end{document} |
155 |
255 |
156 Least squares: Smooth curve fit. |
256 Least squares: Smooth curve fit. |