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% 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}
\author{}
\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}
\begin{document}
\maketitle
\section{Inline Sage, code blocks}
This is an example $2+2=\sage{2+2}$. If you raise the current year mod
$100$ ($\sage{mod(\the\year, 100)}$) 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}
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 = log(sin(x)/x)
\end{sageblock}
The Taylor Series of $f$ is: $\sage{ f.taylor(x, 0, 10) }$.
\section{Plotting}
Here's a plot of the elliptic curve $E$.
\sageplot{E.plot(-3,3)}
\begin{sagesilent}
var('x')
f=-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:
\centerline{\sageplot[scale=.2]{p}}
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,
beause 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}
Indentation and so on works fine.
\begin{sageblock}
s = 7
s2 = 2^s
P.<x> = 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}
\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 2.10.3) can't
produce EPS or PDF files from 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:
\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')}
\end{document}
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