Sage, LaTeX and Friends

AUTHOR: Rob Beezer (2010-05-23)

Sage and the LaTeX dialect of TeX have an intensely synergistic relationship. This section aims to introduce the variety of interactions, beginning with the most basic and proceeding to the more unusual and arcane. (So you may not want to read this entire section on your first pass through this tutorial.)

Overview

It may be easiest to understand the various uses of LaTeX with a brief overview of the mechanics of the three principal methods employed by Sage.

  1. Every “object” in Sage is required to have a LaTeX representation. You can access this representation by executing, in the notebook or at the sage command line, latex(foo) where foo is some object in Sage. The output is a string that should render a reasonably accurate representation of foo when used in TeX’s math-mode (for example, when enclosed between a pair of single dollar signs). Some examples of this follow below.

    In this way, Sage can be used effectively for constructing portions of a LaTeX document: create or compute an object in Sage, print latex() of the object and cut/paste it into your document.

  2. The notebook interface is configured to use MathJax to render mathematics cleanly in a web browser. MathJax is an open source JavaScript display engine for mathematics that works in all modern browsers. It is able to render a large, but not totally complete, subset of TeX. It has no support for things like complicated tables, sectioning or document management, as it is oriented towards accurately rendering “snippets” of TeX. Seemingly automatic rendering of math in the notebook is provided by converting the latex() representation of an object (as described above) into a form of HTML palatable to MathJax.

    Since MathJax uses its own scalable fonts, it is superior to other methods that rely on converting equations, or other snippets of TeX, into static inline images.

  3. At the Sage command-line, or in the notebook when LaTeX code is more involved than MathJax can handle, a system-wide installation of LaTeX can be employed. Sage includes almost everything you need to build and use Sage, but a significant exception is TeX itself. So in these situations you need to have TeX installed, along with some associated conversion utilities, to utilize the full power.

Here we demonstrate some basic uses of the latex() function.

sage: var('z')
z
sage: latex(z^12)
z^{12}
sage: latex(integrate(z^4, z))
\frac{1}{5} \, z^{5}
sage: latex('a string')
\text{\texttt{a{ }string}}
sage: latex(QQ)
\Bold{Q}
sage: latex(matrix(QQ, 2, 3, [[2,4,6],[-1,-1,-1]]))
\left(\begin{array}{rrr}
2 & 4 & 6 \\
-1 & -1 & -1
\end{array}\right)

Basic MathJax functionality is largely automatic in the notebook, but we can partially demonstrate this support with the MathJax class. The eval function of this class converts a Sage object to its LaTeX representation and then wraps it in HTML that invokes the CSS “math” class, which then employs MathJax.

sage: from sage.misc.latex import MathJax
sage: mj = MathJax()
sage: var('z')
z
sage: mj(z^12)
<html><script type="math/tex; mode=display">\newcommand{\Bold}[1]{\mathbf{#1}}z^{12}</script></html>
sage: mj(QQ)
<html><script type="math/tex; mode=display">\newcommand{\Bold}[1]{\mathbf{#1}}\Bold{Q}</script></html>
sage: mj(ZZ[x])
<html><script type="math/tex; mode=display">\newcommand{\Bold}[1]{\mathbf{#1}}\Bold{Z}[x]</script></html>
sage: mj(integrate(z^4, z))
<html><script type="math/tex; mode=display">\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{5} \, z^{5}</script></html>

Basic Use

As indicated in the overview, the simplest way to exploit Sage’s support of LaTeX is to use the latex() function to create legitimate LaTeX code to represent mathematical objects. These strings can then be incorporated into standalone LaTeX documents. This works identically in the notebook and at the Sage command line.

At the other extreme is the view() command. At the Sage command line the command view(foo) will create the LaTeX representation of foo, incorporate this into a simple LaTeX document, and then process that document with your system-wide TeX installation. Finally, the appropriate viewer will be called to display the output from the TeX command. Which version of TeX is used, and therefore the nature of the output and associated viewer, can be customized (see Customizing LaTeX Processing).

In the notebook, the view(foo) command creates the appropriate combination of HTML and CSS so that MathJax will render the LaTeX representation properly in the worksheet. To the user, it simply creates a nicely formatted version of the output, distinct from the default ASCII output of Sage. Not every mathematical object in Sage has a LaTeX representation amenable to the limited capabilities of MathJax. In these cases, the MathJax interpretation can be bypassed, the system-wide TeX called instead, and the subsequent output converted to a graphic image for display in the worksheet. Affecting and controlling this process is discussed below in the section Customizing LaTeX Generation.

The notebook has two other features for employing TeX. The first is the “Typeset” button just above the first cell of a worksheet, to the right of the four drop-down boxes. When checked, any subsequent evaluations of cells will result in output interpreted by MathJax, hence of a typeset quality. Note that this effect is not retroactive – previously evaluated cells need to be re-evaluated. Essentially, checking the “Typeset” button is identical to wrapping the output of each cell in the view() command.

A second feature of the notebook is entering TeX as part of annotating a worksheet. When the cursor is placed between cells of a worksheet so that a blue bar appears, then a shift-click will open a mini-word-processor, TinyMCE. This allows for the entry of text, using a WSIWYG editor to create HTML and CSS command for styled text. So it is possible to add formatted text as commentary within a worksheet. However, text between pairs of dollar signs, or pairs of double dollar signs is interpreted by MathJax as inline or display math (respectively).

Customizing LaTeX Generation

There are several ways to customize the actual LaTeX code generated by the latex() command. In the notebook and at the Sage command-line there is a pre-defined object named latex which has several methods, which you can list by typing latex., followed by the tab key (note the period).

A good example is the latex.matrix_delimiters method. It can be used to change the notation surrounding a matrix – large parentheses, brackets, braces, vertical bars. No notion of style is enforced, you can mix and match as you please. Notice how the backslashes needed in LaTeX require an extra slash so they are escaped properly within the Python string.

sage: A = matrix(ZZ, 2, 2, range(4))
sage: latex(A)
\left(\begin{array}{rr}
0 & 1 \\
2 & 3
\end{array}\right)
sage: latex.matrix_delimiters(left='[', right=']')
sage: latex(A)
\left[\begin{array}{rr}
0 & 1 \\
2 & 3
\end{array}\right]
sage: latex.matrix_delimiters(left='\\{', right='\\}')
sage: latex(A)
\left\{\begin{array}{rr}
0 & 1 \\
2 & 3
\end{array}\right\}

The latex.vector_delimiters method works similarly.

The way common rings and fields (integers, rational, reals, etc.) are typeset can be controlled by the latex.blackboard_bold method. These sets are by default typeset in bold, but may optionally be written in a double-struck fashion as sometimes done in written work. This is accomplished by redefining the \Bold{} macro which is built-in to Sage.

sage: latex(QQ)
\Bold{Q}
sage: from sage.misc.latex import MathJax
sage: mj=MathJax()
sage: mj(QQ)
<html><script type="math/tex; mode=display">\newcommand{\Bold}[1]{\mathbf{#1}}\Bold{Q}</script></html>
sage: latex.blackboard_bold(True)
sage: mj(QQ)
<html><script type="math/tex; mode=display">\newcommand{\Bold}[1]{\mathbb{#1}}\Bold{Q}</script></html>
sage: latex.blackboard_bold(False)

It is possible to take advantage of the extensible nature of TeX by adding in new macros and new packages. First, individual macros can be added so that they are used when MathJax interprets a snippet of TeX in the notebook.

sage: latex.extra_macros()
''
sage: latex.add_macro("\\newcommand{\\foo}{bar}")
sage: latex.extra_macros()
'\\newcommand{\\foo}{bar}'
sage: var('x y')
(x, y)
sage: latex(x+y)
x + y
sage: from sage.misc.latex import MathJax
sage: mj=MathJax()
sage: mj(x+y)
<html><script type="math/tex; mode=display">\newcommand{\Bold}[1]{\mathbf{#1}}\newcommand{\foo}{bar}x + y</script></html>

Additional macros added this way will also be used in the event that the system-wide version of TeX is called on something larger than MathJax can handle. The command latex_extra_preamble is used to build the preamble of a complete LaTeX document, so the following illustrates how this is accomplished. As usual note the need for the double-backslashes in the Python strings.

sage: latex.extra_macros('')
sage: latex.extra_preamble('')
sage: from sage.misc.latex import latex_extra_preamble
sage: print latex_extra_preamble()
\newcommand{\ZZ}{\Bold{Z}}
...
\newcommand{\Bold}[1]{\mathbf{#1}}
sage: latex.add_macro("\\newcommand{\\foo}{bar}")
sage: print latex_extra_preamble()
\newcommand{\ZZ}{\Bold{Z}}
...
\newcommand{\Bold}[1]{\mathbf{#1}}
\newcommand{\foo}{bar}

Again, for larger or more complicated LaTeX expressions, it is possible to add packages (or anything else) to the preamble of the LaTeX file. Anything may be incorporated into the preamble with the latex.add_to_preamble command, and the specialized command latex.add_package_to_preamble_if_available will first check if a certain package is actually available before trying to add it to the preamble.

Here we add the geometry package to the preamble and use it to set the size of the region on the page that TeX will use (effectively setting the margins). As usual, note the need for the double-backslashes in the Python strings.

sage: from sage.misc.latex import latex_extra_preamble
sage: latex.extra_macros('')
sage: latex.extra_preamble('')
sage: latex.add_to_preamble('\\usepackage{geometry}')
sage: latex.add_to_preamble('\\geometry{letterpaper,total={8in,10in}}')
sage: latex.extra_preamble()
'\\usepackage{geometry}\\geometry{letterpaper,total={8in,10in}}'
sage: print latex_extra_preamble()
\usepackage{geometry}\geometry{letterpaper,total={8in,10in}}
\newcommand{\ZZ}{\Bold{Z}}
...
\newcommand{\Bold}[1]{\mathbf{#1}}

A particular package may be added along with a check on its existence, as follows. As an example, we just illustrate an attempt to add to the preamble a package that presumably does not exist.

sage: latex.extra_preamble('')
sage: latex.extra_preamble()
''
sage: latex.add_to_preamble('\\usepackage{foo-bar-unchecked}')
sage: latex.extra_preamble()
'\\usepackage{foo-bar-unchecked}'
sage: latex.add_package_to_preamble_if_available('foo-bar-checked')
sage: latex.extra_preamble()
'\\usepackage{foo-bar-unchecked}'

Customizing LaTeX Processing

It is also possible to control which variant of TeX is used for system-wide invocations, thus also influencing the nature of the output. Similarly, it is also possible to control when the notebook will use MathJax (simple TeX snippets) or the system-wide TeX installation (more complicated LaTeX expressions).

The latex.engine() command can be used to control if the system-wide executables latex, pdflatex or xelatex are employed for more complicated LaTeX expressions. When view() is called from the sage command-line and the engine is set to latex, a dvi file is produced and Sage will use a dvi viewer (like xdvi) to display the result. In contrast, using view() at the Sage command-line, when the engine is set to pdflatex, will produce a PDF as the result and Sage will call your system’s utility for displaying PDF files (acrobat, okular, evince, etc.).

In the notebook, it is necessary to intervene in the decision as to whether MathJax will interpret a snippet of TeX, or if the LaTeX is complicated enough that the system-wide installation of TeX should do the work instead. The device is a list of strings, which if any one is discovered in a piece of LaTeX code signal the notebook to bypass MathJax and invoke latex (or whichever executable is set by the latex.engine() command). This list is managed by the latex.add_to_mathjax_avoid_list and latex.mathjax_avoid_list commands.

sage: latex.mathjax_avoid_list([])
sage: latex.mathjax_avoid_list()
[]
sage: latex.mathjax_avoid_list(['foo', 'bar'])
sage: latex.mathjax_avoid_list()
['foo', 'bar']
sage: latex.add_to_mathjax_avoid_list('tikzpicture')
sage: latex.mathjax_avoid_list()
['foo', 'bar', 'tikzpicture']
sage: latex.mathjax_avoid_list([])
sage: latex.mathjax_avoid_list()
[]

Suppose a LaTeX expression is produced in the notebook with view() or while the “Typeset” button is checked, and then recognized as requiring the external LaTeX installation through the “mathjax avoid list.” Then the selected executable (as specified by latex.engine()) will process the LaTeX. However, instead of then spawning an external viewer (which is the command-line behavior), Sage will attempt to convert the result into a single, tightly-cropped image, which is then inserted into the worksheet as the output of the cell.

Just how this conversion proceeds depends on several factors – mostly which executable you have specified as the engine and which conversion utilities are available on your system. Four useful converters that will cover all eventualities are dvips, ps2pdf, dvipng and from the ImageMagick suite, convert. The goal is to produce a PNG file as the output for inclusion back into the worksheet. When a LaTeX expression can be converted successfully to a dvi by the latex engine, then dvipng should accomplish the conversion. If the LaTeX expression and chosen engine creates a dvi with specials that dvipng cannot handle, then dvips will create a PostScript file. Such a PostScript file, or a PDF file created by an engine such as pdflatex, is then processed into a PNG with the convert utility. The presence of two of these converters can be tested with the have_dvipng() and have_convert() routines.

These conversions are done automatically if you have the necessary converters installed; if not, then an error message is printed telling you what’s missing and where to download it.

For a concrete example of how complicated LaTeX expressions can be processed, see the example in the next section (An Example: Combinatorial Graphs with tkz-graph) for using the LaTeX tkz-graph package to produce high-quality renderings of combinatorial graphs. For other examples, there are some pre-packaged test cases. To use these, it is necessary to import the sage.misc.latex.latex_examples object, which is an instance of the sage.misc.latex.LatexExamples class, as illustrated below. This class currently has examples of commutative diagrams, combinatorial graphs, knot theory and pstricks, which respectively exercise the following packages: xy, tkz-graph, xypic, pstricks. After the import, use tab-completion on latex_examples to see the pre-packaged examples. Calling each example will give you back some explanation about what is required to make the example render properly. To actually see the examples, it is necessary to use view() (once the preamble, engine, etc are all set properly).

sage: from sage.misc.latex import latex_examples
sage: latex_examples.diagram()
LaTeX example for testing display of a commutative diagram produced
by xypic.

To use, try to view this object -- it won't work.  Now try
'latex.add_to_preamble("\\usepackage[matrix,arrow,curve,cmtip]{xy}")',
and try viewing again -- it should work in the command line but not
from the notebook.  In the notebook, run
'latex.add_to_mathjax_avoid_list("xymatrix")' and try again -- you
should get a picture (a part of the diagram arising from a filtered
chain complex).

An Example: Combinatorial Graphs with tkz-graph

High-quality illustrations of combinatorial graphs (henceforth just “graphs”) are possible with the tkz-graph package. This package is built on top of the tikz front-end to the pgf library. So all of these components need to be part of a system-wide TeX installation, and it may be possible that these components may not be at their most current versions as packaged in some TeX implementations. So for best results, it could be necessary or advisable to install these as part of your personal texmf tree. Creating, maintaining and customizing a system-wide or personal TeX installation is beyond the scope of this document, but it should be easy to find instructions. The necessary files are listed in A Fully Capable TeX Installation.

Thus, to start we need to insure that the relevant packages are included by adding them to the preamble of the eventual LaTeX document. The images of graphs do not form properly when a dvi file is used as an intermediate format, so it is best to set the latex engine to the pdflatex executable. At this point a command like view(graphs.CompleteGraph(4)) should succeed at the Sage command-line and produce a PDF with an appropriate image of the complete graph \(K_4\).

For a similar experience in the notebook, it is necessary to disable MathJax processing of the LaTeX code for the graph by using the “mathjax avoid list.” Graphs are included with a tikzpicture environment, so this is a good choice for a string to include in the avoidance list. Now, view(graphs.CompleteGraph(4)) in a worksheet should call pdflatex to create a PDF and then the convert utility will extract a PNG graphic to insert into the output cell of the worksheet. The following commands illustrate the steps to get graphs processed by LaTeX in the notebook.

sage: from sage.graphs.graph_latex import setup_latex_preamble
sage: setup_latex_preamble()
sage: latex.extra_preamble() # random - depends on system's TeX installation
'\\usepackage{tikz}\n\\usepackage{tkz-graph}\n\\usepackage{tkz-berge}\n'
sage: latex.engine('pdflatex')
sage: latex.add_to_mathjax_avoid_list('tikzpicture')
sage: latex.mathjax_avoid_list()
['tikz', 'tikzpicture']

At this point, a command like view(graphs.CompleteGraph(4)) should produce a graphic version of the graph pasted into the notebook, having used pdflatex to process tkz-graph commands to realize the graph. Note that there is a variety of options to affect how a graph is rendered in LaTeX via tkz-graph, which is again outside the scope of this section, see the section of the Reference manual titled “LaTeX Options for Graphs” for instructions and details.

A Fully Capable TeX Installation

Many of the more advanced features of the integration of TeX with Sage requires a system-wide installation of TeX. Many versions of Linux have base TeX packages based on TeX-live, for OSX there is TeXshop and for Windows there is MikTeX. The convert utility is part of the ImageMagick suite (which should be a package or an easy download), and the three programs dvipng, ps2pdf, and dvips may be included with your TeX distribution. The first two may also be obtained, respectively, from http://sourceforge.net/projects/dvipng/ and as part of Ghostscript.

Rendering combinatorial graphs requires a recent version of the PGF library, and the files tkz-graph.sty, tkz-arith.sty and perhaps tkz-berge.sty, all from the Altermundus site.

External Programs

There are three programs available to further integrate TeX and Sage. The first is sagetex. A concise description of sagetex is that it is a collection of TeX macros that allow a LaTeX document to include instructions to have Sage compute various objects and/or format objects using the latex() support built in to Sage. So as an intermediate step of compiling a LaTeX document, all of the computational and LaTeX-formatting features of Sage can be handled automatically. As an example, a mathematics examination can maintain a correct correspondence between questions and answers by using sagetex to have Sage compute one from the other. See Using SageTeX for more information.

tex2sws begins with a LaTeX document, but defines extra environments for the placement of Sage code. When processed with the right tools, the result is a Sage worksheet, with content properly formatted for MathJax and the Sage code incorporated as input cells. So a textbook or article can be authored in LaTeX, blocks of Sage code included, and the whole document can be transformed into a Sage worksheet where the mathematical text is nicely formatted and the blocks of Sage code are “live.” Currently in development, see tex2sws @ BitBucket for more information.

sws2tex reverses the process by beginning with a Sage worksheet and converting it to legitimate LaTeX for subsequent processing with all the tools available for LaTeX documents. Currently in development, see sws2tex @ BitBucket for more information.