General Conventions¶

There are many ways to contribute to Sage including sharing scripts and Sage worksheets that implement new functionality using Sage, improving to the Sage library, or to working on the many underlying libraries distributed with Sage [1]. This guide focuses on editing the Sage library itself.

Sage is not just about gathering together functionality. It is about providing a clear, systematic and consistent way to access a large number of algorithms, in a coherent framework that makes sense mathematically. In the design of Sage, the semantics of objects, the definitions, etc., are informed by how the corresponding objects are used in everyday mathematics.

[1]	See https://www.sagemath.org/links-components.html for a full list of packages shipped with every copy of Sage

To meet the goal of making Sage easy to read, maintain, and improve, all Python/Cython code that is included with Sage should adhere to the style conventions discussed in this chapter.

Python Code Style¶

Follow the standard Python formatting rules when writing code for Sage, as explained at the following URLs:

In particular,

Use 4 spaces for indentation levels. Do not use tabs as they can result in indentation confusion. Most editors have a feature that will insert 4 spaces when the tab key is hit. Also, many editors will automatically search/replace leading tabs with 4 spaces.
Whitespace before and after assignment and binary operator of the lowest priority in the expression:
```
i = i + 1
c = (a+b) * (a-b)
```
No whitespace before or after the = sign if it is used for keyword arguments:
```
def complex(real, imag=0.0):
    return magic(r=real, i=imag)
```
No whitespace immediately inside parenthesis, brackets, and braces:
```
spam(ham[1], {eggs: 2})
[i^2 for i in range(3)]
```
Use all lowercase function names with words separated by underscores. For example, you are encouraged to write Python functions using the naming convention:
```
def set_some_value():
    return 1
```
Note, however, that some functions do have uppercase letters where it makes sense. For instance, the function for lattice reduction by the LLL algorithm is called Matrix_integer_dense.LLL.

Use CamelCase for class names:

class SomeValue(object):
    def __init__(self, x):
    self._x  = 1

and factory functions that mimic object constructors, for example PolynomialRing or:

def SomeIdentityValue(x):
    return SomeValue(1)

Files and Directory Structure¶

Roughly, the Sage directory tree is layout like this. Note that we use SAGE_ROOT in the following as a shortcut for the (arbitrary) name of the directory containing the Sage sources:

SAGE_ROOT/
    sage          # the Sage launcher
    Makefile      # top level Makefile
    build/        # sage's build system
        deps
        install
        ...
        pkgs/     # install, patch, and metadata from spkgs
    src/
        setup.py
        module_list.py
        ...
        sage/     # sage library (formerly devel/sage-main/sage)
        ext/      # extra sage resources (formerly devel/ext-main)
        mac-app/  # would no longer have to awkwardly be in extcode
        bin/      # the scripts in local/bin that are tracked
    upstream/     # tarballs of upstream sources
    local/        # installed binaries

Python Sage library code goes into src/ and uses the following conventions. Directory names may be plural (e.g. rings) and file names are almost always singular (e.g. polynomial_ring.py). Note that the file polynomial_ring.py might still contain definitions of several different types of polynomial rings.

Note

You are encouraged to include miscellaneous notes, emails, design discussions, etc., in your package. Make these plain text files (with extension .txt) in a subdirectory called notes.

If you want to create a new directory in the Sage library SAGE_ROOT/src/sage (say, measure_theory), that directory should contain a file __init__.py that contains the single line import all in addition to whatever files you want to add (say, borel_measure.py and banach_tarski.py), and also a file all.py listing imports from that directory that are important enough to be in the Sage’s global namespace at startup. The file all.py might look like this:

from borel_measure import BorelMeasure
from banach_tarski import BanachTarskiParadox

but it is generally better to use the lazy import framework:

from sage.misc.lazy_import import lazy_import
lazy_import('sage.measure_theory.borel_measue', 'BorelMeasure')
lazy_import('sage.measure_theory.banach_tarski', 'BanachTarskiParadox')

Then in the file SAGE_ROOT/src/sage/all.py, add a line

from sage.measure_theory.all import *

Non-Python Sage source code and supporting files should be placed in appropriate subdirectories of SAGE_ROOT/src/ext/. They will then be automatically copied to the corresponding subdirectories of SAGE_ROOT/local/share/sage/ext/ during the build process and can be accessed at runtime using SAGE_EXTCODE. For example, if file is placed in SAGE_ROOT/src/ext/directory/ it can be accessed with

from sage.env import SAGE_EXTCODE
file = os.path.join(SAGE_EXTCODE, 'directory', 'file')

SAGE_EXTCODE is used because not all distributions have SAGE_ROOT.

Learn by copy/paste¶

For all of the conventions discussed here, you can find many examples in the Sage library. Browsing through the code is helpful, but so is searching: the functions search_src, search_def, and search_doc are worth knowing about. Briefly, from the “sage:” prompt, search_src(string) searches Sage library code for the string string. The command search_def(string) does a similar search, but restricted to function definitions, while search_doc(string) searches the Sage documentation. See their docstrings for more information and more options.

Headings of Sage Library Code Files¶

The top of each Sage code file should follow this format:

r"""
<Very short 1-line summary>

<Paragraph description>

EXAMPLES::

<Lots and lots of examples>

AUTHORS:

- YOUR NAME (2005-01-03): initial version

- person (date in ISO year-month-day format): short desc

"""

# ****************************************************************************
#       Copyright (C) 2013 YOUR NAME <your email>
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 2 of the License, or
# (at your option) any later version.
#                  https://www.gnu.org/licenses/
# ****************************************************************************

As an example, see SAGE_ROOT/src/sage/rings/integer.pyx, which contains the implementation for $\ZZ$. The names of the people who made major contributions to the file appear in the AUTHORS section. You can add your name to the list if you belong to the people, but refrain from being verbose in the description. The AUTHORS section shows very rough overview of the history, especially if a lot of people have been working on that source file. The authoritative log for who wrote what is always the git repository (see the output of git blame).

All code included with Sage must be licensed under the GPLv2+ or a compatible, that is, less restrictive license (e.g. the BSD license).

Documentation Strings¶

The docstring of a function: content¶

Every function must have a docstring that includes the following information. You can use the existing functions of Sage as templates.

A one-sentence description of the function.

It must be followed by a blank line and end in a period. It describes the function or method’s effect as a command (“Do this”, “Return that”), not as a description like “Returns the pathname …”.

For methods of a class, it is recommended to refer to the self argument in a descriptive way, unless this leads to a confusion. For example, if self is an integer, then this integer or the integer is more descriptive, and it is preferable to write
```
Return whether this integer is prime.
```
A longer description.

This is optional if the one-sentence description does not need more explanations.

Start with assumptions of the object, if there are any. For example,
```
The poset is expected to be ranked.
```
if the function raises an exception when called on a non-ranked poset.

Define your terms
```
The lexicographic product of `G` and `H` is the graph with vertex set ...
```
and mention possible aliases
```
The tensor product is also known as the categorical product and ...
```
An INPUT and an OUTPUT block describing the input/output of the function.

The INPUT block describes all arguments that the function accepts.
1. The type names should be descriptive, but do not have to represent the exact Sage/Python types. For example, use “integer” for anything that behaves like an integer, rather than int.
2. Mention the default values of the input arguments when applicable.
```
INPUT:

- ``n`` -- integer

- ``p`` -- prime integer (default: `2`); coprime with ``n``
```
The OUTPUT block describes the expected output. This is required if the one-sentence description of the function needs more explanation.
```
OUTPUT: the plaintext decrypted from the ciphertext ``C``
```
It is often the case that the output consists of several items.
```
OUTPUT: a tuple of

- the reduced echelon form `H` of the matrix `A`

- the transformation matrix `U` such that `UA = H`
```
You are recommended to be verbose enough for complicated outputs.
```
OUTPUT:

The decomposition of the free module on which this matrix `A` acts from
the right (i.e., the action is `x` goes to `xA`), along with whether
this matrix acts irreducibly on each factor. The factors are guaranteed
to be sorted in the same way as the corresponding factors of the
characteristic polynomial.
```
An EXAMPLES block for examples. This is not optional.

These examples are used for documentation, but they are also tested before each release just like TESTS block.

They should have good coverage of the functionality in question.
A SEEALSO block (highly recommended) with links to related parts of Sage. This helps users find the features that interest them and discover the new ones.
```
.. SEEALSO::

    :ref:`chapter-sage_manuals_links`,
    :meth:`sage.somewhere.other_useful_method`,
    :mod:`sage.some.related.module`.
```
See Hyperlinks for details on how to setup link in Sage.

An ALGORITHM block (optional).

It indicates what algorithm and/or what software is used, e.g. ALGORITHM: Uses Pari. Here’s a longer example with a bibliographical reference:

ALGORITHM:

The following algorithm is adapted from page 89 of [Nat2000]_.

Let `p` be an odd (positive) prime and let `g` be a generator
modulo `p`. Then `g^k` is a generator modulo `p` if and only if
`\gcd(k, p-1) = 1`. Since `p` is an odd prime and positive, then
`p - 1` is even so that any even integer between 1 and `p - 1`,
inclusive, is not relatively prime to `p - 1`. We have now
narrowed our search to all odd integers `k` between 1 and `p - 1`,
inclusive.

So now start with a generator `g` modulo an odd (positive) prime
`p`. For any odd integer `k` between 1 and `p - 1`, inclusive,
`g^k` is a generator modulo `p` if and only if `\gcd(k, p-1) = 1`.

The bibliographical reference should go in Sage’s master bibliography file, SAGE_ROOT/src/doc/en/reference/references/index.rst:

.. [Nat2000] \M. B. Nathanson. Elementary Methods in Number Theory.
   Springer, 2000.

A NOTE block for tips/tricks (optional).

.. NOTE::

    You should note that this sentence is indented at least 4
    spaces. Never use the tab character.

A WARNING block for critical information about your code (optional).

For example known situations for which the code breaks, or anything that the user should be aware of.

.. WARNING::

    Whenever you edit the Sage documentation, make sure that
    the edited version still builds. That is, you need to ensure
    that you can still build the HTML and PDF versions of the
    updated documentation. If the edited documentation fails to
    build, it is very likely that you would be requested to
    change your patch.

A TODO block for future improvements (optional).

It can contain disabled doctests to demonstrate the desired feature. Here’s an example of a TODO block:

.. TODO::

    Add to ``have_fresh_beers`` an interface with the faster
    algorithm "Buy a Better Fridge" (BaBF)::

        sage: have_fresh_beers('Bière de l\'Yvette', algorithm="BaBF") # not implemented
        Enjoy !

A PLOT block to illustrate with pictures the output of a function.

Generate with Sage code an object g with a .plot method, then call sphinx_plot(g):
```
.. PLOT::

    g = graphs.PetersenGraph()
    sphinx_plot(g)
```
A REFERENCES block to list related books or papers (optional).

Almost all bibliographic information should be put in the master bibliography file, see below. Citations will then link to the master bibliography where the reader can find the bibliographic details (see below for citation syntax). REFERENCE blocks in individual docstrings are therefore usually not necessary.

Nevertheless, a REFERENCE block can be useful if there are relevant sources which are not explicitly mentioned in the docstring or if the docstring is particularly long. In that case, add the bibliographic information to the master bibliography file, if not already present, and add a reference block to your docstring as follows:
```
REFERENCES:

For more information, see [Str1969]_, or one of the following references:

- [Sto2000]_

- [Voe2003]_
```
Note the trailing underscores which makes the citations into hyperlinks. See below for more about the master bibliography file. For more about citations, see the Sphinx/reST markup for citations. For links to trac tickets or wikipedia, see Hyperlinks.
A TESTS block (highly recommended).

Formatted just like EXAMPLES, containing tests that are not relevant to users. In particular, these blocks are not shown when users ask for help via foo?: they are stripped by the function sage.misc.sagedoc.skip_TESTS_block().

Special and corner cases, like number zero, one-element group etc. should usually go to this block. This is also right place for most tests of input validation; for example if the function accepts direction='up' and direction='down', you can use this block to check that direction='junk' raises an exception.

For the purposes of removal, A “TESTS” block is a block starting with “TEST:” or “TESTS:” (or the same with two colons), on a line on its own, and ending either with a line indented less than “TESTS”, or with a line with the same level of indentation – not more – matching one of the following:
- a Sphinx directive of the form “.. foo:”, optionally followed by other text.
- text of the form “UPPERCASE:”, optionally followed by other text.
- lines which look like a reST header: one line containing anything, followed by a line consisting only of whitespace, followed by a string of hyphens, equal signs, or other characters which are valid markers for reST headers: - = ` : ' " ~ _ ^ * + # < >. However, lines only containing double colons $::$ do not end “TESTS” blocks.

Note about Sphinx directives vs. other blocks¶

The main Sphinx directives that are used in Sage are:

.. MATH::, .. NOTE::, .. PLOT::, .. RUBRIC::, .. SEEALSO::, .. TODO::, .. TOPIC:: and .. WARNING::.

They must be written exactly as above, so for example WARNING:: or .. WARNING :: will not work.

Some other directives are also available, but less frequently used, namely:

.. MODULEAUTHOR::, .. automethod::, .. autofunction::, .. image::, .. figure::.

Other blocks shall not be used as directives; for example .. ALGORITHM:: will not be shown at all.

Sage documentation style¶

All Sage documentation is written in reStructuredText (reST) and is processed by Sphinx. See http://www.sphinx-doc.org/rest.html for an introduction. Sage imposes these styles:

Lines should be shorter than 80 characters. If in doubt, read PEP8: Maximum Line Length.
All reST and Sphinx directives (like .. WARNING::, .. NOTE::, .. MATH::, etc.) are written in uppercase.
Code fragments are quoted with double backticks. This includes function arguments and the Python literals like ``True``, ``False`` and ``None``. For example:
```
If ``check`` is ``True``, then ...
```

Sage’s master BIBLIOGRAPHY file¶

All bibliographical references should be stored in the master bibliography file, SAGE_ROOT/src/doc/en/reference/references/index.rst, in the format

.. [Gau1801] \C. F. Gauss, *Disquisitiones Arithmeticae*, 1801.

.. [RSA1978] \R. Rivest, A. Shamir, L. Adleman,
             "A Method for Obtaining Digital Signatures and
             Public-Key Cryptosystems".
             Communications of the ACM **21** (February 1978),
             120–126. :doi:`10.1145/359340.359342`.

The part in brackets is the citation key: given these examples, you could then use [Gau1801]_ in a docstring to provide a link to the first reference. Note the trailing underscore which makes the citation a hyperlink.

When possible, the key should have this form: for a single author, use the first three letters of the family name followed by the year; for multiple authors, use the first letter of each of the family names followed by the year. Note that the year should be four digits, not just the last two – Sage already has references from both 1910 and 2010, for example.

When abbreviating the first name of an author in a bibliography listing, be sure to put a backslash in front of it. This ensures that the letter (C. in the example above) will not be interpreted as a list enumerator.

For more about citations, see the Sphinx/reST markup for citations.

Template¶

Use the following template when documenting functions. Note the indentation:

def point(self, x=1, y=2):
    r"""
    Return the point `(x^5,y)`.

    INPUT:

    - ``x`` -- integer (default: `1`); the description of the
      argument ``x`` goes here. If it contains multiple lines, all
      the lines after the first need to begin at the same indentation
      as the backtick.

    - ``y`` -- integer (default: `2`); the description of the
      argument ``y``

    OUTPUT: the point as a tuple

    EXAMPLES:

    This example illustrates ... ::

        sage: A = ModuliSpace()
        sage: A.point(2,3)
        xxx

    We now ... ::

        sage: B = A.point(5,6)
        sage: xxx

    It is an error to ... ::

        sage: C = A.point('x',7)
        Traceback (most recent call last):
        ...
        TypeError: unable to convert 'r' to an integer

    .. NOTE::

        This function uses the algorithm of [BCDT2001]_ to determine
        whether an elliptic curve `E` over `Q` is modular.

    ...

    .. SEEALSO::

        :func:`line`

    TESTS::

        sage: A.point(42, 0)  # Check for corner case y=0
        xxx
    """
    <body of the function>

The master bibliography file would contain

.. [BCDT2001] Breuil, Conrad, Diamond, Taylor,
              "Modularity ...."

You are strongly encouraged to:

Use LaTeX typesetting (see LaTeX Typesetting).
Liberally describe what the examples do.

Note

There must be a blank line after the example code and before the explanatory text for the next example (indentation is not enough).
Illustrate the exceptions raised by the function with examples (as given above: “It is an error to [..]”, …)
Include many examples.

They are helpful for the users, and are crucial for the quality and adaptability of Sage. Without such examples, small changes to one part of Sage that break something else might not go seen until much later when someone uses the system, which is unacceptable.

Private functions¶

Functions whose names start with an underscore are considered private. They do not appear in the reference manual, and their docstring should not contain any information that is crucial for Sage users. You can make their docstrings be part of the documentation of another method. For example:

class Foo(SageObject):

    def f(self):
        """
        <usual docstring>

        .. automethod:: _f
        """
        return self._f()

    def _f(self):
         """
         This would be hidden without the ``.. automethod::``
         """

Private functions should contain an EXAMPLES (or TESTS) block.

A special case is the constructor __init__: due to its special status the __init__ docstring is used as the class docstring if there is not one already. That is, you can do the following:

sage: class Foo(SageObject):
....:     # no class docstring
....:     def __init__(self):
....:         """Construct a Foo."""
sage: foo = Foo()
sage: from sage.misc.sageinspect import sage_getdoc
sage: sage_getdoc(foo)              # class docstring
'Construct a Foo.\n'
sage: sage_getdoc(foo.__init__)     # constructor docstring
'Construct a Foo.\n'

LaTeX Typesetting¶

In Sage’s documentation LaTeX code is allowed and is marked with backticks or dollar signs:

`x^2 + y^2 = 1` and $x^2 + y^2 = 1$ both yield $x^2 + y^2 = 1$.

Backslashes: For LaTeX commands containing backslashes, either use double backslashes or begin the docstring with a r""" instead of """. Both of the following are valid:

def cos(x):
    """
    Return `\\cos(x)`.
    """

def sin(x):
    r"""
    Return $\sin(x)$.
    """

MATH block: This is similar to the LaTeX syntax \[<math expression>\] (or $$<math expression>$$). For instance:

.. MATH::

    \sum_{i=1}^{\infty} (a_1 a_2 \cdots a_i)^{1/i}
    \leq
    e \sum_{i=1}^{\infty} a_i

\[\sum_{i=1}^{\infty} (a_1 a_2 \cdots a_i)^{1/i} \leq e \sum_{i=1}^{\infty} a_i\]

The aligned environment works as it does in LaTeX:

.. MATH::

    \begin{aligned}
     f(x) & = x^2 - 1 \\
     g(x) & = x^x - f(x - 2)
    \end{aligned}

\[\begin{split}\begin{aligned} f(x) & = x^2 - 1 \\ g(x) & = x^x - f(x - 2) \end{aligned}\end{split}\]

When building the PDF documentation, everything is translated to LaTeX and each MATH block is automatically wrapped in a math environment – in particular, it is turned into \begin{gather} block \end{gather}. So if you want to use a LaTeX environment (like align) which in ordinary LaTeX would not be wrapped like this, you must add a :nowrap: flag to the MATH mode. See also Sphinx’s documentation for math blocks. :

.. MATH::
   :nowrap:

   \begin{align}
      1+...+n &= n(n+1)/2\\
      &= O(n^2)\\
   \end{align}

\begin{align} 1+...+n &= n(n+1)/2\\ &= O(n^2)\\ \end{align}

Readability balance: in the interactive console, LaTeX formulas contained in the documentation are represented by their LaTeX code (with backslashes stripped). In this situation \\frac{a}{b} is less readable than a/b or a b^{-1} (some users may not even know LaTeX code). Make it pleasant for everybody as much as you can manage.

Commons rings $(\Bold{Z},\Bold{N},...)$: The Sage LaTeX style is to typeset standard rings and fields using the locally-defined macro \\Bold (e.g. \\Bold{Z} gives $\Bold{Z}$).

Shortcuts are available which preserve readability, e.g. \\ZZ ($\ZZ$), \\RR ($\RR$), \\CC ($\CC$), and \\QQ ($\QQ$). They appear as LaTeX-formatted \\Bold{Z} in the html manual, and as Z in the interactive help. Other examples: \\GF{q} ($\GF{q}$) and \\Zmod{p} ($\Zmod{p}$).

See the file SAGE_ROOT/src/sage/misc/latex_macros.py for a full list and for details about how to add more macros.

Writing Testable Examples¶

The examples from Sage’s documentation have a double purpose:

They provide illustrations of the code’s usage to the users
They are tests that are checked before each release, helping us avoid new bugs.

All new doctests added to Sage should pass all tests (see Running Sage’s doctests), i.e. running sage -t your_file.py should not give any error messages. Below are instructions about how doctests should be written.

What doctests should test:

Interesting examples of what the function can do. This will be the most helpful to a lost user. It is also the occasion to check famous theorems (just in case):
```
sage: is_prime(6) # 6 is not prime
False
sage: 2 * 3 # and here is a proof
6
```
All meaningful combinations of input arguments. For example a function may accept an algorithm="B" argument, and doctests should involve both algorithm="A" and algorithm="B".
Corner cases: the code should be able to handle a 0 input, or an empty set, or a null matrix, or a null function, … All corner cases should be checked, as they are the most likely to be broken, now or in the future. This probably belongs to the TESTS block (see The docstring of a function: content).
Systematic tests of all small-sized inputs, or tests of random instances if possible.

Note

Note that TestSuites are an automatic way to generate some of these tests in specific situations. See SAGE_ROOT/src/sage/misc/sage_unittest.py.

The syntax:

Environment: doctests should work if you copy/paste them in Sage’s interactive console. For example, the function AA() in the file SAGE_ROOT/src/sage/algebras/steenrod/steenrod_algebra.py includes an EXAMPLES block containing the following:
```
sage: from sage.algebras.steenrod.steenrod_algebra import AA as A
sage: A()
mod 2 Steenrod algebra, milnor basis
```
Sage does not know about the function AA() by default, so it needs to be imported before it is tested. Hence the first line in the example.
Preparsing: As in Sage’s console, $4/3$ returns $4/3$ and not $1$ as in Python 2.7. Testing occurs with full Sage preparsing of input within the standard Sage shell environment, as described in Sage Preparsing.
Writing files: If a test outputs to a file, the file should be a temporary file. Use tmp_filename() to get a temporary filename, or tmp_dir() to get a temporary directory. An example from SAGE_ROOT/src/sage/plot/graphics.py):
```
sage: plot(x^2 - 5, (x, 0, 5), ymin=0).save(tmp_filename(ext='.png'))
```

Multiline doctests: You may write tests that span multiple lines, using the line continuation marker ....:

sage: for n in srange(1,10):
....:     if n.is_prime():
....:         print(n)
2
3
5
7

Python3 print: even if Python2 syntax for print can still be used in your own code for the moment, Python3 syntax for print must be used in Sage code and doctests. If you use an old-style print in doctests, it will raise a SyntaxError:
```
sage: print "not like that"
Traceback (most recent call last):
...
SyntaxError: ...
sage: print("but like this")
but like this
```
Split long lines: You may want to split long lines of code with a backslash. Note: this syntax is non-standard and may be removed in the future:
```
sage: n = 123456789123456789123456789\
....:     123456789123456789123456789
sage: n.is_prime()
False
```
Doctests flags: flags are available to change the behaviour of doctests: see Special Markup to Influence Doctests.

Special Markup to Influence Doctests¶

Overly complicated output in the example code can be shortened by an ellipsis marker ...:

sage: [ZZ(n).ordinal_str() for n in range(25)]
['0th',
 '1st',
 '2nd',
 '3rd',
 '4th',
 '5th',
 ...
 '21st',
 '22nd',
 '23rd',
 '24th']
sage: ZZ('sage')
Traceback (most recent call last):
...
TypeError: unable to convert 'sage' to an integer

On the proper usage of the ellipsis marker, see Python’s documentation.

There are a number of magic comments that you can put into the example code that change how the output is verified by the Sage doctest framework. Here is a comprehensive list:

random: The line will be executed, but its output will not be checked with the output in the documentation string:
```
sage: c = CombinatorialObject([1,2,3])
sage: hash(c)  # random
1335416675971793195
sage: hash(c)  # random
This doctest passes too, as the output is not checked
```
However, most functions generating pseudorandom output do not need this tag since the doctesting framework guarantees the state of the pseudorandom number generators (PRNGs) used in Sage for a given doctest.

When possible, avoid the problem, e.g.: rather than checking the value of the hash in a doctest, one could illustrate successfully using it as a key in a dict.
long time: The line is only tested if the --long option is given, e.g. sage -t --long f.py.

Use it for doctests that take more than a second to run. No example should take more than about 30 seconds:
```
sage: E = EllipticCurve([0, 0, 1, -1, 0])
sage: E.regulator()        # long time (1 second)
0.0511114082399688
```
tol or tolerance: The numerical values returned by the line are only verified to the given tolerance. It is useful when the output is subject to numerical noise due to system-dependent (floating point arithmetic, math libraries, …) or non-deterministic algorithms.
- This may be prefixed by abs[olute] or rel[ative] to specify whether to measure absolute or relative error (see the Wikipedia article Approximation_error).
- If none of abs/rel is specified, the error is considered to be absolute when the expected value is zero, and is relative for nonzero values.
```
sage: n(pi)  # abs tol 1e-9
3.14159265358979
sage: n(pi)  # rel tol 2
6
sage: n(pi)  # abs tol 1.41593
2
sage: K.<zeta8> = CyclotomicField(8)
sage: N(zeta8)  # absolute tolerance 1e-10
0.7071067812 + 0.7071067812*I
```
Multiple numerical values: the representation of complex numbers, matrices, or polynomials usually involves several numerical values. If a doctest with tolerance contains several numbers, each of them is checked individually:
```
sage: print("The sum of 1 and 1 equals 5")  # abs tol 1
The sum of 2 and 2 equals 4
sage: e^(i*pi/4).n() # rel tol 1e-1
0.7 + 0.7*I
sage: ((x+1.001)^4).expand() # rel tol 2
x^4 + 4*x^3 + 6*x^2 + 4*x + 1
sage: M = matrix.identity(3) + random_matrix(RR,3,3)/10^3
sage: M^2 # abs tol 1e-2
[1 0 0]
[0 1 0]
[0 0 1]
```
The values that the doctesting framework involves in the error computations are defined by the regular expression float_regex in sage.doctest.parsing.
not implemented or not tested: The line is never tested.

Use it for very long doctests that are only meant as documentation. It can also be used for todo notes of what will eventually be implemented:
```
sage: factor(x*y - x*z)    # todo: not implemented
```
It is also immediately clear to the user that the indicated example does not currently work.
Note

Skip all doctests of a file/directory
- file: If one of the first 10 lines of a file starts with any of r""" nodoctest (or """ nodoctest or # nodoctest or % nodoctest or .. nodoctest, or any of these with different spacing), then that file will be skipped.
- directory: If a directory contains a file nodoctest.py, then that whole directory will be skipped.
Neither of this applies to files or directories which are explicitly given as command line arguments: those are always tested.
py2 or py3: Run the line on Python 2 only or Python 3 only respectively. Generally this should be avoided as code should be tested on both Python 2 and Python 3, but there are on occasion tests that are simply inapplicable on one or the other, such as tests that rely on optional features that are only available on one Python version or the other.
optional: A line flagged with optional - keyword is not tested unless the --optional=keyword flag is passed to sage -t (see Run Optional Doctests). The main applications are:
- optional packages: When a line requires an optional package to be installed (e.g. the sloane_database package):
```
sage: SloaneEncyclopedia[60843]    # optional - sloane_database
```
- internet: For lines that require an internet connection:
```
sage: oeis(60843)                 # optional - internet
A060843: Busy Beaver problem: a(n) = maximal number of steps that an
n-state Turing machine can make on an initially blank tape before
eventually halting.
```
- bug: For lines that describe bugs. Alternatively, use # known bug instead: it is an alias for optional bug.
```
The following should yield 4.  See :trac:`2`. ::

    sage: 2+2  # optional: bug
    5
    sage: 2+2  # known bug
    5
```
Note
- Any words after # optional are interpreted as a list of package names, separated by spaces.
- Any punctuation (periods, commas, hyphens, semicolons, …) after the first word ends the list of packages. Hyphens or colons between the word optional and the first package name are allowed. Therefore, you should not write optional: needs package CHomP but simply optional: CHomP.
- Optional tags are case-insensitive, so you could also write optional: chOMP.
indirect doctest: in the docstring of a function A(...), a line calling A and in which the name A does not appear should have this flag. This prevents sage --coverage <file> from reporting the docstring as “not testing what it should test”.

Use it when testing special functions like __repr__, __add__, etc. Use it also when you test the function by calling B which internally calls A:
```
This is the docstring of an ``__add__`` method. The following
example tests it, but ``__add__`` is not written anywhere::

    sage: 1+1 # indirect doctest
    2
```
32-bit or 64-bit: for tests that behave differently on 32-bit or 64-bit machines. Note that this particular flag is to be applied on the output lines, not the input lines:
```
sage: hash(2^31 + 2^13)
-2147475456               # 32-bit
2147491840                # 64-bit
```

Using search_src from the Sage prompt (or grep), one can easily find the aforementioned keywords. In the case of todo: not implemented, one can use the results of such a search to direct further development on Sage.

Running Automated Doctests¶

This section describes Sage’s automated testing of test files of the following types: .py, .pyx, .sage, .rst. Briefly, use sage -t <file> to test that the examples in <file> behave exactly as claimed. See the following subsections for more details. See also Documentation Strings for a discussion on how to include examples in documentation strings and what conventions to follow. The chapter Running Sage’s doctests contains a tutorial on doctesting modules in the Sage library.

Testing .py, .pyx and .sage Files¶

Run sage -t <filename.py> to test all code examples in filename.py. Similar remarks apply to .sage and .pyx files:

$ sage -t [--verbose] [--optional]  [files and directories ... ]

The Sage doctesting framework is based on the standard Python doctest module, but with many additional features (such as parallel testing, timeouts, optional tests). The Sage doctester recognizes sage: prompts as well as >>> prompts. It also preparses the doctests, just like in interactive Sage sessions.

Your file passes the tests if the code in it will run when entered at the sage: prompt with no extra imports. Thus users are guaranteed to be able to exactly copy code out of the examples you write for the documentation and have them work.

For more information, see Running Sage’s doctests.

Testing reST Documentation¶

Run sage -t <filename.rst> to test the examples in verbatim environments in reST documentation.

Of course in reST files, one often inserts explanatory texts between different verbatim environments. To link together verbatim environments, use the .. link comment. For example:

EXAMPLES::

        sage: a = 1

Next we add 1 to ``a``.

.. link::

        sage: 1 + a
        2

If you want to link all the verbatim environments together, you can put .. linkall anywhere in the file, on a line by itself. (For clarity, it might be best to put it near the top of the file.) Then sage -t will act as if there were a .. link before each verbatim environment. The file SAGE_ROOT/src/doc/en/tutorial/interfaces.rst contains a .. linkall directive, for example.

You can also put .. skip right before a verbatim environment to have that example skipped when testing the file. This goes in the same place as the .. link in the previous example.

See the files in SAGE_ROOT/src/doc/en/tutorial/ for many examples of how to include automated testing in reST documentation for Sage.

General Coding Style Regarding Whitespace¶

Use spaces instead of tabs for indentation. The only exception is for makefiles, in which tabs have a syntactic meaning different from spaces.

Do not add trailing whitespace.

Sage provides editor configuration for Emacs, using the file .dir-locals.el, to use spaces instead of tabs. Regarding trailing whitespace, see https://www.emacswiki.org/emacs/DeletingWhitespace for various solutions.

If you use another editor, we recommend to configure it so you do not add tabs to files.

Global Options¶

Global options for classes can be defined in Sage using GlobalOptions.

Miscellanous minor things¶

Some decisions are arbitrary, but common conventions make life easier.

Non-ASCII names in identifiers:
- Translate ä and ö to ae and oe, like moebius_function for Möbius function.
- Translate á to a, like lovasz_number for Lovász number.
Common function keyword arguments:

This is a list of some keyword arguments that many functions and methods take. For consistency, you should use the keywords from the list below with the meaning as explained here. Do not use a different keyword with the same meaning (for example, do not use method; use algorithm instead).
- algorithm, a string or None: choose between various implementation or algorithm. Use None as a default that selects a sensible default, which could depend on installed optional packages.
- certificate, a Boolean with False as default: whether the function should return some kind of certificate together with the result. With certificate=True the return value should be a pair $(r, c)$ where $r$ is the result that would be given with certificate=False and $c$ is the certificate or None if there is no meaningfull certificate.
- proof, a Boolean with True as default: if True, require a mathematically proven computation. If False, a probabilistic algorithm or an algorithm relying to non-proved hypothesis like RH can be used.
- check, a Boolean: do some additional checks to verify the input parameters. This should not otherwise influence the functioning of the code: if code works with check=True, it should also work with check=False.
- coerce, a Boolean: convert the input parameters to a suitable parent. This is typically used in constructors. You can call a method with coerce=False to skip some checks if the parent is known to be correct.
- inplace, a Boolean: whether to modify the object in-place or to return a copy.