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% General example LaTeX file for including Sage calculations and plots
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\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.<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}

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}

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 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.

% FIXME: not sure this works with remote sagetex

\begin{sagesilent}
  x, y = var('x y')
\end{sagesilent}

\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} variable to
True:
\renewcommand{\sageexampleincludetextoutput}{True}
\begin{sageexample}
  sage: 1+1
  2
  sage: factor(x^2 + 2*x + 1)
  (x + 1)^2
\end{sageexample}
Multiline statements are support, 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}
When typesetting your document, the validity of the outputs is not
checked (go ahead and try changing something above) but it should be
possible to run the usual Sage doctest mechanism on the generated
\texttt{.sage} file---or perhaps the \texttt{.py} file. Running
doctests on files outside the main Sage library does not always work,
so contact 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:
\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)
  1/5*x^5 - 1/4*x^4 + 1/3*x^3 - 1/2*x^2 + 5*x + 1  
\end{sageexample}


\end{document}