% 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{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} 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} \tikzstyle{every picture}=[] Here's some more graphs, plotted using the usual plot routines. \sageplot[scale=.5]{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[scale=.5]{G4plot, axes=False} \subsection{3D plotting} 3D plotting right now (Sage version 4.3.4) is problematic because there's no convenient way to produce vector graphics. We can make PNGs, though, so if you pass \verb|sageplot| a graphics object that cannot be saved to EPS or PDF format, we will automatically save to a PNG file, which can be used when typesetting a PDF file, but not when creating a DVI file. However, you can specify the \texttt{imagemagick}'' option, which will use the Imagemagick \texttt{convert} utility to make EPS files. See the documentation for details. % FIXME: not sure this works with remote sagetex \begin{sagesilent} x, y = var('x y') \end{sagesilent} Here's a 3D plot whose format we do not specify; it will automatically get saved as a PNG file and won't work when using \texttt{latex} to make a DVI file. \sageplot[scale=.5]{plot3d(sin(pi*(x^2+y^2))/2,(x,-1,1),(y,-1,1))} Here's the (perhaps-not-so-) 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()} \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. Just write a Sage/Python function that outputs a string of \LaTeX{} code, and use \verb|\sagestr|. Here's how to make Pascal's triangle. \begin{sageblock} def pascals_triangle(n): # start of the table s = [r"\begin{tabular}{cc|" + "r" * (n+1) + "}"] s.append(r" & & $k$: & \\") # second row, with k values: s.append(r" & ") for k in [0..n]: s.append("& {0} ".format(k)) s.append(r"\\") # the n = 0 row: s.append(r"\hline" + "\n" + r"$n$: & 0 & 1 & \\") # now the rest of the rows for r in [1..n]: s.append(" & {0} ".format(r)) for k in [0..r]: s.append("& {0} ".format(binomial(r, k))) s.append(r"\\") # add the last line and return s.append(r"\end{tabular}") return ''.join(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} \section{Include doctest-like examples in your document} Here are some examples of using the \texttt{sageexample} environment: \begin{sageexample} sage: 1+1 2 sage: factor(x^2 + 2*x + 1) (x + 1)^2 \end{sageexample} If you want to see the plain-text output as well as the typeset output, renew the \texttt{sageexampleincludetextoutput} command to True: \begin{verbatim} \renewcommand{\sageexampleincludetextoutput}{True} \end{verbatim} \renewcommand{\sageexampleincludetextoutput}{True} This can be useful to check that the two outputs are consistent. When this environment is near the bottom of the page, it may look like the page number is the output of a command, when in fact the real output is on the next page. If the output of a command below looks like \thepage, don't worry, that's just the page number. \begin{sageexample} sage: 1+1 2 sage: factor(x^2 + 2*x + 1) (x + 1)^2 \end{sageexample} Multiline statements are supported, as are triple-quoted strings delimited by single quotes: \renewcommand{\sageexampleincludetextoutput}{False} \begin{sageexample} sage: def f(a): ... '''This function is really quite nice, ... although perhaps not very useful.''' ... print "f called with a = ", a ... y = integrate(SR(cyclotomic_polynomial(10)) + a, x) ... return y + 1 sage: f(x) f called with a = x 1/5*x^5 - 1/4*x^4 + 1/3*x^3 + x + 1 \end{sageexample} Note that the $f$ called with\ldots'' stuff doesn't get typeset, since when running Sage on \texttt{example.sage}, that gets printed to the terminal. When typesetting your document, the validity of the outputs is not checked. In fact, the provided outputs are completely ignored: \renewcommand{\sageexampleincludetextoutput}{True} \begin{sageexample} sage: is_prime(57) toothpaste \end{sageexample} \renewcommand{\sageexampleincludetextoutput}{False} However, typesetting your document produces a file named \texttt{example\_doctest.sage} containing all the doctest-like examples, and you can have Sage check them for you with: \begin{verbatim} $sage -t example_doctest.sage \end{verbatim} You should get one doctest failure from the toothpaste'' line above. Please look into this file for the original line numbers. Beware that \texttt{sage -t} does not handle well file names with special characters in them, particularly dashes, dots, and spaces---this ultimately comes from the way Python interprets \texttt{import} statements. Also, running doctests on files outside the main Sage library does not always work, so contact \texttt{sage-support} if you run into troubles. Some more examples. This environment is implemented a little bit differently than the other environments, so it's good to make sure that definitions are preserved across multiple uses. This will correctly define$a$, but not print its output because the statement is made up of a sequence of expressions. \begin{sageexample} sage: 1; 2; a=4; 3; a 1 2 3 4 \end{sageexample} After that, Sage should remember that$a = \sage{a}\$ and be able to use that in future \texttt{sageexample} blocks: \begin{sageexample} sage: f(a) f called with a = 4 1/5*x^5 - 1/4*x^4 + 1/3*x^3 - 1/2*x^2 + 5*x + 1 \end{sageexample} \section{Plotting functions in Ti\emph{k}Z with Sage\TeX} The wonderful graphics package TikZ has the ability to plot functions by reading in a sequence of points from an external file---see chapter 18, page 193 of the TikZ manual. This facility is designed around files produced by Gnuplot, but the file format is so simple that it's very easy to use Sage\TeX{} to generate them. First you need a function that will evaluate functions and write the results into a file: % set up plotting stuff \begin{sageblock} def gnuplot(x, y, tvals_, fn): """ Write out a gnuplot-style file of points x(t), y(t). """ tvals = list(tvals_) lines = ['#This is a gnuplot-style file written by SageTeX.', '#x: {0}'.format(x), '#y: {0}'.format(y), '#Curve 0, {0} points'.format(len(tvals)), '#x y type'] fmt = lambda _: _.n().str(no_sci=2) for t in tvals: try: lines.append('{0} {1} i'.format(fmt(x(t)), fmt(y(t)))) except ValueError, ZeroDivisonError: pass with open(fn, 'w') as f: f.write('\n'.join(lines) + '\n') \end{sageblock} There probably should be some more exceptions in that list, and the above code doesn't check to make sure it's writing real values, but then again, this is just a file of examples! Then you define callable functions x and y and pass them in, along with a sequence of values and a file name. Here's a plot that I used on a calculus exam: \begin{sageblock} r(t) = 1 - 2*sin(3*t) x(t) = r(t)*cos(t) y(t) = r(t)*sin(t) gnuplot(x, y, srange(0, 2*pi + .05, .05), 'example-tikz1.table') \end{sageblock} (Usually you would do that in sagesilent environments, I guess.) Then you call TikZ with your plot. \begin{tikzpicture} \draw[very thin,->] (-3.25,0) -- (3.25,0); \draw[very thin,->] (0,-3.25) -- (0,3.25); \draw[smooth] plot file {example-tikz1.table}; \end{tikzpicture} For regular Cartesian plots, just pass in the identity function for x: \begin{sageblock} x = lambda t: t y(t) = t*sin(1/t) gnuplot(x, y, [0.01, 0.02..(0.5)] + [0.55, 0.6..2], 'example-tikz2.table') \end{sageblock} \begin{tikzpicture} \draw[very thin,->] (-0.25,0) -- (2,0); \draw[very thin,->] (0,-1/3) -- (0,1); \draw[smooth, red] plot file {example-tikz2.table}; \end{tikzpicture} This style of plotting will become even more useful and powerful when the new TikZ Data Visualization library is available---you will be able to feed TikZ a bunch of data points, and it automatically make a very nice plot for you, including axes, labels, and so on. \section{The \texttt{sagecommandline} environment} When writing a \TeX{} document about Sage, you may want to show some examples of commands and their output. But naturally, you are lazy and don't want to cut and paste the output into your document. Why should I have to do anything? Why can't Sage and \TeX{} cooperate and do it for me?'' you may cry. Well, they \emph{can} cooperate: \begin{sagecommandline} sage: 1+1 sage: is_prime(57) \end{sagecommandline} Note that the output of the commands is not included in the source file, but are included in the typeset output. Because of the way the environment is implemented, not everything is exactly like using Sage in a terminal: the two commands below would produce some output, but don't here: \begin{sagecommandline} sage: x = 2010; len(x.divisors()) sage: print 'Hola, mundo!' \end{sagecommandline} The difference lies in the Python distinction between statements and expressions. One nice thing is that you can set labels by using an @ sign: \begin{sagecommandline} sage: l = matrix([[1,0,0],[3/5,1,0],[-2/5,-2,1]]) sage: d = diagonal_matrix([15, -1, 4]) #@\label{diagonal} sage: u = matrix([[1,0,1/3],[0,1,2],[0,0,1]]) #@\label{anotherlabel} \# foo sage: l*d*u # this is a comment \end{sagecommandline} And then refer to that label: it was on line \ref{diagonal}, which is on page \pageref{diagonal}. Note that the other text after the hash mark on that line does not get typeset as a comment, and that you cannot have any space between the hash mark and the~@. You can also typeset the output: \renewcommand{\sagecommandlinetextoutput}{False} \begin{sagecommandline} sage: l*d*u sage: x = var('x') sage: (1-cos(x)^2).trig_simplify() \end{sagecommandline} \renewcommand{\sagecommandlinetextoutput}{True} The Sage input and output is typeset using the \texttt{listings} package with the styles \texttt{SageInput} and \texttt{SageOutput}, respectively. If you don't like the defaults you can change them. It is recommended to derive from \texttt{DefaultSageInput} and \texttt{DefaultSageOutput}, for example\ldots \lstdefinestyle{SageInput}{style=DefaultSageInput,basicstyle={\color{red}}} \lstdefinestyle{SageOutput}{style=DefaultSageOutput,basicstyle={\color{green}}} makes things overly colorful: \begin{sagecommandline} sage: pi.n(100) \end{sagecommandline} \lstdefinestyle{SageInput}{style=DefaultSageInput} \lstdefinestyle{SageOutput}{style=DefaultSageOutput} Plotting things doesn't automatically pull in the plot, just the text representation of the plot (the equivalent of applying \texttt{str()} to it): \begin{sagecommandline} sage: plot(sin(x), (x, 0, 2*pi)) \end{sagecommandline} You can include output, but it will be ignored. This is useful for doctesting, as all the \texttt{sagecommandline} environment things get put into the `\texttt{\_doctest.sage}'' file. \begin{sagecommandline} sage: factor(x^2 + 2*x + 1) (x + 999)^2 \end{sagecommandline} \end{document}