# SageTeX / example.tex

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 % General example LaTeX file for including Sage calculations and plots % Build with: % % (pdf)latex example.tex; sage example.sage; pdflatex example.tex % % Please read README and the documentation of the SageTeX package for % more information! \documentclass{article} \title{Examples of embedding Sage in \LaTeX{} with \textsf{Sage\TeX}} \author{Dan Drake and others} %\usepackage{amsmath} \usepackage{hyperref} \usepackage{tkz-berge} \usepackage{sagetex} % % If you want SageTeX to use Imagemagick's convert' utility to make eps % files from png files when generating a dvi file, add the "imagemagick" % option above: % % \usepackage[imagemagick]{sagetex} \setlength{\sagetexindent}{10ex} \begin{document} \maketitle \section{Inline Sage, code blocks} This is an example $2+2=\sage{2+2}$. If you raise the current year mod $100$ (which equals $\sage{mod(\the\year, 100)}$) to the power of the current day ($\the\day$), you get $\sage{Integer(mod(\the\year, 100))^\the\day}$. Also, $\the\year$ modulo $42$ is $\sage{\the\year \percent 42}$. Code block which uses a variable \texttt{s} to store the solutions: \begin{sageblock} 1+1 var('a,b,c') eqn = [a+b*c==1, b-a*c==0, a+b==5] s = solve(eqn, a,b,c) \end{sageblock} Solutions of $\mbox{eqn}=\sage{eqn}$: $\sage{s[0]}$ $\sage{s[1]}$ Now we evaluate the following block: \begin{sageblock} E = EllipticCurve("37a") \end{sageblock} You can't do assignment inside \verb|\sage| macros, since Sage doesn't know how to typeset the output of such a thing. So you have to use a code block. The elliptic curve $E$ given by $\sage{E}$ has discriminant $\sage{E.discriminant()}$. You can do anything in a code block that you can do in Sage and/or Python. Here we save an elliptic curve into a file. \begin{sageblock} try: E = load('E2') except IOError: E = EllipticCurve([1,2,3,4,5]) E.anlist(100000) E.save('E2') \end{sageblock} The 9999th Fourier coefficient of $\sage{E}$ is $\sage{E.anlist(100000)[9999]}$. The following code block doesn't appear in the typeset file\dots \begin{sagesilent} e = 2 e = 3*e + 1 \end{sagesilent} but we can refer to whatever we did in that code block: $e=\sage{e}$. \begin{sageblock} var('x') f(x) = log(sin(x)/x) \end{sageblock} The Taylor Series of $f$ begins: $\sage{ f.taylor(x, 0, 10) }$. \section{Plotting} Here's a plot of the elliptic curve $E$. \sageplot{E.plot(-3,3)} \begin{sagesilent} # the var line is unecessary unless you've defined x to be something # other than a symbolic variable var('x') f(x) = -x^3+3*x^2+7*x-4 \end{sagesilent} You can use variables to hold plot objects and do stuff with them. \begin{sageblock} p = plot(f, x, -5, 5) \end{sageblock} Here's a small plot of $f$ from $-5$ to $5$, which I've centered: \begin{center} \sageplot[scale=.2]{p} \end{center} On second thought, use the default size of $3/4$ the \verb|\textwidth| and don't use axes: \sageplot{p, axes=False} Remember, you're using Sage, and can therefore call upon any of the software packages Sage is built out of. \begin{sageblock} f = maxima('sin(x)^2*exp(x)') g = f.integrate('x') \end{sageblock} Plot $g(x)$, but don't typeset it. \begin{sagesilent} # g is a Maxima thingy, it needs to get converted into a Sage object plot1 = plot(g.sage(),x,-1,2*pi) \end{sagesilent} You can specify a file format and options for \verb|includegraphics|. The default is for EPS and PDF files, which are the best choice in almost all situations. (Although see the section on 3D plotting.) \sageplot[angle=45, width=.5\textwidth][png]{plot1} If you use regular \verb|latex| to make a DVI file, you'll see a box, because DVI files can't include PNG files. If you use \verb|pdflatex| that will work. See the documentation for details. When using \verb|\sageplot|, you can pass in just about anything that Sage can call \verb|.save()| on to produce a graphics file: \begin{center} \sageplot{plot1 + plot(f.sage(),x,-1,2*pi,rgbcolor=hue(0.4)), figsize=[1,2]} \end{center} \sageplot{graphs.FlowerSnark().plot()} \begin{sageblock} G4 = DiGraph({1:[2,2,3,5], 2:[3,4], 3:[4], 4:[5,7], 5:[6]},\ multiedges=True) G4plot = G4.plot(layout='circular') \end{sageblock} \sageplot{G4plot, axes=False} To fiddle with aspect ratio, first save the plot object: \begin{sageblock} p = plot(x, 0, 1) + circle((0,0), 1) p.set_aspect_ratio(1) \end{sageblock} Now plot it and see the circular circle and nice 45 degree angle: \sageplot[scale=.33]{p} Indentation and so on works fine. \begin{sageblock} s = 7 s2 = 2^s P. = GF(2)[] M = matrix(parent(x),s2) for i in range(s2): p = (1+x)^i pc = p.coeffs() a = pc.count(1) for j in range(a): idx = pc.index(1) M[i,idx+j] = pc.pop(idx) matrixprogram = matrix_plot(M,cmap='Greys') \end{sageblock} And here's the picture: \sageplot{matrixprogram} Reset \texttt{x} in Sage so that it's not a generator for the polynomial ring: \sage{var('x')} \subsection{Plotting (combinatorial) graphs with TikZ} \label{sec:plotting-graphs-with} Sage now includes some nice support for plotting graphs using \href{http://www.texample.net/tikz/}{TikZ}. Here, we mean things with vertices and edges, not graphs of a function of one or two variables. First define our graph: \begin{sageblock} g = graphs.PetersenGraph() g.set_latex_options(tkz_style='Art') \end{sageblock} Now just do \verb|\sage{}| on it to plot it. You'll need to use the \href{http://altermundus.com/pages/graphtheory.html}{\texttt{tkz-berge}} package for this to work; that package in turn depends on \href{http://altermundus.com/pages/graph.html}{\texttt{tkz-graph}} and TikZ. See \href{http://sagemath.org/doc/reference/sage/graphs/graph_latex.html}{\LaTeX{} Options for Graphs''} in the Sage reference manual for more details. \begin{center} \sage{g} \end{center} The above command just outputs a \texttt{tikzpicture} environment, and you can control that environment using anything supported by TikZ---although the output of \verb|\sage{g}| explicitly hard-codes a lot of things and cannot be flexibly controlled in its current form. \tikzstyle{every picture}=[rotate=45, scale=1/2] \begin{center} \sage{g} \end{center} \subsection{3D plotting} 3D plotting right now is problematic because there's no convenient way to produce vector graphics. We can make PNGs, though, and since the \verb|sageplot| command defaults to EPS and PDF, \emph{you must specify a valid format for 3D plotting}. Sage right now (version 4.2.1) can't produce EPS or PDF files from \texttt{plot3d} objects, so if you don't specify a valid format, things will go badly. You can specify the \texttt{imagemagick}'' option, which will use the Imagemagick \texttt{convert} utility to make EPS files. See the documentation for details. Here's the famous Sage cube graph in 3D. \begin{sageblock} G = graphs.CubeGraph(5) \end{sageblock} % need empty [] so sageplot knows you want png format, and aren't % passing an option to includegraphics \sageplot[][png]{G.plot3d(engine='tachyon')} And here's a regular sort of 3D plot. Since \texttt{plot3d} objects don't properly support the kind of \texttt{.save()} method that we need, so we have to work around it a bit and do things manually. Note that we can't use \verb|\jobname| below. The \texttt{sage.misc.viewer.BROWSER} bit tells Sage to not pop up a viewer program; otherwise, when you run the \texttt{.sage} script, it will try to start a viewer program on the resulting image, which we don't want. % FIXME: get this working with remote sagetex \begin{sageblock} sage.misc.viewer.BROWSER='' x, y = var('x y') g = plot3d(sin(pi*(x^2+y^2))/2,(x,-1,1),(y,-1,1)) g.show(filename='sage-plots-for-example.tex/my-3d-plot', viewer='tachyon') \end{sageblock} \IfFileExists{sage-plots-for-\jobname.tex/my-3d-plot.png}{ \includegraphics[scale=.5]{sage-plots-for-\jobname.tex/my-3d-plot.png}} {Hey! Run Sage so that this plot exists!} \section{Pausing Sage\TeX} \label{sec:pausing-sagetex} Sometimes you want to pause'' for a bit while writing your document if you have embedded a long calculation or just want to concentrate on the \LaTeX{} and ignore any Sage stuff. You can use the \verb|\sagetexpause| and \verb|\sagetexunpause| macros to do that. \sagetexpause A calculation: $\sage{factor(2^325 + 1)}$ and a code environment that simulates a time-consuming calculation. While paused, this will get skipped over. \begin{sageblock} import time time.sleep(15) \end{sageblock} Graphics are also skipped: \sageplot{plot(2*sin(x^2) + x^2, (x, 0, 5))} \sagetexunpause \section{Make Sage write your \LaTeX{} for you} With \textsf{Sage\TeX}, you can not only have Sage do your math for you, it can write parts of your \LaTeX{} document for you! For example, I hate writing \texttt{tabular} environments; there's too many fiddly little bits of punctuation and whatnot\ldots and what if you want to add a column? It's a pain---or rather, it \emph{was} a pain. Here's how to make Pascal's triangle. It requires the \texttt{amsmath} package because of what Sage does when producing a \LaTeX{} representation of a string. (It puts it inside a \verb|\text| macro.) \begin{sageblock} def pascals_triangle(n): # start of the table s = r"\begin{tabular}{cc|" + "r" * (n+1) + "}" s += r" & & $k$: & \\" # second row, with k values: s += r" & " for k in [0..n]: s += "& %d " % k s += r"\\" # the n = 0 row: s += r"\hline" + "\n" + r"$n$: & 0 & 1 & \\" # now the rest of the rows for r in [1..n]: s += " & %d " % r for k in [0..r]: s += "& %d " % binomial(r, k) s += r"\\" # add the last line and return s += r"\end{tabular}" return s # how big should the table be? n = 8 \end{sageblock} Okay, now here's the table. To change the size, edit \texttt{n} above. If you have several tables, you can use this to get them all the same size, while changing only one thing. \begin{center} \sagestr{pascals_triangle(n)} \end{center} \end{document} `